I.S.F.A.

École Doctorale Sciences Économiques et de Gestion

Etude des marchés d’assurance non-vie
à l’aide d’équilibres de Nash et demodèles de risques avec dépendance

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THÈSE
Numéro d’ordre 70-2012

présentée et soutenue publiquement le 31 Mai 2012pour l’obtention duDoctorat de l’Université Claude Bernard Lyon I(mathématiques appliquées)
parChristophe Dutang
Composition du juryPrésident :Jean-Noël Bacro, Professeur à l’Université Montpellier 2

Rapporteurs :Etienne Marceau, Professeur à l’Université Laval de Québec
Patrice Bertail, Professeur à l’Université Paris X

Examinateurs :Stéphane Loisel, Professeur à l’Université Lyon 1 (co-directeur de thèse)Véronique Maume-Deschamps, Professeur à l’Université Lyon 1 (directrice de thèse)
Christian-Yann Robert, Professeur à l’Université Lyon 1

Laboratoire Science Actuarielle Financière — EA 2429

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Remerciements
Je voudrais commencer par remercier l’initiateur de ce projet de thèse, Stéphane Loisel,sans qui rien n’aurait été possible. Un jour de décembre 2007, Stéphane m’a fait une propositionde thèse CIFRE, me permettant de garder un pied en entreprise et l’autre en université, que je
ne pouvais tout simplement pas refuser. Je le remercie tout particulièrement pour les travauxque nous avons effectués ensemble et pour nos discussions sans fin sur l’actuariat non-vie. Jetiens aussi à remercier mon autre directrice de thèse, Véronique Maume-Deschamps, toujours
disponible pour répondre à mes questionnements. J’espère que ma collaboration avec mes
directeurs de thèse ne s’arrêtera pas avec ma soutenance.
Je suis très honoré qu’Etienne Marceau et Patrice Bertail aient accepté d’être les rapporteurs de ma thèse. Merci à eux pour le temps consacré à parcourir ce manuscrit. Je tiens à
remercier également Christian-Yann Robert et Jean-Noël Bacro d’avoir accepté de faire partie
de mon jury.
Un grand merci à Hansjoerg Albrecher pour m’avoir aidé quand j’en avais besoin. Il ne
m’a pas abandonné sur le projet de théorie des jeux et cycles de marché, malgré mon longapprentissage du sujet. De plus, je remercie Claude Lefèvre pour avoir lancé l’idée du sujet dethéorie de la ruine. Nous avons eu d’agréables moments de discussion ensemble, tant sur ce
projet que sur d’autres sujets.
Je tiens aussi à remercier mes chefs successifs au GRM d’AXA : Emmanuel Pierron et ses
citations mythiques, Valérie Gilles et son sang-froid à toute épreuve, et Gilles Hornecker et son
sens de l’humour. Ils m’ont chacun aidé, à leur manière, à la réalisation de cette thèse. J’en
profite aussi pour remercier les membres du GRM non-vie PAP Valeur que j’ai pu côtoyer : lesdeux Mathieu, Jean, Francis, Nicolas, Antoine, Maria, Jérôme, Amélie, Yannick, Cyril, Pierre,Sara, Valérie et Jean-François. J’espère que la bonne ambiance de ce côté du plateau perdurera
après mon départ. De plus, j’ai beaucoup apprécié les liens qui se sont créés avec les autres
thésards CIFRE. Merci à Nabil, Théodora et Madeleine-Sophie pour nos longues discussions
et nos repas trimestriels dont l’organisation nécessitait des doodles ! Un grand merci à toutel’équipe du GRM non-vie qui m’a toujours bien accueilli.
Je n’oublierai pas d’adresser un petit clin d’oeil aux autres thésards, à commencer par ceuxde ma génération Xavier, Yahia, Elena, Manel, Abdou, Alain, Anisa, Soffana, à ceux qui m’ont
précédé, Romain, Mathieu, Areski, Florent, et à ceux qui me suivent, les deux Julien, Andrès,
Erwan, Jean-Charles et à ceux que j’aurais oubliés. Je remercie tout particulièrement l’équipede recherche du laboratoire SAF au plot 4, ainsi que les membres du personnel administratif
du plot 1, toujours prêts à faciliter mes recherches et mes démarches.
On dit souvent qu’une thèse est un long périple. Pour ma part, cela a pris la forme d’unlong voyage de 154 671 kilomètres au prix de 9,94 centimes le kilomètre... Un grand merci
donc à la SNCF pour maintenir l’illusion que Paris est proche de toutes les grandes villes deFrance. Par monts et par vaux, j’ai pu travailler sur ma thèse dans ses trains, sans subir perte
de temps et fatigue sur les routes.
Un grand merci au créateur de TEX, Donald Knuth, pour avoir inventé ce fabuleux logiciel
dans les années 70, qui fut plus tard repris par Leslie Lamport pour devenir LATEX. Je n’ose
pas compter les heures de souffrance évitées grâce à LATEX comparé aux logiciels de type
WYSIWYG. Sur le même thème, je remercie également Martin Maechler d’avoir encouragé
Robert Gentleman et Ross Ihaka de rendre libre les sources de R en 1995. J’ai énormément
utilisé ce logiciel gratuit et libre durant toute ma thèse.

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De plus, je tiens à remercier tous mes amis, à commencer par ceux de Grenoble, Jérémy,
Quentin, Benoit, Fabrice, Vincent, Olivier, Horté, Julie. Les moments de détente en leurcompagnie ont été très bénéfiques pour ne jamais perdre le moral ! Je remercie aussi les ρ⊕de Paris, Onur, Fabrice, Jean-François, Tugdual, Romain, Marie, Julien, Samuel, Guillaume,
avec qui j’ai passé des dimanches soirs tout simplement mémorables.
Merci à ma belle-famille pour avoir établi un pont aérien entre l’Ariège et Paris 13ème afin
de m’envoyer à flot quasi-continu des spécialités vernaculaires ! Cela m’a permis de rester enforme durant toute cette thèse. Un grand merci aussi à Alain et Christine pour leur soutienet leur relecture.
J’ai une pensée particulière pour mon frère, qui m’a accueilli de nombreuses fois dans son
appartement les veilles de séminaire Lyon-Lausanne. Je suis aussi pour toujours redevable à
mes parents, qui m’ont inculqué les valeurs morales pour que je réussisse dans la vie. Merci à
eux pour leurs relectures durant le dernier mois.
Le mot de la fin est adressé à ma tendre Isabelle, à qui je dédie cette thèse. Elle a toujours
été là à mes côtés pour me soutenir et m’encourager durant ces trois longues années. Je nepourrai jamais assez la remercier.

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À Isabellevvi

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Résumétel-00703797, version 2 - 7 Jun 2012

Etude des marchés d’assurance non-vie à l’aide d’équilibres de Nash et demodèles de risques avec dépendance
L’actuariat non-vie étudie les différents aspects quantitatifs de l’activité d’assurance. Cette
thèse vise à expliquer sous différentes perspectives les interactions entre les différents agentséconomiques, l’assuré, l’assureur et le marché, sur un marché d’assurance. Le chapitre 1 souligne à quel point la prise en compte de la prime marché est importante dans la décision del’assuré de renouveler ou non son contrat d’assurance avec son assureur actuel. La nécessitéd’un modèle de marché est établie.
Le chapitre 2 répond à cette problématique en utilisant la théorie des jeux non-coopératifs
pour modéliser la compétition. Dans la littérature actuelle, les modèles de compétition se
réduisent toujours à une optimisation simpliste du volume de prime basée sur une vision d’unassureur contre le marché. Partant d’un modèle de marché à une période, un jeu d’assureurs
est formulé, où l’existence et l’unicité de l’équilibre de Nash sont vérifiées. Les propriétésdes primes d’équilibre sont étudiées pour mieux comprendre les facteurs clés d’une position
dominante d’un assureur par rapport aux autres. Ensuite, l’intégration du jeu sur une période
dans un cadre dynamique se fait par la répétition du jeu sur plusieurs périodes. Une approche
par Monte-Carlo est utilisée pour évaluer la probabilité pour un assureur d’être ruiné, de rester
leader, de disparaître du jeu par manque d’assurés en portefeuille. Ce chapitre vise à mieux
comprendre la présence de cycles en assurance non-vie.
Le chapitre 3 présente en profondeur le calcul effectif d’équilibre de Nash pour n joueurs
sous contraintes, appelé équilibre de Nash généralisé. Il propose un panorama des méthodes
d’optimisation pour la résolution des n sous-problèmes d’optimisation. Cette résolution se
fait à l’aide d’une équation semi-lisse basée sur la reformulation de Karush-Kuhn-Tucker duproblème d’équilibre de Nash généralisé. Ces équations nécessitent l’utilisation du Jacobien
généralisé pour les fonctions localement lipschitziennes intervenant dans le problème d’optimisation. Une étude de convergence et une comparaison des méthodes d’optimisation sontréalisées.
Enfin, le chapitre 4 aborde le calcul de la probabilité de ruine, un autre thème fondamental
de l’assurance non-vie. Dans ce chapitre, un modèle de risque avec dépendance entre lesmontants ou les temps d’attente de sinistre est étudié. De nouvelles formules asymptotiquesde la probabilité de ruine en temps infini sont obtenues dans un cadre large de modèle de risquesavec dépendance entre sinistres. De plus, on obtient des formules explicites de la probabilité deruine en temps discret. Dans ce modèle discret, l’analyse structure de dépendance permet dequantifier l’écart maximal sur les fonctions de répartition jointe des montants entre la version
continue et la version discrète.

Mots-clés: Comportement client, cycles de marché, théorie de la ruine, actuariat non-vie,
théorie des jeux, calcul d’équilibre de Nash généralisé, montants de sinistres dépendants

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Abstract

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Studying non-life insurance markets with Nash equilibria and dependentrisk models
In non-life actuarial mathematics, different quantitative aspects of insurance activity are studied. This thesis aims at explaining interactions among economic agents, namely the insured,
the insurer and the market, under different perspectives. Chapter 1 emphasizes how essential
the market premium is in the customer decision to lapse or to renew with the same insurer.
The relevance of a market model is established.
In chapter 2, we address this issue by using noncooperative game theory to model competition. In the current literature, most competition models are reduced to an optimisationof premium volume based on the simplistic picture of an insurer against the market. Startingwith a one-period model, a game of insurers is formulated, where the existence and uniqueness of a Nash equilibrium are verified. The properties of premium equilibria are examined
to better understand the key factors of leadership positions over other insurers. Then, thederivation of a dynamic framework from the one-period game is done by repeating of the
one-shot game over several periods. A Monte-Carlo approach is used to assess the probabilityof being insolvent, staying a leader, or disappearing of the insurance game. This gives further
insights on the presence of non-life insurance market cycles.
A survey of computational methods of a Nash equilibrium under constraints is conducted
in Chapter 3. Such generalized Nash equilibrium of n players is carried out by solving asemismooth equation based on a Karush-Kuhn-Tucker reformulation of the generalized Nashequilibrium problem. Solving semismooth equations requires using the generalized Jacobianfor locally Lipschitzian function. Convergence study and method comparison are carried out.
Finally, in Chapter 4, we focus on ruin probability computation, another fundemantal
point of non-life insurance. In this chapter, a risk model with dependence among claim
severity or claim waiting times is studied. Asymptotics of infinite-time ruin probabilities
are obtained in a wide class of risk models with dependence among claims. Furthermore,
we obtain new explicit formulas for ruin probability in discrete-time. In this discrete-timeframework, dependence structure analysis allows us to quantify the maximal distance between
joint distribution functions of claim severity between the continuous-time and the discretetime versions.

Keywords: Customer behavior, market cycles, ruin theory, non-life insurance, game theory,
generalized Nash equilibrium computation, dependent claim severity models

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Table des matières
Remerciementsiii

Résumé

vii

Tables des matières

xi

Introduction générale

Introduction

3

Modèles de résiliation et cycles de marché . . . . . . . . . . . . . . . . . . . . . .

4

Théorie des jeux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Théorie de la ruine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Principaux résultats

39

Comportement d’un client . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Compétition et cycles en assurance non-vie . . . . . . . . . . . . . . . . . . . . . 39
Calcul d’équilibres de Nash généralisés . . . . . . . . . . . . . . . . . . . . . . . . 40
Asymptotique de la probabilité de ruine . . . . . . . . . . . . . . . . . . . . . . . 40

Modèles de régressionxi

Table des matières
Chapitre 1 Sur la nécessité d’un modèle de marché

45

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.2

GLMs, a brief introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.3

Simplistic applications and biased business conclusions . . . . . . . . . . . . 52

1.4

Incorporating new variables in the regression . . . . . . . . . . . . . . . . . 57

1.5

Testing asymmetry of information . . . . . . . . . . . . . . . . . . . . . . . 62

1.6

Other regression models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

1.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

1.8

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Théorie des jeux

Chapitre 2 Theorie des jeux et cycles de marché

93

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.2

A one-period model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.3

Refinements of the one-period model . . . . . . . . . . . . . . . . . . . . . . 107

2.4

Dynamic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

2.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.6

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Chapitre 3 Calcul d’équilibre de Nash généralisé

137

3.1

Problem reformulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.2

Methods to solve nonlinear equations . . . . . . . . . . . . . . . . . . . . . 143

3.3

Numerical results

3.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.5

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
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Chapitre 4 Asymptotiques de la probabilité de ruine

173

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.2

Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.3

Asymptotics – the A + B/u rule . . . . . . . . . . . . . . . . . . . . . . . . 181

4.4

Focus on the dependence structure . . . . . . . . . . . . . . . . . . . . . . . 195

4.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

4.6

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Conclusion

Conclusion et perspectives

217

Bibliographie

219

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Table des matièresxiv

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Introduction générale

1

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Introduction

Tout ce qui augmente la liberté
augmente la responsabilité.
Victor Hugo (1802-1885)

L’actuariat non-vie étudie les différents aspects mathématiques de l’activité d’assurance.
L’objet de cette thèse est d’expliquer sous différentes perspectives les interactions entre lesdifférents agents économiques, l’assuré, l’assureur et le marché, sur un marché d’assurance, tant
au niveau de la modélisation des primes que des sinistres. Cette introduction vise à présenteret à remettre dans leur contexte quatre articles en cours de soumission, qui constituent leschapitres de cette thèse. Ceux-ci couvrent trois grands sujets de l’activité d’assurance : lamodélisation des résiliations en prenant en compte le marché, la modélisation des primes dansun environnement de compétition et l’évolution de la richesse d’une compagnie d’assurance.
En souscrivant une police d’assurance, un individu souhaite se prémunir contre les conséquences d’évènements extérieurs (incendies, accidents, etc. . .) envers un de ses biens (voiture,logement, etc. . .) ou sa personne (responsabilité civile). En contrepartie de cette assurance,l’assuré paie une prime d’assurance en début de période. L’assureur quant à lui peut êtreamené à fournir une prestation si un certain type de sinistre survient pendant la période
considérée. A ces deux agents économiques s’ajoutent une troisième composante, impersonnelle, le marché. Dans le cadre de cette thèse, nous excluons la réassurance de notre étude.
Dans ce schéma à trois agents, l’assureur fait donc face en premier lieu au risque d’avoir peu
ou pas d’assurés, dans le cas où ses prix sont excessifs ou simplement très supérieurs à ceux
des autres assureurs. Le marché agit à la fois sur l’assuré en pouvant l’inciter à résilier son
contrat d’assurance pour se couvrir chez un autre assureur, et sur l’assureur en le contraignant
dans une certaine mesure à rendre ses primes d’assurance à des niveaux acceptables. Le risquede prime regroupe donc deux composantes : les résiliations et la compétition. Les modèles derésiliation reposent sur les modèles statistiques de régression dont le plus connu est le modèle
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Introduction
linéaire généralisé. Cette thèse propose une revue de tels modèles et étudie leur pertinencedans le schéma à trois agents.
La compétition sur les marchés d’assurance peut être modélisée de deux façons : uneapproche agrégée modélisant le marché dans sa globalité, une vision plus fine visant à modéliser
chacun des assureurs composant le marché. La première approche consiste à étudier l’évolution
de variables macroéconomiques telles que la prime moyenne marché, le ratio sinistre sur prime
du marché. La seconde approche repose sur l’utilisation de la théorie des jeux pour modéliser
les interactions entre assureurs. En théorie des jeux, le concept de solution est soustrait au
concept d’équilibre. Un des apports de cette thèse est de proposer un modèle de théorie desjeux pour comprendre les interactions entre assureurs et assurés sur un marché d’assurance.
En plus de l’application d’un jeu pour le marché d’assurance, nous proposons un panorama
des méthodes de calcul d’équilibre de la théorie des jeux.
Par ailleurs, les assureurs sont confrontés au risque propre d’assurance une fois des contrats
souscrits, en plus du risque de prime. En effet, l’assureur est tenu de respecter ses engagements
envers les assurés. Pour ce faire, il détient un capital initial auquel se rajoutent les primes et se
retranchent les montants des sinistres au cours du temps. La théorie de la ruine s’intéresse àl’évolution du niveau de richesse d’une compagnie d’assurances à l’aide de processus stochastiques. Parmi les mesures généralement considérées, nous nous intéressons à la probabilité deruine en temps infini, qui dépend du capital initial de l’assureur. Un dernier apport de la thèse
est de proposer de nouvelles formules asymptotiques de probabilités de ruine, dans un cadrede dépendance entre les sinistres, lorsque le capital initial est élevé. De plus, nous obtenons
des formules explicites pour la probabilité de ruine en temps discret.
La suite de cette introduction va développer les points évoqués ci-dessus, en commençantpar décrire les modèles de résiliation et de cycles de marché, puis en présentant les modèlesde théories des jeux usuels et enfin les modèles de théorie de la ruine.

Modèles de résiliation et cycles de marché
En assurance non-vie, l’individu se prémunit contre les conséquences financières d’un risqueenvers un de ses biens ou sa personne en achetant une police d’assurance. En contrepartie decette assurance, l’assuré paie une prime d’assurance en début de période de couverture, parexemple un an en assurance automobile. En France et dans de nombreux pays d’Europe
continentale, les contrats d’assurance sont renouvelés par tacite reconduction, c’est à dire si
l’assuré ne manifeste pas son intention de mettre un terme à son contrat d’assurance, celui-ci
est automatiquement renouvelé pour la même durée de couverture.
D’une part, la compétition sur les marchés d’assurance empêche de demander des prixtrès élevés pour une couverture d’assurance. En effet, la prime d’assurance est calculée en
fonction des caractéristiques propres de l’assuré, mais dépend tout autant des conditions demarché. Il n’y a rien de plus naturel que l’assuré soit tenté de résilier son contrat d’assurance
s’il peut trouver moins cher chez un autre assureur. A couverture équivalente, il semble donc
raisonnable de supposer que les individus vont chercher à s’assurer à moindre prix. Notons unpremier biais de perception de la part de l’assuré, les couvertures proposées par les assureurs
ne sont pas forcément équivalentes, franchise et limite pouvant être différentes, sans parler
des couvertures additionnelles comme l’assistance, la perte de valeur, la garantie du contenu,
etc. . . Constatons donc que le prix de l’assurance n’explique pas entièrement le comportementd’un client.
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D’autre part, les assureurs, les mutuelles et autres structures d’assurances cherchent dansune certaine mesure à maximiser leur volume de primes, défini comme la somme des primesémises sur l’ensemble des contrats pendant une année. Au premier ordre, le volume de prime
peut être approché par le produit du nombre de contrats et de la prime moyenne. Il paraît
logique que ces deux grandeurs évoluent en sens inverse : une prime moyenne élevée entraînera
une diminution du nombre de contrats souscrits, et vice versa.
Une autre caractéristique de l’assurance non-vie est que certaines assurances sont obligatoires, par exemple, la garantie responsabilité civile en assurance automobile, qui permetl’indemnisation des dommages causés par un automobiliste aux tiers. Ainsi, certains produitsd’assurance non-vie sont plus concurrentiels que d’autres, du fait de leur caractère obligatoire. La résiliation d’un contrat d’assurance résulte donc de plusieurs décisions dépendant dedifférents agents : l’assuré, son assureur actuel et les concurrents.
A première vue, la survie d’un assureur est menacée par deux phénomènes : une sinistralité
accrue qui pourrait le rendre insolvable ou même le ruiner, et une baisse d’attractivité auprès
des individus cherchant à s’assurer. Etant nécessaire à toutes études prospectives sur le volume
de prime, le taux de résiliation est donc une variable clé pour les assureurs. Le taux derésiliation peut se définir soit en nombre de contrats, soit en volume de prime, ou encore en
nombre de risques. Cette thèse ciblant l’assurance de particuliers, nous choisirons comme tauxde résilitation, le nombre de contrats résiliés (sur une période) divisé par le nombre total decontrats en début de période.
Ce constat étant établi, il apparait tout à fait pertinent de chercher à modéliser la résiliation des assurés et les cycles de marché. Les deux modélisations sont généralement réalisées
indépendamment l’une de l’autre, comme nous allons le présenter dans cette partie de l’introduction. Le chapitre 2 de cette thèse vise à modéliser conjointement le comportement desassurés et des assureurs à l’aide de la théorie des jeux.

Causalités de la résiliation et comportement des clients
Dans cette thèse, nous nous concentrons sur la résiliation des contrats d’assurance parl’assuré au moment du renouvellement du contrat. Cela exclut donc les résiliations avant
terme, par exemple, la disparition du risque suite à la revente du véhicule en assurance auto.
Nous écartons aussi les résiliations du contrat par l’assureur.
Tout d’abord, les études purement descriptives, voir Bland et al. (1997); Kelsey et al.
(1998), révèlent le premier constat : pour une génération donnée de polices, le taux de résiliation décroît au cours du temps. Autrement dit, plus l’assuré reste longtemps en portefeuille,plus sa probabilité de résiliation diminue.
Une autre variable clé de la résiliation est sans grande surprise la prime proposée. Ainsi lapolitique de prix de l’assureur, c’est à dire la combinaison d’un tarif technique, d’un éventuelrabais commercial et de négotiation, va faire évoluer la prime d’assurance. Par conséquent, larésiliation est très fortement impactée par la politique tarifaire de l’assureur.
On peut légitemement supposer qu’un assuré résilie s’il a trouvé mieux ailleurs, soit enterme de prix soit en terme de couverture. La comparaison avec la concurrence est indissociable
du processus de résiliations. Par conséquent, malgré internet, le géomarketing et l’information
visible sur la concurrence ne sont pas à négliger : par exemple, un assureur spécialisé pour les
zones rurales est moins inquiété par un assureur généraliste des zones urbaines qu’un assureur
spécialisé ciblant aussi cette portion du marché. Ainsi si les garanties sont adaptées à saclientèle, l’assureur spécialisé n’a pas besoin d’être le moins cher pour garder ses clients en
5

Introduction
portefeuille.
A cela se rajoutent l’inertie des habitudes, la subjectivité de la couverture d’assurance
et l’image de marque de l’assureur. Lorsque les assurés comparent les primes d’assurance, lacomparaison est souvent biaisée du fait de la segmentation et de la valeur du risque assuré.
Cependant, l’arrivée des comparateurs de prix sur internet ajoute un peu de transparence surles prix proposés par les assureurs.
A garantie et profil de risque comparables, la résiliation dépend majoritairement de l’élasticité de l’assuré au prix. Cette sensibilité au prix dépend du caractère psychologique desprix en premier lieu. Sur cet aspect, la loi psychophysique de Webner-Fechner précise que lasensation varie comme le logarithme de l’excitation. Sa transcription en termes d’élasticité
prix nous laisse penser qu’une hausse successive des prix entraîne moins de résiliations qu’une
hausse brutale et unique des prix. Vice versa, une baisse successive de prix devrait favoriser
le renouvellement des contrats par rapport à une baisse unique des prix.

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Modèle logistique
Au delà de ces considérations économiques et de marketing, il convient de vérifier a posteriori les causalités envisagées pour la résiliation et l’élasticité prix des clients. La résiliation
s’exprime à l’aide d’une variable aléatoire de Bernoulli Yi valant 1 si l’assuré i résilie son
contrat, et 0 s’il le renouvelle. Notons Xi le vecteur des variables explicatives de l’assuré i.
Un des modèles les plus simples est le modèle logistique qui repose sur l’équation suivante :P (Yi = 1) =1T
1 + e−β Xi,(1)où β est le vecteur de paramètres et M T est la transposée de la matrice M . Il est facile devérifier que le membre de droite, la fonction logistique, reste toujours compris dans l’intervalle[0,1].
La loi de probabilité étant spécifiée, nous pouvons donc estimer le paramètre β par maximum de vraisemblance. Le modèle logistique fait partie de la grande classe des modèles linéaires généralisés introduits par Nelder et Wedderburn (1972), pour lesquels de nombreuses
propriétés ont été démontrées. Le modèle logistique est donc un choix de modèles très naturel. Le chapitre 1 analyse les modèles de régression pour expliquer la résiliation en assurance
non-vie, notamment les modèles linéaires généralisés et une de leurs extensions les modèles
additifs généralisés. Ce premier chapitre souligne tout particulièrement les variables explicatives indispensables pour obtenir des résultats cohérents en terme de prédiction des taux derésiliation et met en exergue le role clé de la prime moyenne marché. Le chapitre conclut surla pertinence de modéliser le marché.
Modèles de choix
La modélisation du marché peut se faire de deux façons différentes : (i) soit nous modélisonsle marché comme un seul compétiteur, (ii) soit nous modélisons l’ensemble des assureurs
constituant le marché. Dans le premier cas, le modèle logistique suffit (i.e variable de décision
Yi ∈ {0, 1}) puisque si l’assuré résilie, c’est pour aller vers le marché. Il suffit donc de modéliser
le marché à l’aide de séries chronologiques (cf. la sous-section suivante), les résiliations à l’aide
d’un modèle logistique et d’utiliser la théorie du contrôle optimal pour déterminer une prime
répondant à certains critères. Dans le second cas, chaque assureur présent sur le marché va être6

tel-00703797, version 2 - 7 Jun 2012

modélisé, ainsi la variable de décision des clients n’aura plus deux modalités. Ainsi, il nous fautun modèle de choix (entre chaque assureur) pour modéliser une variable Yi ∈ {0, . . . , c − 1}.
L’étude des modèles de choix est un thème bien connu en économétrie. Le livre de Manski
et McFadden (1981) constitue un ouvrage de référence dans ce domaine. McFadden (1981) ∗présente en profondeur les modèles de choix dans le chapitre 5 de ce livre. C’est une extension
probabiliste du modèle de l’homo economicus de l’économie classique.
Les modèles de choix reposent sur deux composantes : (i) un système de probabilité dechoix et (ii) un cadre de maximisation d’utilité aléatoire. Soit P la probabilité de choix pour
un individu. P est une fonction de l’ensemble I × B × S dans l’intervalle [0,1], où I désignel’ensemble des alternatives, B ⊂ I l’ensemble des choix possibles offerts à l’individu et s une
caractéristique mesurée de l’individu.
P (i| B, s) désigne la probabilité de choisir l’alternative i parmi la sélection B pour unindividu de caractéristique s. De plus, nous rajoutons une fonction d’attribut observé ξ : I 7→
Z, telle que ξ(i) représente les attributs observés. Le système de probabilité de choix est donc
le vecteur (I, Z, ξ, B, S, P ). Sur ce système, deux hypothèses sont faites : (i) la sommation∀B ∈ B, P (B|B, s) = 1, (ii) la caractérisation totale ∀B = {i1 , . . . , in }, ∀B̃ = {ĩ1 , . . . , ĩn },ξ(ik ) = ξ(ĩk ) entraîne P (ik |B, s) = P (ĩk |B, s).
Outre le système de probabilité de choix, McFadden (1981) se place dans un cadre de
maximisation d’utilité aléatoire. En effet, comme l’utilité d’un individu n’est pas quelquechose de très facilement mesurable, il est cohérent de considérer son caractère aléatoire d’unpoint de vue de l’observateur.
La deuxième composante des modèles de choix, une hypothèse de maximisation d’utilité
aléatoire, est définie par un vecteur (I, Z, ξ, S, µ) où (I, Z, ξ, S) vient du système de probabilité et µ est la mesure de probabilité sur l’espace des fonctions d’utilités définies sur I,
dépendant s ∈ S. µ(., s) représente donc la loi de probabilité des “goûts” pour la populationde caractéristique s. Ainsi, la probabilité P (ik |B, s) de choisir l’alternative ik s’écrit
µ({U ∈ RI |∀j = 1, . . . , n, U (ik ) ≥ U (ij )}, s).
Des hypothèses additionnelles complètent la composante (I, Z, ξ, S, µ) pour que l’équation
précédente soit toujours définie.
Deux modèles paramétriques sont très utilisés dans ce cadre d’étude : les modèles de Luce
(1959) et de Thurstone (1927). Luce (1959) considère la forme paramétrique suivante pour laprobabilité de choisir l’alternative i parmi B
T

P (i|z B , β) = P

eβ z i

j∈B e

βT zj

,

où z B = (z 1 , . . . , z m ) correspond au vecteur d’attributs observés pour les alternatives dans B
et β un vecteur de paramètres. Cette forme paramétrique présuppose l’indépendance des alternatives non pertinentes, c’est à dire, pour tout i ∈ A ⊂ B, P (i|z B , β) = P (i|z A , β)P (A|z B , β).
Très souvent, une catégorie i0 de référence est considérée pour laquelle z i0 = 0 entrainant l’apparition de 1 dans la fraction ci-dessus. Le modèle logistique est un cas particulier de ce modèle
avec deux alternatives.
∗. Daniel L. McFadden a reçu le prix Nobel d’économie pour ces travaux sur les choix discrets le 10 décembre
2000.

7

Introduction
Thurstone (1927) considère la forme suivante
P (i|z B , β) = Φ0,Ω (−z B−i β),

tel-00703797, version 2 - 7 Jun 2012où z B−i = (z 1 − z i , . . . , z i−1 − z i , z i+1 − z i , . . . , z m − z i ) et Φ0,Ω est la fonction de répartition de la loi normale multivariée de moyenne 0 et de matrice de variance-covarianceΩ = z B−i AAT z TB−i . Ici, la catégorie de référence est i. Ces deux paramétrisations sont généralement appelées modèles logit multinomial et probit multinomial pour leur lien avec lesmodèles linéaires généralisés multivariés.
Par la suite, le chapitre 2, centré sur un modèle de compétition entre assureurs, modélise
le choix des assurés par une paramétrisation logit multinomiale. L’ensemble des alternatives
est l’ensemble des I assureurs B = {1, . . . , I}, tandis que les caractéristiques de l’individu selimiteront au numéro de son assureur actuel j et au vecteur des prix proposés par chaqueassureur x = (x1 , . . . , xI ). Ainsi, P (i|(x, j), β) représentera la probabilité de choisir l’assureur
i sachant que l’assuré est chez l’assureur j, une gamme de prix x et un vecteur de paramètre
β. Nous renvoyons au chapitre 2 pour plus de détails.

Dynamique de prix et cycles de marché
La dynamique des prix a un grand impact sur le comportement des assurés, aussi bien envaleur absolue qu’en valeur relative par rapport au marché. Les modèles de cycles de marché
sont donc très utilisés par les praticiens pour estimer le prix du marché de l’année prochaine.
Puis, chaque assureur va essayer de se positionner par rapport au prix du marché attendu.
En plus du prix de marché attendu, le ratio sinistre sur prime espéré doit être en adéquation
avec les réserves courantes en capital. Il est temps maintenant d’expliquer les raisons de laprésence de cycles sur les marché d’assurance non-vie. Ensuite, nous présenterons les modèles
de séries temporelles les plus utilisés, avant de conclure sur les problèmes de cette approche.
Causes du cycle
L’étude des cycles de marché est un thème depuis longtemps débattu en économie d’assurance. Les articles fondamentaux remontent à Venezian (1985) et Cummins et Outreville(1987). Dans cette introduction, nous nous basons sur la revue bibliographique très exhaustive
de Feldblum (2001).
Cet article traite de diverses théories classiques expliquant la causalité des cycles de marchéen assurance non-vie. L’auteur crédibilise et décrédibilise une à une les principales théories.
Au final, il conclut que la présence de cycles est dû l’effet conjugué de quatre causes : (i)la tarification actuarielle, (ii) la philosophie de la souscription, (iii) les fluctuations des tauxd’intérêts et (iv) la stratégie compétitive. Nous précisons ici chacune des causes avant deprésenter les modèles de séries temporelles utilisés pour montrer la pertinence d’une cause
plutôt qu’une autre.
La tarification actuarielle peut engendrer des cycles résultant de l’effet combiné de l’incertitude sur les sinistres et de la contre-cyclicité des coûts réels. L’incertitude est inhérenteà toute activité d’assurance et est due à l’inversion de cycle de production. Donc sans mêmeévoquer les sinistres les plus volatiles commes les catastrophes naturelles, l’assureur ne connaîtra jamais avec certitude le montant de la charge sinistre à venir au moment où il tarifie ses
contrats. La contre-cyclicité provient du décalage entre la survenance des sinistres et leur intégration dans les nouveaux tarifs. Ce retard d’information dû à des contraintes légales ettechniques est généralement de 2 ou 3 ans.
8

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L’antithèse de cette vision est que les cycles de marché ne sont généralement pas contracycliques aux conditions macroéconomique (prospérité vs. récession). Les actuaires sont capables
d’apprendre de leurs erreurs passées et d’intégrer prospectivement les tendances actuelles dansleur tarif. Les cycles de souscription ne seraient pas dus à la tarification actuarielle mais à larésistance des souscripteurs à appliquer de nouveaux tarifs via les rabais commerciaux qu’ils
peuvent accorder.
Une psychologie de masse des souscripteurs crée des effets d’emballement. En phase profitable, les assureurs sont optimistes et se font la guerre des prix en quête de parts de marché.
A l’opposé, lorsque les résultats de souscription se dégradent, les assureurs sont contraints aubout d’un certain temps de remonter les prix de manière à retrouver un niveau de profitabilité
acceptable. Le jeu consiste à détecter le plus tôt possible les changements de cycles pour entirer profit, car malgré une psychologie de masse, les souscripteurs n’agissent pas de manière
concertée.
Et, c’est là le problème de cette thèse : si la compétition est aussi intense que l’on sousentend, l’offre et la demande ne devraient-elles pas converger vers un équilibre ? La coordination entre assureurs étant interdite, les assureurs sont incapables de déterminer le prix et laquantité d’assurance d’une demande globale en perpétuel déséquilibre. Les consensus sur leschangements de phase restent toujours un mystère et pourtant les résultats de souscriptiond’un assureur à l’autre sont très corrélés.
Une caractéristique fondamentale du marché de l’assurance est le temps assez long s’écoulant entre l’encaissement des primes et le paiement effectif des sinistres (tout particulièrement
sur les garanties de responsabilité civile). La fluctuation des intérêts n’est qu’une couche supplémentaire aux contraintes que subissent les assureurs. En période de taux d’intérêts forts,
les souscripteurs peuvent se permettre de souscrire au coût attendu des sinistres et espérer les
profits sur l’actualisation des sinistres. D’un autre côté, en période de taux bas, les assureurssont forcés de remonter les prix.
Or, les cycles ne perdent rarement de leur intensité lorsque les taux d’intérêts fluctuent
peu. De plus, les taux augmentent à la fois les primes et les coûts des sinistres, donc lesbénéfices de souscription devraient être faiblement impactés par ses changements de taux.
L’expertise financière des assureurs rend l’hypothèse de fluctuations des taux d’intérêts peu
crédible, car elle suppose des méthodes de tarification naïves, des souscripteurs simplistes, desrégulateurs rigides et des gestionnaires de fond décevants. En environnement compétitif, lesjoueurs irrationnels disparaissent.
Enfin, la théorie économique sur la compétition garantit que les firmes vendent au coûtmarginal. Cependant de nombreux effets contrecarrent cette belle théorie, les assureurs ne
connaissent ni l’offre ni la demande en assurance et encore moins le coût d’un produit d’assurance (au moment de sa vente). A cela, se rajoutent les barrières à l’entrée non pas financières
mais opérationnelles et techniques : coût de distribution, définition d’un standard de souscription, segmentation, etc. . . Comme le souligne Feldblum (2001), il est facile d’entrer sur unmarché d’assurance, mais il est nettemment plus difficile d’y entrer avec succès.
Comme déjà énoncé, la faible différentiation des produits et la relative inélasticité desclients rend difficile l’arrivée de nouveaux entrants. Les caractéristiques intrinsèques de l’assurance rendent les stratégies compétitives à la fois indispensables et stabilisantes. De plus,
sur les marchés matures, seules les fusions-acquisitions permettent de gagner de grosses partsde marché sans pratiquer une brutale chute des prix. Par conséquent, le nombre de risque parassureur reste en général stable en régime de croisière.
9Introduction
Modèles de séries temporelles
Nous présentons dans cette sous-section les modèles de séries temporelles les plus classiques
utilisées dans la littérature des cycles de marché pour affirmer ou infirmer une conjecture. L’objet des séries temporelles est l’étude des processus stochastiques en temps discret (Xt )t∈N . Le
modèle de base est le modèle autorégressif d’ordre p. Pour un processus faiblement stationnaire (Xt ) (c’est à dire espérance constante et fonction d’autocovariance dépendant seulement
de l’écart de temps), (Xt )t est dit autorégressif d’ordre p s’il existe des coefficient a1 , . . . , ap
et un processus bruit blanc (Et ) tels que
Xt =

p
X

ai Xt−i + Et .

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i=1

Pour le cas particulier p = 1, on retrouve une marche aléatoire, tandis que pour p = 2, (Xt )peut être périodique si a2 < 0 et a21 + 4a2 < 0 avec une périodea1√
P = 2π arccos.
2 −a2Malgré sa simplicité et le caractère très fort des hypothèses, les modèles autoregressifs ont été
appliqués dans beaucoup d’articles de cycles de marché, par exemple Cummins et Outreville(1987). Cummins et Outreville (1987) cherchent à montrer l’hypothèse de tarification actuarielle avec Xt le résultat de souscription de l’année t. Les périodes de cycles oscillent entre 6
et 10 ans suivant les pays.
Une hypothèse centrale des modèles autorégressifs est la stationnarité du processus. Ellepeut par exemple être testée par le test de racine unitaire de Dickey-Fuller (voir, par exemple,
Gourieroux et Monfort (1997)). En pratique, l’hypothèse de stationnarité est rarement vérifiée.
Pour compenser cela, Fields et Venezian (1989) et Gron (1994a) vont introduire des variables
explicatives pour obtenir une régression temporelle du type
Xt = a1 Xt−1 + b0 Yt + c0 Zt + Et ,où Yt , Zt sont des indicateurs temporels indépendants et a0 , b0 , c0 6= 0. Une modélisation plus
intéressante est proposée par Haley (1993) en utilisant les modèles de cointégration.
Les modèles cointégrés ont été proposés par Granger et Engle (1987) ∗ , voir aussi Committee (2003). Deux processus (Xt )t et (Yt )t sont cointégrés s’il existe une combinaison linéaire
de ces deux processus qui est stationnaire, c’est à dire (αXt + βYt )t est stationnaire pour uncouple (α, β) non nul. Cette notion de cointégration est cruciale car elle permet de modéliser
des tendances long-terme entre les deux séries (Xt )t et (Yt )t .
Prenons l’exemple de Hamilton (1994) avec
Xt = Xt−1 + Et et Yt = 2Xt + Eet .
où Et , Eet sont deux bruits blancs. Il est facile de voir que (Yt − 2Xt ) est un processus stationnaire. Néanmoins, les trajectoires des processus (Xt )t et (Yt )t sont fortement corrélées,comme l’illustre la figure 1a. Sur la figure 1, on a aussi tracé l’évolution de la prime moyennemarché et l’inverse du ratio sinistre sur prime (S/P) marché pour le marché auto français. Le
graphique 1b montre à quel point ces deux grandeurs sont liées et valide le fait que la prime
marché est inversement proportionnelle au ratio S/P.
∗. Clive W.J. Granger et Robert F. Engle ont reçu le prix Nobel d’économie pour leurs travaux sur lesséries temporelles macroéconomiques et financières le 10 décembre 2003.

10

Marché français auto
1.25

Exemples modèles cointégrés

1.15
1.10
1.00

1.05

Indices marché

10
5

0.95

0

Yt

Accrois. prime défl.
Inverse S/P

1.20

Y1t
Y2t

0

10

20

30

40

50

1970 1975 1980 1985 1990 1995 2000 2005

tel-00703797, version 2 - 7 Jun 2012

Temps tAnnée(a) Exemple de modèles cointégrés

(b) Marché auto français

Figure 1 – Séries temporelles

Si on désigne l’opérateur différence par ∆, le système d’équations ci-dessus peut se réécrire∆Xt = Et et ∆Yt = 2Xt − Yt + 2Et + Eet .
L’exemple précédent est en fait un cas particulier du théorème de réprésentation de Granger.
Pour deux séries Xt , Yt cointégrées d’ordre 1, il existe α1 , α2 non tous nul et β 6= 0, telle quesoit ∆Xt − α1 (Yt−1 − βXt−1 ) soit ∆Yt − α2 (Yt−1 − βXt−1 ) sont stationnaires.
Plus généralement, les modèles autorégressifs vectoriels avec cointégration d’ordre p admettent la représentation suivante :∆Xt = AB T Xt−1 +pX

Γj ∆Xt−j + Et ,j=1

où Xt est un processus de dimension n, AB T une matrice n×n produit de matrices n×r, Γj desmatrices n × n et Et un bruit blanc corrélé de dimension n. Ces modèles sont largement utilisésen économétrie. Dans le cadre des modèles de cycles de marché, ils sont aussi abondamment
utilisés (voir, par exemple, Haley (1993); Grace et Hotchkiss (1995); Doherty et Garven (1995);
Blondeau (2001)). Typiquement, nous modélisons la prime moyenne marché en fonction desratios sinistres sur primes, de l’inflation et d’autres indicateurs macroéconomiques comme lestaux d’intérêts court et long termes.
Problèmes de cette approche
Pour prendre des actions pertinentes, l’assureur doit savoir se positionner par rapport
au marché. L’utilisation de modèles cointégrés apparaît nécessaire, car les modèles de séries
chronologiques auto-régressifs sont rarement adaptés pour modéliser la prime marché ou leratio marché sinistre sur prime. Une fois après avoir calibré un modèle de séries chronologiques
11Introduction
et un modèle de résiliation, l’assureur peut donc juger en théorie de sa politique tarifaire pourl’année suivante.
En pratique, il ne suffit pas de modéliser la prime moyenne marché pour faire un tarif.
L’assureur doit être capable de décliner les changements de tarif par segment. C’est à dire,il doit pouvoir modéliser les plus grands assureurs de la place (par exemple les 5 premiers)
individuellement et pas seulement comme un seul acteur. Et c’est là que l’approche par série
temporelle est problématique : il faut avoir un historique assez important par assureur pourla variable considérée. En pratique, c’est rarement le cas voire impossible.
L’objectif du chapitre 2 est de proposer une réponse à ce problème. Nous modélisonsconjointement le comportement des assurés et la compétition entre assureurs. Le comportement des assurés sera modélisé par une paramétrisation multinomiale logit, tandis que lacompétition sera modélisée par la théorie des jeux non-coopérative, que nous présentons dansla prochaine section.

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Théorie des jeux et modèles de compétition
La théorie des jeux est l’étude des interactions entre plusieurs agents (hommes, entreprises,
animaux, etc. . .) et regroupe l’ensemble des outils mathématiques nécessaires à la compréhension du phénomène de prise de décision pour un problème donné. Le principe fondamental
sous-jacent à la théorie des jeux est que les joueurs tiennent compte, d’une manière ou d’une
autre, des comportements des autres joueurs dans leur prise de décision, à l’opposé d’une
vision individualiste de la théorie du contrôle optimal.
La théorie des jeux prend ses racines dans les études économiques d’oligopoles réalisées par
Cournot (1838); Edgeworth (1881) et Bertrand (1883). Elle a été popularisée et est devenue
une discipline à part entière grâce au livre de von Neumann et Morgenstern (1944), qui pose
les bases des jeux à somme nulle à plusieurs joueurs, non coopératifs et coopératifs. Quelquesannées plus tard, Nash (1950a,b, 1951, 1953) ∗ a transformé la théorie des jeux en proposant
un nouveau concept d’équilibre et étudié l’existence de tels équilibres. Depuis, la théorie desjeux n’a cessé de croître dans de multiples directions.
Le champ d’application de la théorie des jeux ne se restreint pas à l’économie. Elle s’applique notamment à la biologie, l’ingénierie, les transports, les réseaux, etc. . . La présentation,qui suit, se base sur les ouvrages de référence suivants : Fudenberg et Tirole (1991), Basar et
Olsder (1999), Osborne et Rubinstein (2006).
Un jeu est une description formelle d’une interaction entre plusieurs joueurs. Il est constituéd’un ensemble de joueurs E = {1, . . . , I}, d’une fonction objective ou d’une fonction coût pourchacun des joueurs Oi : X 7→ R, et d’un ensemble d’actions possibles par joueur Xi ⊂ Rni
pour i ∈ E, où X = X1 × · · · × XI . Notons que l’ensemble des actions Xi du joueur i n’est
pas nécessairement fini ni nécessairement discret. Un profil d’action x regroupe un ensembled’actions xi des I joueurs. Un concept de solution va spécifier un critère selon lequel un profild’action x est plus préférable que y pour un joueur.
Il existe de multiples classes de jeux permettant de préciser le type d’intéractions étudiées :les actions des joueurs sont elles simultanées ou séquentielles (description normale ou extensive
des jeux) ; cherche-t-on à maximiser un bien-être global (coopération) ou les joueurs sont-ils
non coopératifs ; l’information entre les joueurs est-elle parfaite (chaque joueur connait les objectifs de ses compétiteurs) ou seule une partie de l’information est révélée aux concurrents ;∗. John F. Nash a reçu le prix Nobel d’économie pour ces travaux en théorie des jeux le 8 décembre 1994.

12

joue-t-on sur une ou plusieurs périodes ; la fonction objective dépend-elle d’un phénomène aléatoire ? Nous ne présenterons dans cette introduction que les jeux simultanés non-coopératifs
déterministes à information parfaite.

Jeux statiques
Nous supposons que l’information est parfaite, c’est à dire chaque joueur i connait les
fonctions objectives/coûts Oj des autres joueurs j 6= i. Les joueurs choisissent leur action
simultanément : personne ne peut tirer profit en jouant après les autres. De plus, chaque joueurcherche son propre bien-être et ne peut coopérer, c’est à dire on exclut les jeux coopératifs.

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Jeux finis
Considérons le cas où l’ensemble des actions possibles est fini, c’est à dire les ensembles
Xi sont discrets. L’ensemble des actions possibles est donc fini et contient Card(X1 ) × · · · ×
Card(XI ) éléments. Pour toutes ces possibilités, on peut calculer la valeur des fonctions objectives pour chacun des joueurs. Ainsi, la fonction objective de chaque joueur peut être décritedans un tableau multidimensionnel. Par simplification, on se restreint aux jeux à deux joueurs,
I = 2.
Commençons par un exemple, le dilemne du prisonnier. Deux suspects (complices d’undélit) sont retenus dans des cellules séparées, dans lesquelles ils ne peuvent pas communiquer.
Les enquêteurs leurs proposent de passer aux aveux pour réduire leur éventuelle peine deprison. Si un et seulement un des deux prisonniers dénonce l’autre, il est remis en liberté alors
que le second écopera de la peine maximale (par exemple 10 ans). Si les deux se dénoncent
entre eux, ils seront condamnés à une peine plus légère (par exemple 5 ans). Enfin si les deux
refusent de se dénoncer, la peine sera minimale (par exemple 6 mois), faute d’éléments au
dossier.
Les coûts des joueurs sont représentés par la double matrices suivantes, où le joueur 1 joue
sur les lignes et le joueur 2 sur les colonnes.
J1 | J2

se taitdénoncese taitdénonce(-1/2, -1/2)(0, -10)(-10, 0)(-5, -5)

S’ils coopéraient, les deux joueurs écoperaient seulement de 6 mois de prison. Mais comme ils
ne peuvent coopérer, chacun va chercher à minimiser sa peine potentielle, c’est à dire le joueur
1 cherche le minimum des maximum des lignes, tandis que le joueur 2 cherche le minimum desmaximum des colonnes. Par conséquent, chaque joueur va choisir de dénoncer l’autre joueur.
Les jeux finis à deux joueurs sont appelés les jeux bimatrices. En effet, les coûts des deux
joueurs sont représentés par deux matrices A et B de taille Card(X1 ) × Card(X2 ). Danscette configuration aij et bij représentent le coût des joueurs lorsque pour un profil d’action(i, j) où i (respectivement j) désigne le ième (j ème ) élément de l’ensemble fini X1 (X2 ). Nous
introduisons maintenant l’équilibre de Nash.
Définition. Une paire de stratégies (i? , j ? ) constitue un équilibre de Nash au jeu bimatrice
A, B si les inégalités suivantes sont vérifiées
ai? j ? ≤ aij ? et bi? j ? ≤ bi? j(2)pour tout i = 1, . . . , Card(X1 ) et j = 1, . . . , Card(X2 ).
13

Introduction
Si A représente les gains plutôt que les coûts, il suffit d’inverser les inégalités. Un équilibrede Nash s’interprète comme un point où aucun des joueurs n’a d’intérêt à changer d’actions
tant que son opposant ne change pas.
Une question légitime qui peut se poser maintenant est l’existence d’équilibre de Nash pour
toutes matrices A, B. Malheureusement, il existe des matrices pour lesquelles aucun équilibren’existe. Par exemple, pour




1 0
3 2
A=
et B =.
2 −1
0 1
La raison de l’inexistence d’équilibre de Nash est la discontinuité des actions i = 1, . . . , Card(X1 )et j = 1, . . . , Card(X2 ).
Un cas particulier des jeux finis à deux joueurs est le cas où les objectifs des deux joueurssont antagonistes, c’est à dire B = −A. Ces sont les jeux à somme nulle. L’équation (2) del’équilibre de Nash se réduit à la double inégalité suivantetel-00703797, version 2 - 7 Jun 2012

ai? j ≤ ai? j ? ≤ aij ? .
L’équilibre de Nash (i? , j ? ) est appelé point col. Définissons le minimax et le maximin par
V (A) = maxj mini aij et V (A) = mini maxj aij . Si V (A) = V (A) alors il existe un équilibrede Nash. Pour les jeux à somme non nulle, on a vu que ce n’était pas aussi simple.
L’astuce proposée par Nash lui-même pour garantir l’existence est de considérer des stratégies mixtes, où les joueurs choisissent leur action en fonction d’un événement aléatoire. Par
exemple, dans l’exemple précédent, le joueur 1 peut choisir 2 fois sur 3 de jouer la première
action et 1 fois sur 3 la deuxième. Une telle stratégie est notée (2/3, 1/3). Les stratégies mixtes
du joueur i sont par définition une loi de probabilité parmi les actions de l’ensemble Xi . Pourne pas confondre, les stratégies non mixtes sont appelées stratégies pures et sont des cas particuliers des stratégies mixtes où la loi de probabilité est dégénérée, par exemple (1,0) dansl’exemple précédent.
Définition. Une paire de stratégies (x? , y ? ) constitue un équilibre de Nash au jeu bimatrice(A, B) en stratégie mixte si pour tous vecteurs de probabilité x, y, on ax?T Ay ? ≤ xT Ay ? et x?T By ? ≤ x?T By.
On peut maintenant énoncer un théorème d’existence.
Théorème. Tous les jeux bimatrices admettent un équilibre de Nash en stratégie mixte.
La démonstration est basée sur le théorème de point de fixe de Brouwer, qui suit.
Théorème (Brouwer (1912)). Soient B n la boule unité d’un espace euclidien de dimension n
et T : B n 7→ B n une application. Si T est continue, alors T admet au moins un point fixe.
L’ensemble B n peut être remplacé par n’importe quel ensemble compact, convexe, nonvide. Dans notre cas, on considèrera le simplexe de dimension 2 ou supérieure. Il est assez
facile de comprendre pourquoi l’équilibre de Nash en stratégies pures n’existe pas forcément :l’ensemble X1 × X2 n’est pas convexe.
Le calcul d’équilibre en stratégie mixte est assez complexe. Néanmoins, on peut le reformuler au problème d’optimisation bilinéaire
min xT Ay + xT By + p + q,

x,y,p,q14

sous contrainte
Ay ≥ −p1, B T x ≥ −q1, x ≥ 0, y ≥ 0, xT 1 = 1, y T 1 = 1,p, q ∈ R sont des variables auxiliaires telles que si (x? , y ? , p? , q ? ) sont solutions du problème
précédent alors p? = x?T Ay ? et q ? = x?T By ? , voir la section 3.6 de Basar et Olsder (1999).
Une autre approche basée sur l’itération des stratégies à l’aide de pivots est l’algorithme de
Lemke-Howson.
Les jeux finis à deux joueurs peuvent être généralisés à des jeux à I joueurs. Les matrices
se transforment en tableaux à I dimensions et les deux inégalités définissant un équilibre
deviennent I inégalités. Nous renvoyons le lecteur intéressé vers les ouvrages de référence
précédemment listés.

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Jeux continus
Traitons maintenant le cas des jeux continus où les ensembles de stratégies Xi sont continus
et non plus discrets. On omet volontairement le cas des jeux où l’espace Xi est dénombrable
et infini, où N, car il ne représente d’utilité dans le cadre de cette thèse. On peut par exemple
penser à un intervalle de prix, un intervalle de quantités, etc. . . Les ensembles Xi sont généralement supposés compact, convexe et non vide. L’équilibre de Nash se définit de la manière
suivante.
Définition. Pour un jeu à deux joueurs où O1 , O2 désignent le coût des joueurs, un couple destratégie (x?1 , x?2 ) ∈ X1 × X2 est un équilibre de Nash si les inégalités suivantes sont respectéesO1 (x?1 , x?2 ) ≤ O1 (x1 , x?2 ) et O1 (x?1 , x?2 ) ≤ O1 (x?1 , x2 ),(3)

pour tout (x1 , x2 ) ∈ X1 × X2 .
Lorsqu’on travaille avec des fonctions de gains O1 , O2 plutôt que des fonctions de coûts, il
suffit de renverser les inégalités. Si le jeu est à somme nulle, c’est à dire O2 = −O1 , alors unéquilibre de Nash (équation (3)) est un point colO1 (x?1 , x2 ) ≤ O1 (x?1 , x?2 ) ≤ O1 (x1 , x?2 ).
Pour un jeu à I joueurs, on introduit les notations suivantes. Soit i ∈ E un joueur : xi
désigne l’action du joueur i, tandis que x−i = (x1 , . . . , xi−1 , xi+1 , . . . , xI ) les actions des autres
joueurs. L’équilibre de Nash se définit comme suit.
Définition. Pour un jeu à I joueurs où Oi , i ∈ E désignent le coût du joueur i, un vecteurde stratégie (x?1 , . . . , x?I ) ∈ X est un équilibre de Nash si pour tout i ∈ E, on aOi (x?i , x?−i ) ≤ Oi (xi , x?−i ), pour tout xi ∈ Xi .

(4)

Pour mieux comprendre les théorèmes d’existence qui suivent, il faut comprendre quel’équation (4) est en fait un problème d’optimisation. Un équilibre de Nash x? vérifie les I
sous-problèmes d’optimisation suivant
x?i ∈ arg min Oi (xi , x?−i ).
xi ∈Xi15

Introduction
Le problème d’optimisation ci-dessus admet (au moins) une solution si la fonction xi 7→
Oi (xi , x?−i ) est quasiconvexe. Une fonction f : R 7→ R est quasiconvexe si pour tout x, y ∈ R,
et pour tout λ ∈]0, 1[, on a f (λx + (1 − λ)y) ≤ max(f (x), f (y)). Géométriquement parlant,
une fonction univariée quasiconvexe est unimodale, par exemple monotone ou décroissante etcroissante.
On énonce maintenant un premier théorème d’existence.

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Théorème (Nikaido et Isoda (1955)). Soit un jeu à I joueurs où les espaces de stratégie Xisont non-vides, convexes et compacts. Supposons que les fonctions de coût Oi : X 7→ R sontcontinus. Si les fonctions xi 7→ Oi (xi , x−i ) sont quasiconvexes, alors il existe un équilibre de
Nash (en stratégie pure).
Si on travaille avec des fonctions de gain, alors la quasiconvexité devient la quasiconcavité,
définie par f (λx + (1 − λ)y) ≥ min(f (x), f (y)), pour tout λ ∈]0, 1[.
Le concept de quasiconvexité est plus faible que celui de convexité. En fait, il existe plusieurs variantes allant de la quasiconvexité à la stricte convexité. Nous rappellons ci-dessous
certains concepts, renvoyons le lecteur vers Diewert et al. (1981) détaillant les neuf sortes dequasiconvexité. Soit f : Rn 7→ R une fonction. On dit que– f est quasiconvexe : ∀x, y ∈ R, ∀λ ∈]0, 1[, on a f (λx + (1 − λ)y) ≤ max(f (x), f (y)).
– f est convexe : ∀x, y ∈ R, ∀λ ∈]0, 1[, on a f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y).
– f est strictement convexe : ∀x, y ∈ R, ∀λ ∈]0, 1[, on a f (λx + (1 − λ)y) < λf (x) + (1 −λ)f (y).
Un concept manquant est la pseudoconvexité, mais qui requiert une fonction au moins différentiable directionnellement. Pour une fonction C1 , on a
– f est quasiconvexe : ∀x, y ∈ R, on a f (x) ≥ f (y) ⇒ ∇f (x)T (y − x) ≤ 0.
– f est pseudoconvexe : ∀x, y ∈ R, on a f (x) > f (y) ⇒ ∇f (x)T (y − x) < 0.
– f est convexe : ∀x, y ∈ R, on a f (y) − f (x) ≤ ∇f (x)T (y − x).
– f est strictement convexe : ∀x, y ∈ R, on a f (y) − f (x) < ∇f (x)T (y − x).
Pour une fonction C2 , on a
– f est quasiconvexe : ∀x ∈ R, ∀d ∈ Rn , dT ∇f (x) = 0 ⇒ dT ∇2 f (x)d ≥ 0.
– f est pseudoconvexe : ∀x ∈ R, ∀d ∈ Rn , dT ∇f (x) = 0 ⇒ dT ∇2 f (x)d > 0.
– f est convexe : ∀x ∈ R, ∀d ∈ Rn , dT ∇2 f (x)d ≥ 0, c’est à dire ∇2 f semidéfinie positive.
– f est strictement convexe : ∀x ∈ R, ∀d ∈ Rn , dT ∇2 f (x)d > 0, c’est à dire ∇2 f définie
positive.
Toutes les définitions sont incrémentales, ainsi la stricte convexité implique la convexité, impliquant la pseudoconvexité, impliquant la quasiconvexité. Par conséquent, on constate que lethéorème d’existence d’équilibre de Nash de Nikaido et Isoda (1955) requiert une des conditions
les plus faibles de convexité sur la fonction xi 7→ Oi (xi , x−i ).
Pour avoir l’unicité de l’équilibre de Nash, il faut cependant requérir beaucoup plus quela quasiconvexité. Le théorème 2 de Rosen (1965) donne un résultat d’unicité dans un cadre
légèrement plus général que l’équation (4). Il considère le problème suivant
min Oi (xi , x?−i ) tel que g i (xi ) ≤ 0,xi ∈Xi(5)

où gi : xi 7→ g i (xi ) est la fonction contrainte du joueur i supposée continue. L’ensemble deei = {xi ∈ Xi , g i (xi ) ≤ 0}.
stratégies possibles se réduit donc à l’ensemble X
16

Théorème (Rosen (1965)). Soit un jeu continu à I joueurs avec des espaces Xi non-vides,
convexes, compacts, des fonctions contraintes g i convexes et des fonctions coûts Oi telles que
xi 7→ Oi (x) sont convexes.
S’il existe r > 0 telle que la fonction gO : Rn × RI 7→ Rn définie par
r1 ∇x1 O1 (x)..
gO (x, r) = ,.
rI ∇xI OI (x)vérifie la propriété suivantetel-00703797, version 2 - 7 Jun 2012(x − y)T gO (y, r) + (y − x)T gO (x, r) > 0,

(6)

alors l’équilibre de Nash vérifiant l’équation (5) est unique.
Une condition suffisante pour l’équation (6) est que la matrice par bloc suivante soit définie
positive
r1 ∇x1 ∇x1 O1 (x) . . . r1 ∇xI ∇x1 O1 (x)


..
..
GO (x, r) = .
.
.
rI ∇x1 ∇xI OI (x) . . .

rI ∇xI ∇xI OI (x)L’équation (6) se réécritIX

ri (xi − yi )T ∇xi Oi (y) +

i=1

I
X

ri (yi − xi )T ∇xi Oi (x) > 0.

i=1

Rappelons que pour une fonction f strictement convexe, on a ∇f (x)T (y − x) > f (y) − f (x),ce qui est équivalent à ∇f (y)T (x − y) > f (x) − f (y). Ainsi, une condition suffisante (maispas nécessaire) pour que l’équation (6) soit vérifée est la stricte-convexité des fonctions xi 7→
Oi (xi , x−i ).
Une autre condition garantissant cette fois-ci le caractère définie positive de la matrice
GO (x, r) est la dominance diagonale , c’est à dire ∀i ∈ E, ∀m = 1, . . . , ni ,I

nj

X X ∂ 2 Oi (x)∂ 2 Oi (x)>.
∂xjk ∂xim
∂x2im
j=1
k=1

Ainsi, guarantir l’unicité de l’équilibre de Nash requiert la convexité des fonctions xi 7→ Oi (x)et que l’équation (6) soit vérifiée, tandis que l’existence nécessite seulement la quasiconvexitéde ces fonctions.
Les méthodes de calcul d’équilibre de Nash sont complexes, puisqu’il ne suffit pas de réaliser
I optimisations basées sur l’équation (5). Les équations sont toutes liées puisque les fonctions
objectives Oi dépendent des actions des autres joueurs. Nous présentons ici le cas plus simple
des jeux à deux joueurs et renvoyons au chapitre 3 pour le cas général.
Notons R1 (resp. R2 ) la fonction de meilleure réponse du joueur 1 (joueur 2) pour lesactions du joueur 2 (joueur 1) : R1 (x2 ) = {x1 ∈ X1 , ∀y1 ∈ X1 , O1 (x1 , x2 ) ≤ O1 (y1 , x2 )} (resp.
R2 (x1 ) = {x2 ∈ X2 , ∀y2 ∈ X2 , O2 (x2 , x1 ) ≤ O2 (y2 , x1 )}). Dans les définitions précédentes,
on suppose implicitement que la réaction d’un joueur par rapport à l’autre soit unique, parexemple lorsque les fonctions objectives sont convexes. Un équilibre de Nash est un point
17

Introductiond’intersection des courbes (x1 , R2 (x1 )) et (R1 (x2 ), x2 ) pour x1 ∈ X1 et x2 ∈ X2 . C’est à direun point fixe de l’équation x1 = R1 (R2 (x1 )).
De la même manière que pour les jeux finis, on peut définir des stratégies mixtes pourles jeux continus. Les actions sont des fonctions de répartition µi et les fonctions objectives
deviennent
ZZ
Oi (µ1 , . . . , µI ) =...
Oi (x1 , . . . , xI )dµ1 (x1 ) . . . dµI (xI ).
X1

XI

On peut étendre la définition d’un équilibre de Nash aux stratégies mixtes.
Définition. Pour un jeu à I joueurs où Oi , i ∈ E désignent le coût du joueur i, un vecteurde probabilité (µ?1 , . . . , µ?I ) ∈ X est un équilibre de Nash en stratégie mixte si pour tout i ∈ E,et pour toute fonction de répartion µi sur Xi , on aOi (µ?i , µ?−i ) ≤ Oi (µi , µ?−i ).

(7)

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Nous donnons ci-dessous le théorème de Glicksberg (1950).
Théorème. Soit un jeu à I joueurs où les espaces de stratégie Xi sont compacts. Si les
fonctions de coût Oi : X 7→ R sont continues, alors il existe un équilibre de Nash en stratégie
mixte.
Jeux généralisés
Les jeux généralisés proposent une extension des équilibres de Nash suggérée dans l’équation (5). Pour un jeu à I joueurs, nous introduisons une fonction contrainte rendant les actions possibles d’un joueur dépendantes non seulement de son action mais aussi des actions
des autres joueurs. Soit g i : X 7→ Rmi la fonction contrainte d’un joueur telle que les actions
possibles du joueur i appartiennent à l’ensemble{xi ∈ Xi , g i (xi , x−i ) ≤ 0}.
L’équilibre de Nash généralisé se définit comme suit.
Définition. Pour un jeu à I joueurs où Oi , i ∈ E désignent le coût du joueur i, un vecteurde stratégie (x?1 , . . . , x?I ) ∈ X est un équilibre de Nash généralisé si pour tout i ∈ E, on aOi (x?i , x?−i ) ≤ Oi (xi , x?−i ), pour tout xi ∈ Xi , g i (xi , x−i ) ≤ 0.

(8)

La différence entre les équations (5) et (8) est le fait que la fonction contrainte dépend desactions de tous les joueurs et pas seulement de l’action xi du joueur i. Un équilibre de Nashgénéralisé x? vérifie donc I sous-problèmes d’optimisation
min Oi (xi , x?−i ) such that g i (xi , x?−i ) ≤ 0,xi ∈Xi(9)pour tout i ∈ E. Pour donner des théorèmes d’existence d’équilibres généralisés, nous introduisons les correspondances.
Une correspondance F : X 7→ 2Y est une application telle que ∀x ∈ X, F (x) est un sousensemble de Y . Les correspondances sont parfois notées F : X 7→ P(Y ) ou encore F : X ⇒ Y .
Pour les correspondances,
le domaine de F se définit par dom(F ) = {x ∈ X, F (x) 6= ∅}, laS
portée par rg(F ) = x F (x), le graphe de F par Gr(F ) = {(x, y) ∈ X × Y, y ∈ F (x)}. Deux18

exemples typiques de correspondances sont F : x 7→ [−|x|, |x|] ; l’inverse d’une fonction f ,F : x 7→ f −1 (x).
Maintenant, nous définissons deux types de continuités pour les correspondances : semicontinuité inférieure et supérieure, abrégées l.s.c. et u.s.c.. Dans la littérature, deux définitions
s’opposent : la semicontinuité au sens de Berge (voir (Berge, 1963, page 109)) et la semicontinuité au sens de Hausdorff (voir (Aubin et Frankowska, 1990, page 38-39)).
Cependant, ces définitions sont équivalentes si la correspondance F est à valeur compacte.
Dans ce cas, les semicontinuités u.s.c/l.s.c se caractérisent sur le graphe de F . Nous rapportons
ici ces définitions, voir Hogan (1973).

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Définition. F est semicontinue supérieurement (u.s.c.) en x, si ∀(xn ) ∈ X N , xn → x, ∀yn ∈T (xn ), et ∀y ∈ Y,
yn → y ⇒ y ∈ T (x).
F est semicontinue inférieurement (l.s.c.) en x, si ∀(xn )n ∈ X N , xn → x, ∀y ∈ T (x), ∃(yk ) ∈Y N et ∀k ∈ N,
yk ∈ T (xk ) et yk → y.
F est semicontinue supérieurement (resp. inférieurement) sur X, si F est semicontinue supérieurement (inférieurement) en tout point de X.
Introduisons maintenant la correspondance de contraintes liés aux équilibres de Nash généralisés Ci : X−i 7→ 2Xi représentant les contraintes du joueur i par
Ci (x−i ) = {xi ∈ Xi , g i (xi , x−i ) ≤ 0}.
Nous pouvons maintenant énoncer le théorème d’existence.
Théorème (Ichiishi (1983)). Soit un jeu à I joueurs caractérisé par des espaces de stratégies
Xi ⊂ Rni , des correspondances de contrainte Ci et des fonctions objectives Oi : Rni 7→ R. Si
pour tout joueur i, on a
– Xi est non vide, convexe et compact,– Ci est u.s.c. et l.s.c. sur X−i ,– ∀x−i ∈ X−i , Ci (x−i ) est non vide, fermé et convexe,– Oi est continue sur le graphe Gr(Ci ),– ∀x ∈ X, xi 7→ Oi (xi , x−i ) est quasiconcave sur Ci (x−i ),Alors il existe un équilibre de Nash généralisé.
La démonstration repose sur le théorème de point fixe pour les correspondances ∗ de Kakutani (Kakutani (1941)) et sur le théorème du maximum de Berge. Nous renvoyons le lecteurvers Ichiishi (1983), Aubin (1998) ou Ok (2005) pour une démonstration de ce théorème.
Nous analysons maintenant les conséquences de la semicontinuité l.s.c. et u.s.c. de la correspondance Ci sur les fonctions contraintes. Une propriété de Rockafellar et Wets (1997)
permet d’obtenir facilement la semicontinuité u.s.c. lorsque les fonctions g i sont continues.
Proposition (Rockafellar et Wets (1997)). Soit Ci : X−i 7→ 2Xi la correspondance decontraintes définie précédemment. Si les ensembles Xi sont fermés et que toutes les composantes gji sont continues sur Xi × X−i , alors Ci est une u.s.c. sur X−i .
∗. L’existence d’équilibre de Nash sans contrainte repose sur le théorème de Brouwer.

19

Introduction
Cependant, il est plus ardu de montrer la semicontinuité inférieure d’une correspondance.
Rockafellar et Wets (1997) suppose l’existence d’un point à l’intérieur du domaine de contraintes,
c’est à dire ∃(x̄i , x̄−i ) ∈ Xi × X−i , g i (x̄i , x̄−i ) > 0. Mais en utilisant le théorème 13 de Hogan(1973), nous avons une condition plus faible.

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Proposition (Hogan (1973)). Soit Ci : X−i 7→ 2Xi la correspondance de contrainte du jeu
ei définie par C
ei (x−i ) = {xi ∈ Xi , g i (xi , x−i ) > 0}. Si les composantes de g i
généralisé. Soit C
ei (x̄−i )), alorssont semicontinues (c’est à dire fermeture de l’épigraphe) et si Ci (x̄−i ) ⊂ cl(CCi est l.s.c.
Par conséquent, si les fonctions contraintes g i sont continues alors Ci est bien semicontinue
inférieurement et supérieurement. Néanmoins, nous devons aussi garantir que Ci renvoie desensembles convexes, fermés et non-vides. Si les fonctions xi 7→ g i (xi , x−i ) sont quasiconvexes,
alors la convexité est garantie. En effet, la quasiconvexité d’une fonction f est équivalente à ceque tous les ensembles Uf (r) = {x ∈ X, f (x) ≥ r} soient convexes pour tout r, voir Diewert
et al. (1981). La continuité de g i va garantir la fermeture des ensembles. Mais, il est difficile
de trouver des conditions garantissant que les ensembles Ci (x̄−i ) soient non-vides, autres quede garantir l’existence d’un point (x̄i , x̄−i ) ∈ Xi × X−i , g i (x̄i , x̄−i ) > 0 pour tout x−i .
L’unicité de l’équilibre de Nash généralisé est un sujet nettement plus complexe que pourles équilibre de Nash standard. Pour appréhender ce problème, nous devons introduire lesconditions d’optimisation du premier ordre des I sous-problèmes.
En supposant que les fonctions objectives Oi et contraintes g i soient continûment différentiable, les conditions nécessaires de Karush-Kuhn-Tucker (KKT) pour le sous-problèmed’équation (9) sont données ci-dessous. Si x? résout le problème (9) pour tout i ∈ E et quepour chaque joueur, une qualification des contraintes est satisfaite, alors pour tout i ∈ E, il
existe un multiplicateur de Lagrange λi? ∈ Rmi tel queXi?
∇xi θi (x? ) +λi?
(∈ Rni ).
j ∇xi gj (x ) = 0
1≤j≤mi(10)i?
i ?
i ? T i?
mi0 ≤ λ , −g (x ) ≥ 0, g (x ) λ = 0(∈ R ).
Pour que les conditions KKT soient aussi suffisantes, il faut requérir des conditions supplémentaires. Celles-ci sont données dans le théorème 4.6 de Facchinei et Kanzow (2009).
Théorème. Soit un problème d’équilibre de Nash généralisé vérifiant l’équation (8) et telles
que les fonctions objective et contrainte soient continûment différentiable.
(i) Si x? est un équilibre de Nash généralisé et que tous les sous-problèmes (9) satisfassent
une qualification de contrainte, alors il existe λ? ∈ Rm tel que x? , λ? résolvent les I
systèmes (10).
(ii) Si x? , λ? résolvent les I systèmes (10), que les fonctions xi 7→ Oi (x) sont pseudoconvexes et que les ensembles Ci (x−i ) sont fermés et convexes, alors x? résout un équilibrede Nash généralisé.
Jusqu’ici nous n’avons pas explicité les contraintes de qualification, nous le faisons cidessous. Les contraintes de qualification ont pour but d’assurer que la version linéariséede l’ensemble contraint est une bonne approximation locale de l’ensemble original (nonlinéaire) contraint. L’ensemble des contraintes actives au point x se définit par Ai (x) = {j =1, . . . , mi , gji (x) = 0}. Deux contraintes de qualification sont très utilisées, nous les énonçonsci-dessous.
20La qualification de contrainte (CQ) d’indépendance linéaire (LICQ) est satisfaite lorsquel’ensemble des gradients des contraintes actives, {∇gji (x), i ∈ Ai (x)}, est linéairement indépendant. La qualification de contrainte de Slater (SCQ) est satisfaite lorsque toutes les contraintesi
actives sont strictement actives, c’est à dire λi?
j > 0 et gj (x) = 0 pour tout j ∈ Ai (x).
En pratique, des critères simples permettent de vérifier de telles conditions : (i) les contraintes
sont toutes linéaires, (ii) les contraintes sont convexes et (iii) les contraintes ont un gradientnon nul lorsqu’elles sont actives. Nous renvoyons le lecteur vers le chapitre 12 de Nocedal et
Wright (2006) et Arrow et Enthoven (1961).
Maintenant, nous avons les éléments pour présenter un résultat d’unicité pour une sousclasse d’équilibres de Nash. Rosen (1965) s’intéresse aux jeux conjointement convexes où lesfonctions contraintes g i sont communes à tous les joueurs, c’est à dire g 1 = · · · = g I = g 0 :R 7→ Rm0 . Ainsi, les ensembles de stratégies sont tels que pour tout i ∈ E,
Ci (x−i ) = {xi ∈ Xi , g 0 (x1 , . . . , xi , . . . , xI ) ≤ 0}.

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L’ensemble global des actions possibles se simplifie
K = {x ∈ X, ∀i ∈ E, xi ∈ Ci (x−i )} = {x ∈ X, g 0 (x) ≤ 0}.
De plus, Rosen (1965) suppose que la fonction g 0 est convexe pour garantir la convexitéde cet ensemble K. Les I systèmes (10) pour ce cas particulier se simplifient légèrement enremplaçant g i par g 0 et λi? ∈ Rm0 . Rosen (1965) définit un équilibre de Nash normalisé pourles jeux conjointement convexes lorsque x? vérifie les I systèmes (10) tels qu’il existe λ? ∈ Rm0
et ri > 0,λi? = λ0? /ri .
(11)
En d’autres termes, les multiplicateurs de Lagrange λi? de chaque joueur i sont reliés par unseul multiplicateur de Lagrange λ0? commun à tous les joueurs et le paramètre r ∈]0, +∞[I .
r s’interprète comme un paramètre d’échelle sur les fonctions objectives Oi .
Théorème (Rosen (1965)). Soit un jeu conjointement convexe à I joueurs, où la fonction
contrainte g 0 est convexe. Si les fonctions objectives Oi sont convexes alors pour tout r ∈]0, +∞[I , il existe un équilibre de Nash généralisé vérifiant (11).
Si de plus, pour r = r̄ > 0 donné, l’inéqualité suivante est vérifiée(x − y)T gO (y, r̄) + (y − x)T gO (x, r̄) > 0, ∀x, y ∈ Rn ,pour gO définie par
r1 ∇x1 O1 (x)..
gO (x, r) = ,.
rI ∇xI OI (x)alors l’équilibre de Nash généralisé vérifiant (11) est unique pour r = r̄.
Ce théorème de Rosen (1965) garantissant l’unicité d’équilibre de Nash généralisé est très
similaire au théorème équivalent pour les équilibres de Nash simples, mais à une différence
importante, l’équilibre de Nash généralisé dépend de la valeur du coefficient r. Cette classed’équilibre de Nash vérifiant l’équation vérifiant (11) est appelée l’ensemble des équilibres
normalisés. Depuis l’introduction des équilibres normalisés, un consensus sur le choix du paramètre r semble être formé (voir, par exemple, Harker (1991); Facchinei et Kanzow (2009);
21

Introduction
von Heusinger et al. (2010), pour lesquels les équilibres normalisés sont calculés pour r = 1).
Les équilibres normalisés ont une interprétation particulière via les inégalités variationnelles,
voir Facchinei et al. (2007).
Dans le chapitre 2, nous utiliserons des équilibres de Nash simples et généralisés. Nous
verrons à quel point les équilibres de Nash généralisés sont plus difficiles à manier du fait
qu’ils ne sont pas uniques.

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Jeux dynamiques
Dans cette sous-section, nous portons une brève attention aux jeux dynamiques, bien quedans le chapitre 2 nous utilisions un jeu statique. Dans la sous-section précédente, nous avonsprésenté des jeux statiques, mais dans beaucoup de cas, cela ne reflète pas la réalité. Les agents
prennent une suite d’action au cours du temps plutôt qu’une seule. Il existe quatre grandes
classes de jeux dynamiques : les jeux répétés, les jeux dynamiques à variable d’état (en temps
discret ou en temps continu) et les jeux de la théorie de l’évolution.
Nous nous concentrons sur les jeux répétés et les jeux à équation d’état en temps discret.
Les premiers trouvent leurs applications dans les jeux à réputation, par exemple, Alesina(1987) ou dans la définition de politique publique, par exemple Sleet (2001). Les seconds
ont été utilisés pour modéliser l’allocation de ressource en eau Ganji et al. (2007); Krawczyk
et Tidball (2005), d’émission carbone Haurie et Viguier (2002), ou de ressource en énergie
Bompard et al. (2008); Genc et Sen (2008).
Jeux répétés
Les jeux répétés s’intéressent aux interactions à long terme entre des joueurs au coursde la répétition d’un jeu ordinaire de période en période. Les conditions du jeu (nombre
de joueurs, ensemble de stratégies, fonctions objectives) sont constantes au cours du temps.
Notons I le nombre de joueurs, Xi , i ∈ E les ensembles de stratégies supposés finis et Oi les
fonctions objectives des joueurs, c’est à dire Oi (x1 , . . . , xI ) représente le gain du joueur i pourx ∈ X. Contrairement au jeu statique, les objectifs des joueurs dans les jeux répétés sontmajoritairement présentés en terme de gain plutôt qu’en terme de coût.
Définition. Un jeu répété basé sur le jeu ordinaire caractérisé par (I, (Xi )i , (Oi )i ) est une
forme extensive d’un jeu avec information parfaite et actions simultanées, telle que les actions
se définissent en profil de stratégies au cours du temps σi = (xi,1 , . . . , xi,t , . . . ) ∈ X ∞ et que lejoueur i peut comparer la suite de gains (Oi (x1,t , . . . , xI,t ))t pour deux profils différents σi , σ̃i .
Une stratégie pour le joueur i est donc une règle de décision permettant de choisir une
suite d’actions σi = (xi,1 , . . . , xi,t , . . . ) dépendant de l’histoire passée du jeu au temps t. Nous
pouvons en imaginer trois grands types : les stratégies à boucle ouverte (ou open-loop) danslesquelles la suite d’actions ne tient pas compte de l’histoire du jeu, les stratégies feedback ou
markoviennes où les actions en t ne dépendent que des actions passées en t − 1 et enfin les
stratégies à boucle fermée (ou closed-loop) dans lesquelles les joueurs utilisent toute l’histoire
passée à n’importe quelle période.
Pour comparer deux stratégies σi , σ̃i , nous utilisons la somme actualisée des gains
Gi (σ1 , . . . , σI ) =

T
X
t=0

22

δ t Oi (x1,t , . . . , xI,t ),où δ est facteur d’actualisation. Cette somme permet de caractériser différentes situations
suivant la valeur du facteur d’actualisation δ < 1 vs. δ = 1 et le nombre de période T < ∞ou T = ∞. Un équilibre de Nash pour le jeu répété est un ensemble de profils (σ1? , . . . , σI? ) tel
que pour tout joueur i
Gi (σ1? , . . . , σi? , . . . , σI? ) ≥ Gi (σ1? , . . . , σi , . . . , σI? ),pour tout profil σi . Les jeux répétés possèdent néanmoins des difficultés qui leur sont propres :les profils de stratégie σi appartiennent à un espace de dimension infinie, les équilibres de Nashdu jeu ordinaire (le constituant) ne sont pas forcément des équilibres pour les jeux répétés.
La litérature académique s’intéresse à caractériser l’ensemble des coûts totaux possibles Gi .
On définit pour ce faire l’ensemble des gains possibles par l’enveloppe convexe de l’ensemble
des gains possibles

tel-00703797, version 2 - 7 Jun 2012

co (O1 , . . . , OI ) ∈ RI , ∀i ∈ E, ∀xi ∈ Xi , Oi = Oi (x1 , . . . , xI ),et l’ensemble des gains individuellement rationnels
R=
gi ∈ R, gi ≥minmax Oi (xi , m−i ) ,m−i ∈M (Xi ) xi ∈Xioù M (Xi ) représente l’ensemble des stratégies mixtes sur l’ensemble fini Xi et mi une stratégie
mixte, c’est à dire un vecteur de probabilité. Maintenant, nous pouvons présenter les “folk”
théorèmes.
Théorème (Folk théorème). Pour un jeu répété infiniment et sans actualisation, c’est àdire T = ∞ et δ = 1, l’ensemble des gains d’équilibre est l’ensemble des gains possibles etindividuellement rationnels.
Des versions du “folk” théorème existent dans le cas d’un jeu actualisé représentant desjoueurs plus ou moins impatients et/ou d’un jeu répété un nombre fini de fois, voir Osborne
et Rubinstein (2006); Tomala et Gossner (2009).
Jeux à temps discretEnfin, nous présentons les jeux dynamiques en temps discret avec équation d’état en se
basant sur le chapitre 5 de Basar et Olsder (1999). Pour définir de tels jeux, nous introduisonsles notations suivantes : un nombre de joueurs I, un nombre d’étapes T , un espace d’étatX ⊂ Rd , des espaces d’actions Uti ⊂ Rmi . Dans cette sous-section, les actions des joueurs nesont plus notées xi mais uti ∈ Uit pour la période t.
Définition. Un jeu dynamique en temps discret est caractérisé par une équation d’état initialisée par x1 ∈ X
xt+1 = ft (xt , u1t , . . . , uIt ),pour une fonction ft : X × Ut1 × · · · × UtI 7→ X, des fonctions coûts Li : S1 × · · · × ST 7→ R,
où St = X × Ut1 × . . . UtI , une structure d’information ηit ⊂ {x11 , . . . , xIt , u11 , . . . , uIt−1 }, et un
ensemble de fonctions γti : X × S1 × · · · × St−1 7→ Uti .
23

Introduction
Des exemples de structure d’information sont similaires à ceux définis pour les jeux répétés :boucle ouverte (ou open-loop) ηti = {x1 }, feedback ou markovien ηti = {x1 , xt } et boucle fermée(ou closed-loop) ηti = {x1 , . . . , xt }. Une stratégie pour le joueur i est donc un ensemble defonctions (γti )t spécifiant l’action à jouer γti (ηit ) en t pour une information ηit .
Pour simplifier, la fonction de coût Li a généralement une forme additive
Li ((u1t , . . . , uNt )t ) =TX

gti (xt+1 , u1t , . . . , uNt , xt ).

t=1

Un équilibre de Nash dans un tel jeu est un ensemble de fonctions γ ? tel que pour toutηit ∈ X It × U11 × · · · × UtI , et pour toute fonction γti : X × S1 × · · · × St−1 7→ Uti ,tel-00703797, version 2 - 7 Jun 2012

Li



 
t
γt1? η1t , . . . , γti? ηit , . . . , γtN ? ηN
≤ Li
t



 
t
.
γt1? η1t , . . . , γti ηit , . . . , γtN ? ηNt

Nous parlons d’équilibre de Nash open-loop, feedback ou closed-loop suivant la structure d’information choisie. Dans le cas d’équilibre de Nash open-loop, le jeu se réduit à un jeu statique
puisque la variable d’état xt n’a pas d’incidences sur les actions choisies. La stratégie optimale
est obtenue à l’aide de la théorie du contrôle optimal et de la programmation dynamique, voir
théorème 6.1 de Basar et Olsder (1999). Dans le cas des stratégies feedback et closed-loop, deséquations rétrogrades du même type donnent des conditions d’optimalités, voir théorèmes 6.5
et 6.6 de Basar et Olsder (1999).

Modèle de compétition en assurance non-vie
Nous présentons dans cette sous-section brièvement le jeu répété du chapitre 2. Considérons un marché d’assurance non-vie composé de I assureurs. Chaque assureur propose descouvertures d’assurance à une population de n  I clients. Connaissant la sinistralité passée,au temps t, le jeu consiste à fixer un prix de police. Notons xj,t le prix proposé par l’assureur
j au temps t et nj,t le nombre de clients en portefeuille pour la période t. La séquence de jeu
pour la période t est la suivante
1. Les assureurs maximisent leur fonction objective
sup Oj,t (xj,t , x−j,t ) tel que gj,t (xj,t ) ≥ 0,xj,toù gj,t (xj,t ) ≥ 0 représente la contrainte de solvabilité, fonction du capital Kj,t−1 ..
2. Une fois la prime d’équilibre calculée x?t , les assurés choisissent de résilier ou de renouveler
leur contrat selon une loi multinomiale logit de vecteur de probabilité pl→j (x?t ). Uneréalisation nj,t de la taille de portefeuille est obtenue.
3. Ensuite, les sinistres pour chaque assuré sont tirés aléatoirement selon un modèle fréquence – sévérité.
4. Enfin, on détermine le résultat de souscription et en déduit le nouveau capital disponible
Kj,t .
Le chapitre 2 analyse les propriétés statiques et dynamiques de ce jeu répété. Nous renvoyons
au prochain chapitre de l’introduction pour plus de détails sur les résultats obtenus.
24

tel-00703797, version 2 - 7 Jun 2012

Théorie de la ruine
La théorie du risque s’intéresse à tous les aspects d’un portefeuille d’assurance non-vie,
tarification, provisionnement, gestion du risque, etc. . . , voir, par exemple,Bowers et al. (1997),
Marceau (2012). La théorie de la ruine se concentre sur la solvabilité à moyen et long termed’un assureur. Nous présentons ci-dessous les grands résultats de la théorie de ruine sanspreuve et renvoyons le lecteur vers les ouvrages de référence : Grandell (1991),Rolski et al.
(1999),Asmussen (2000),Asmussen et Albrecher (2010). Nous suivrons plus particulièrement
la présentation d’Asmussen et Albrecher (2010).
L’étude du niveau de richesse d’une compagnie d’assurance, introduite par Lundberg(1903), est une problématique centrale de la théorie de la ruine. Au début du XXème siècle,l’école Suédoise pose les fondamentaux de cette théorie, sous l’impulsion de Filip Lunderg puisd’Harald Cramér. Cramér (1930) propose le modèle collectif (plus tard appelé le modèle deCramér-Lundberg) dans lequel la richesse de l’assureur (Ut )t au temps t est modélisée par leprocessus stochastique suivant
Nt
X
Ut = u + ct −
Xi ,
(12)
i=1

où u > 0 est le capital initial, c > 0 le taux de prime par unité de temps, (Nt )t≥0 représentant
P t
le nombre de sinistres au temps t et Xi le montant du ième sinistre. Notons St = N
i=1 Xi laperte agrégée au temps t.
Dans le modèle de Cramér-Lundberg, les hypothèses suivantes sont faites : (i) les montants des sinistres (Xi )i sont indépendants et identiquement distribués, (ii) les montants sontindépendants de Nt , (iii) (Nt )t≥0 un processus de Poisson (d’intensité λ). Notons (Ti )i les
temps d’attente entre deux sinistres. Pour un processus de Poisson, les temps (Ti )i sont delois exponentielles E(λ).
Définition. La probabilité de ruine (en temps infini) est définie comme étant le premier
instant où le processus de richesse (Ut )t est strictement négatifψ(u) = P (∃t ≥ 0, Ut < 0).

(13)

De manière similaire, on définit la probabilité de ruine en temps fini parψ(u, T ) = P (∃t ∈ [0, T ], Ut < 0),(14)

où T > 0 est l’horizon de gestion.
Un exemple de trajectoire du processus (Ut )t est donné en figure 2, où les temps d’interoccurrence sont de loi exponentielle E(3), les sinistres de loi exponentielle E(2) et le taux deprime c = 2. Le capital initial u est le point de départ du processus, la pente est donnée parle taux de prime c, représentant l’acquisition des primes au cours du temps. Ensuite chaquesinistre Xi produit un saut vers le bas. Sur cet exemple, la ruine intervient au bout du 6èmesinistre.
Le modèle de Cramér-Lundberg a rapidement été généralisé en considérant des processus
de renouvellement pour (Nt )t≥0 par Andersen (1957), plus tard appelé modèle de Sparre
Andersen. Ainsi, les temps d’inter-occurrence ne sont plus nécessairement de loi exponentiellemais simplement indépendants et identiquement distribués. Le lien avec la théorie des filesd’attente est encore plus clair que dans le modèle de Cramér-Lundberg.
25Introduction
Surplus Ut

X4

X2

X5

X3
X1

T1

T2

T3

T5

T4

T6

X6

Time ttel-00703797, version 2 - 7 Jun 2012

Figure 2 – Une trajectoire du processus (Ut )t

Processus de risque à accroissements indépendants et stationnaires
La théorie de la ruine cherche à déterminer le taux de prime c et le niveau de capital u
répondant à un critère de ruine. Pour éviter la ruine certaine, le taux de prime c doit déjà
vérifier la contrainte dite de profit net. Celle-ci est déduite de la proposition suivante, voir laproposition IV.1.2 d’Asmussen et Albrecher (2010).
Proposition (Dérive et oscillation). Dans le modèle de Cramér-Lundberg avec des sinistresindépendants et identiquement distribués d’espérance E(X), notons ρ = c − λE(X) le gain
espéré par unité de temps. On a presque sûrement
lim

t→+∞

Ut − u= ρ.
tSi ρ > 0, alors presque sûrement
lim Ut = +∞,t→+∞c’est à dire la ruine n’est pas certaine, ψ(u) < 1. Dans le cas contraire, si ρ < 0, alors presquesûrementlim Ut = −∞,t→+∞c’est à dire la ruine est certaine ψ(u) = 1. Dans le cas particulier, où ρ = 0, la ruine est aussi
certaine, puisque le processus (Ut )t vérifie
lim sup Ut = +∞, et lim inf Ut = −∞.
t→+∞

t→+∞

La condition de profit net ρ > 0 peut se réécrire c = (1 + η)λE(X) avec un chargement
η > 0. Nous pouvons maintenant énoncer la première formule fermée pour la probabilité deruine dans le modèle de Cramér-Lundberg, voir le corollaire IV.3.2 d’Asmussen et Albrecher(2010).
26

Théorème. Dans le modèle de Cramér-Lundberg avec des sinistres de loi exponentielle E(1/µ)(de moyenne µ),λµ −u(1/µ−λ/c)ψ(u) =
e,csi la condition de profit net est vérifiée ρ = c − λµ > 0.
Cette formule de probabilité de ruine a la caractéristique de décroitre exponentiellement
en fonction du capital initial u. Cette propriété est vérifiée pour une large classe de modèlesde sinistres. Plus précisément, la décroissance exponentielle est encore valide si la loi dessinistres (Xi )i possède une fonction génératice des moments MX (t) = E(etX ) pour t > 0.
Nous regroupons ici les théorèmes IV.5.2 et IV.5.3 d’Asmussen et Albrecher (2010).
Théorème (Borne et approximation de Cramér-Lundberg). Soit γ la solution strictement
positive de l’équation (en r) de Lundberg

tel-00703797, version 2 - 7 Jun 2012

MX (r)

λ= 1.
λ + rcOn a alors pour tout u ≥ 0
ψ(u) ≤ e−γu et ψ(u)

∼

u→+∞

Ce−γu ,

0 (γ) − c), e−γu est appelée borne deoù la constante C est donnée par C = (c − λµ)/(λMX−γuLundberg, Ceapproximation de Lundberg et γ coefficient d’ajustement.

La décroissance exponentielle est aussi constatée dans le modèle de Sparre Andersen oùle processus du nombre de sinistres (Nt )t≥0 (de l’équation (12)) est un processus de renouvellement. Les temps d’inter-occurrence des sinistres (Ti )i sont indépendants et identiquement
distribués selon une variable générique T . Nous rapportons ci-dessous une extension du théorème de Cramér-Lundberg pour le modèle de Sparre Andersen, voir, par exemple, théorème
6.5.4 de Rolski et al. (1999).
Théorème. Dans le modèle de Sparre Andersen, notons Y les incréments de la perte agrégée
Y = X − cT . x0 est défini comme le supremum de l’ensemble {x, FY (x) < 1}. Pour u ≥ 0,nous disposons de l’encadrement suivant
b− e−γu ≤ ψ(u) ≤ b+ e−γu ,où γ est solution de l’équation MX (r)MT (−rc) = 1, les constantes b− , b+ ont pour expressionb− =eγx F̄Y (x)eγx F̄Y (x)inf R +∞
et b+ = sup R +∞.
x∈[0,x0 [eγy dF̄Y (y)eγy dF̄Y (y)x∈[0,x0 [ xx

En plus de ces asymptotiques, d’autres formules explicites de probabilité de ruine sontdisponibles pour d’autre lois de sinistres, notamment les mélanges de lois exponentielles, leslois Erlang (c’est à dire loi gamma avec un paramètre de forme entier), les mélanges de lois
Erlang. Ces lois font partie de la grande classe des lois phase-type introduite par Neuts (1975),
et popularisée dans la théorie des files d’attente par notamment Neuts (1981).
Soit m ∈ N? un entier positif. Considérons un processus de Markov (Mt )t en temps continu
et à valeurs dans l’ensemble fini {0, 1, . . . , m}, où 0 est un état absorbant. Une loi phase-type27

Introduction

tel-00703797, version 2 - 7 Jun 2012

est la loi du temps d’absorption du processus (Mt )t dans l’état 0 partant d’un état initial del’ensemble {1, . . . , m}.
Ces lois sont paramétrées par une matrice de sous-intensité J, une dimension m et un
vecteur de probabilité initial π ∈ [0, 1]m . La matrice d’intensité Λ du processus sous-jacent(Mt )t est donnée par la matrice par bloc0 0
Λ = (λij )ij =,j0 J
où j0 est le vecteur des intensités de sortie j0 = −J1m et 1m le vecteur rempli de 1 de Rm .
Cela signifie que les probabilités de changement d’état du processus (Mt )t sont donnés par
P (Mt+h = j/Mt = i) = λij h + o(h) si i 6= j et 1 + λii h + o(h) si i = j avec des probabilités
initiales P (M0 = i) = πi .
Pour de telles lois phase-type P H(π, J, m), les fonctions de répartition et de densité sontdonnées par
F (x) = 1 − πeJx 1m , et f (x) = πeJx j0 ,P+∞ T n xn
où eJx correspond à l’exponentielle de matrice définie la série
n=0 n! , voir Moler et
Van Loan (2003) pour une revue récente de son calcul. La loi exponentielle E(λ) est obtenue par la paramétrisation P H(1, λ, 1), le mélange de n lois exponentielles est obtenue
par P H(m, π, J) où m = n, π = (p1 , . . . , pn ), et une matrice de sous-intensité diagonale
J = −diag[(λ1 , . . . , λn )].
Les lois phase-type P H(π, J, m) font partie des lois à queue de distribution légère, aumême titre que la loi exponentielle, la loi gamma, au vue de la décroissance exponentielle desa queue de distribution. Ainsi, la fonction génératrice des moments et le moment d’ordre npossèdent des formules explicites. Dans ce contexte, Asmussen et Rolski (1991) proposent desformules explicite de probabilité de ruine lorsque les montants des sinistres (Xi )i et les temps
d’inter-occurrence (Ti )i sont de lois phase type.
Théorème (Asmussen et Rolski (1991)). Dans le modèle de Cramér-Lundberg, lorsque lesmontants des sinistres sont phase-type P H(π, J, m), la probabilité de ruine est donnée parψ(u) = π+ eQu 1m ,où la matrice s’écrit Q = J +j0 π+ , le vecteur π+ = −λ/cπJ −1 et j0 = −J1m est le vecteur destaux de sortie. En d’autre termes, la probabilité de ruine admet une représentation phase-type
P H(π+ , Q, m).
Dans le modèle de Sparre Andersen, lorsque les temps d’inter-occurrence ont une fonction de répartition FT , la probabilité de ruine admet toujours une représentation phase-type
P H(π+ , Q, m) mais π+ est la solution de l’équation de point fixe
π+ = πMT (J + j0 π+ ),où MT correspond la fonction génératrice des moments (avec un argument matriciel).
Une loi de probabilité admet une représentation phase-type s’il existe une fonction génératrice des moments rationnelle, voir, par exemple, Hipp (2005). Nécessairement, une loi
phase-type a une queue de distribution légère. Ainsi, le théorème montre pour la grande classe
des lois phase-type ∗ de montant de sinistre à queue distribution légère, que la probabilité de∗. Comme l’ensemble des lois phase-type est dense dans l’ensemble des lois de probabilité à support positif,
il est possible en théorie approcher à n’importe quel lois à support positif pour un degré de précision donné.

28

tel-00703797, version 2 - 7 Jun 2012

ruine décroît exponentiellement vite. Une question légitime est donc de savoir si ce principe
est- toujours respecté pour des lois à queue de distribution plus épaisse.
Jusqu’ici
les lois de sinistre X étaient telles que la fonction génératrice des moments
R +∞MX (t) = 0 etx dF̄X (x) existait pour certains t > 0. Cette classe est appelée la classe
des lois à queue de distribution légère. De nombreuses lois n’appartiennent pas à cette classe,
c’est à dire MX (t) est infini pour t > 0, par exemple, la loi lognormale, la loi de Weibull ou
encore la loi de Pareto.
Comme la classe des lois pour lesquelles il n’existe pas de lois de fonction de génératrice demoments est vaste et peu explicite, la classe de lois sous-exponentielles a été introduite. Une
fonction de répartition FX appartient à la famille sous-exponentielle si pour deux variables
aléatoires X1 , X2 indépendantes et identiquement distribuées de fonction de répartition FX ,elles vérifient
P (X1 + X2 > x)
−→ 2.
x→+∞
P (X1 > x)
Pour mieux comprendre cette définition, il est de bon rappeler que pour toute variable aléatoire
positive X, on a P (max(X1 , X2 ) > x) ∼ 2F̄X (x) lorsque x → +∞. Par conséquent, la classe
des lois sous-exponentielle est telle que P (X1 + X2 > x) ∼ P (max(X1 , X2 ) > x) pour desgrandes valeurs de x. La propriété se généralise pour une somme de n variables indépendantes.
En théorie de la ruine, l’application des lois sous-exponentielle a été faite par Teugels et
Veraverbeke (1973) ou encore Embrechts et Veraverbeke (1982). Pour cette classe de montant
de sinistre, la probabilité de ruine décroît comme l’inverse d’un polynome. Nous rapportonsci-dessous la version proposée dans Asmussen et Albrecher (2010).
Théorème (Embrechts et Veraverbeke (1982)). Dans le modèle de Sparre Andersen, où lesespérances des montants (Xi )i et desR x temps d’attente des sinistres (Ti )i sont finis et tels que
E(X) < cE(T ). Notons FX,0 (x) = 0 F X (y)dy/E(X). Si FX et FX,0 appartiennent à la classe
sous-exponentielle, on a alorsZ +∞1ψ(u) ∼F̄X (y)dy.
u→+∞ cE(T ) − E(X) u
Ce théorème donne lieu aux cas particuliers suivant. Considérons des montants de sinistre
Pareto Pa(k, α), c’est à dire P (X > x) = (k/x)α avec α > 1. On a alors α−1kk
ψ(u) ∼.
u→+∞ cE(T )(α − 1) − αkuDe manière similaire, pour les lois à variations régulières dont les queues de distribution
vérifient P (X > x) ∼ L(x)/xα pour des grandes valeurs de x et L une fonction à variation
lente, telle que L(xt)/L(x) → 1 pour t > 0 et x → +∞, nous obtenons
ψ(u)

∼

1

u→+∞ cE(T ) − E(X)

×

L(u)
.
(α − 1)uα−1

Lorsque X suit une loi de Weibull avec P (X > x) = exp(−xβ ) (resp. une loi lognormale P (X >2βx) = 1 − Φ((log x − µ)/σ)), alors on a ψ(u) ∼ u1−β e−u (resp. ψ(u) ∼ ue− log (u) / log2 (u)).
Toutes ces formules, sauf celle pour le cas Weibull, présentent une décroissance en puissancede u du type C/uα , qui contraste nettement avec une décroissance exponentielle pour les lois
à queue légère Ce−γu .
29Introduction

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Jusqu’à maintenant, le seul indicateur de risque introduit est la probabilité de ruine. De
nombreuses autres quantités sont intéressantes à étudier. Notons τ = inf(t > 0, u+Ct−St < 0)le (premier) temps de ruine pour le processus de richesse (Ut )t de valeur initiale u. La valeur duniveau de richesse juste avant la ruine et le déficit au moment de la ruine, respectivement Uτ −et |Uτ |, sont des exemples de mesures de ruine, voir Dickson (1992), Lin et Willmot (2000).
Gerber et Shiu (1998) proposent un cadre unique d’étude pour ces mesures de ruine parla fonction de pénalité suivante (plus tard appelée fonction de Gerber-Shiu)himδ (u) = E e−δτ w(Uτ − , |Uτ |)11(τ <+∞) |U0 = u ,où δ est le taux d’intérêt et w une fonction de pénalité positive. Pour w(x, y) = 1 et δ = 0,
la fonction de Gerber-Shiu mδ (u) se réduit à la probabilité de ruine en temps infini. Notons
que δ peut jouer le role d’une variable pour la transformée de Laplace du terme w(Uτ − , |Uτ |)conditionnelement à un temps de ruine fini.
Dans le modèle de Cramér-Lundberg, la fonction de Gerber-Shiu vérifie l’équation intégralesuivantemδ = mδ ∗ g + h,avec ∗ le produit de convolution et g, h données parZZZ
λ +∞ −γz
λ +∞ −γ(z−x) +∞g(x) =
e fX (x + z)dz et h(x) =ew(z, x − z)fX (x)dxdz,
c 0
c x
z
c
où γ est la solution positive de l’équation de Lundberg δ + λ − cξ = λfc
X (ξ) et fX est latransformée de Laplace de la densité fX . En prenant la transformée de Laplace de l’équation
précédente, l’équation se résout facilement par mb δ (ξ) = bh(ξ)/(1−bg (ξ)). Avec cette formulation,il est possible d’obtenir des asymptotiques de la fonction de pénalité pour des grandes valeursde u. Gerber et Shiu (1998) ont introduit cette fonction de pénalité dans le modèle de CramérLundberg, mais ensuite l’approache a été généralisée dans d’autres modèles, voir, par exemple,
Gerber et Shiu (2005) et Song et al. (2010).
Avant de présenter des modèles de risque avec dépendance, nous listons d’autres extensions
possibles du modèle de Sparre Andersen : intégration de la réassurance, voir, par exemple,
Centeno (2002a),Centeno (2002b),Barges et al. (2012), modélisation de plusieurs de lignes
d’affaire, par exemple, Collamore (1996),Cai et Li (2005),Biard et al. (2010), ou encore lamodélisation des crises de corrélation, Biard et al. (2008).

Processus de risque avec dépendance
Jusqu’à présent, nous avons travaillé avec trois hypothèses d’indépendance : (i) une indépendance entre les montants des sinistres (X1 , X2 , . . . ), (ii) une indépendance entre les temps
d’attente (T1 , T2 , . . . ) et (iii) une indépendance entre le montant Xi et le temps d’attente Ti
du ième sinistre. Cela résulte du fait que le processus d’arrivée des sinistres (Nt )t≥0 est supposéà acroissements indépendants et stationnaires, puisque c’est un processus de renouvellement.
Une première extension consiste à considérer un processus de Poisson non homogène(Nt )t≥0 de paramètre (λt )t pour le processus de risque de l’équation (12). Lu et Garrido (2005)
ont montré l’intérêt de tels processus dans la modélisation des ouragans aux Etats-Unis, oùdeux fonctions déterministes pour λt sont testées : une fonction simplement périodique et une
doublement périodique.
30

Albrecher et Asmussen (2006) utilisent les processus de Poisson non-homogènes où le paramètre d’intensité (λt )t est aussi un processus stochastique. Plus précisement, il suppose que leprocessus d’arrivée des sinistres (Nt )t≥0 a pour paramètre d’intensité le processus stochastiquesuivantX
λt = λ +h(t − Un , Yn ) + νt ,
(15)
n∈N

tel-00703797, version 2 - 7 Jun 2012où λ > 0 une constante, (Un )n sont les temps d’occurrence d’un processus de Poisson deparamètre ρ, (Yn )n une suite de variables aléatoires positives indépendantes et identiquement
distribuées, h(., .) une fonction positive et (νt )t est un processus stochastique représentant lespertubations du passé.
Albrecher et Asmussen (2006) obtiennent différents résultats pour la probabilité de ruineen temps fini et infini avec des sinistres à queue de distribution lourde et légère. Nous donnons
ici que deux de leurs résultats (théorèmes 4.2 et 5.2) et renvoyons le lecteur vers leur article
pour plus de détails.
Théorème (Albrecher et Asmussen (2006)). Considérons le processus de risque de l’équation(12) où le processus d’arrivée (Nt )t≥0 a pour paramètre d’intensité le processus de l’équation(15). Nous supposons que la condition
de profit net est vérifiée par c > E(X)µ, où µ =Rt
λ + ρE(H(∞, Y )) et H(t, y) = 0 h(s, y)ds.
Supposons que les montants des sinistres (Xi )i possèdent une fonction génératrice desmoments MX (α) pour α > 0 proche de 0, que le processus (νt )t vérifie
Rt
log E exp((MX (α) − 1) 0 νs ds)→ 0,t
lorsque t tend vers +∞, et que E(exp(αH(∞, Y ))) existe pour α > 0. On a alors
ψ(u)

∼

u→+∞

e−γu ,où γ est la solution positive d’une certaine équation κ(α) = 0 avec κ une fonction de MX (α).
Si les montants des sinistres (Xi )i appartiennent
à une classe sous-exponentielle et que
Rtpour α > 0, E(exp(αH(∞, Y ))) et E(exp(α 0 νs ds)) existent, alors on aψ(u)µu→+∞ (c − µ)E(X)

Z +∞

∼

F̄X (x)dx.
uMalgré l’augmentation de la variabilité sur le processus de risque et bien que la probabilitéde ruine ait augmentée, la forme des asymptotiques demeurent inchangée par rapport aumodèle de Sparre Andersen.
Dans le même esprit, Asmussen (1989), Asmussen et Rolski (1991), Asmussen et al. (1994)
considèrent un processus d’arrivée des sinistres controlé par un processus Markovien (Jt )t à
valeurs finies. Conditionnellement à Jt = i, le taux de prime est ci , la loi des sinistres est Fi,X ,et le taux d’arrivées de sinistre λi . Cela correspond aussi un processus de Poisson doublement
stochastique de processus d’intensité (λJt )t .
Notons ψi (u) la probabilité de ruine sachant J0 = i. Asmussen (1989),Asmussen et Rolski(1991) fournissent des formules exactes (à l’aide des lois phase-type) et des asymptotiques dansle cas de lois de sinistres à queue de distribution légère, tandis que Asmussen et al. (1994)31

Introduction
traitent le cas des lois à queue lourde. Les asymptotiques dans le cas des lois à queue légère
sont du typeψi (u) ∼ Ci e−γu ,u→+∞tandis que pour les lois de la classe sous-exponentielle avec Fi,X = FXZ +∞ψi (u)∼u→+∞F̄X (x)dx.

aiu

Nous renvoyons le lecteur vers les articles pour les expressions des constantes ai et Ci .
Les extensions précédentes au modèle de Sparre-Andersen se sont portées sur la modification du processus de sinistres (Nt )t≥0 . Nous présentons maintenant les extensions où on
suppose explicitement une dépendance entre le montant Xi et le temps d’attente Ti du ième
sinistre. Albrecher et Boxma (2004) est un premier exemple où la densité du temps d’attente
du i + 1ème sinistre dépend du montant du ième sinistre. Ils supposent que

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fTi+1 (x) = P (Xi > τi )λ1 e−λ1 x + P (Xi ≤ τi )λ2 e−λ2 x ,où τi est une variable aléatoire représentant un seuil de gravité modifiant le temps d’attente
du sinistre. En d’autres termes, Ti est un mélange de loi exponentielle E(λ1 ), E(λ2 ) dont laprobabilité de mélange est P (Xi > τi ). Les variables de seuil (τi ) forment une suite de variables
aléatoires indépendantes et identiquement distribuées. Le processus de risque considéré est
maintenant un processus Markovien puisque le temps d’attente du i + 1ème sinistre dépend
(uniquement) du montant du ième sinistre, c’est à dire les incréments cTi − Xi ne sont plus
stationnaires ou indépendents.
La condition de profit net est E(X) < c(P (X > τ )/λ1 + P (X ≤ τ )/λ2 ), où X, τ sont lesvariables génériques. Dans le cas d’une loi de sinistre à queue de distribution légère et si lepremier temps d’attente est de loi E(λj ), nous pouvons obtenir une expression explicite de latransformée de Laplace de la probabilité de survie conditionnelle 1 − ψj (u) sous la forme d’unratio de fonctions. Une inversion numérique de la transformée de Laplace est possible si latransformée de Laplace est rationnelle. Le fait que la transformée de Laplace ait une uniquesolution à partie réelle positive garantit une décroissance exponentielle de la probabilité deruine du type e−σu .
Dans la même idée, Boudreault et al. (2006) considèrent une structure de dépendance danslaquelle les incréments cTi − Xi sont toujours indépendants et stationnaires. Mais les variables(Xi , Ti ) ne sont plus indépendantes : ils supposent que
TifX(x) = e−βTi f1 (x) + (1 − e−βTi )f2 (x),ioù f1 , f2 sont deux densités.
Toujours en travaillant avec des lois de sinistre à queue légère, Boudreault et al. (2006)
expriment explicitement la transformée de Laplace en termes de ratios. Ils obtiennent une
expression explicite pour la fonction de Gerber-Shiu sous forme de combinaisons exponentielles
eRi u où Ri correspondent aux racines du dénominateurs de la transformée de Laplace.
Dépendance à l’aide des copules
Albrecher et Teugels (2006) s’intéressent aussi à une modélisation directe du couple (Xi , Ti )et à l’impact sur la probabilité de ruine en temps fini et infini. La dépendance est modélisée à32l’aide d’une copule bivariée (u1 , u2 ) 7→ C(u1 , u2 ). Pour une dimension d fixée, une copule est
une fonction multivariée C : [0, 1]d 7→ [0, 1] vérifiant certaines propriétés de manière à ce que C
puisse être interprétée comme une fonction de répartition d’un vecteur aléatoire (U1 , . . . , Ud )
à marginale uniforme.
Définition. Soit C une fonction multivariée de [0, 1]d 7→ [0, 1], où d ≥ 2 est une dimension
fixée. C est une copule si la fonction vérifie les propriétés suivantes :1. ∀i ∈ {1, . . . , d}, ∀u ∈ [0, 1]d , C(u1 , . . . , ui−1 , 0, ui+1 , . . . , ud ) = 0,
2. ∀i ∈ {1, . . . , d}, ∀u ∈ [0, 1]d , C(1, . . . , 1, ui , 1, . . . , 1) = ui ,3. ∀i ∈ {1, . . . , d}, ∀u ∈ [0, 1]d , ∀(ai ≤ bi )i , ∆1a1 ,b1 . . . ∆dad ,bd C(u) ≥ 0, où ∆iai ,bi est la
différence d’ordre i, c’est à dire∆iai ,bi C(u) = C(u1 , . . . , ui−1 , bi , ui+1 , . . . , ud ) − C(u1 , . . . , ui−1 , ai , ui+1 , . . . , ud ).
La propriété 3 est appelée croissance d’ordre d.

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L’interprétation probabiliste d’une copule est la suivanteC(u1 , . . . , ud ) = P (U1 ≤ u1 , . . . , Ud ≤ ud ),où (Ui )i sont des variables aléatoires uniformes U(0, 1). Pour toute copule C, on a l’inégalitésuivante
W (u) = max(u1 + · · · + ud − (d − 1), 0) ≤ C(u1 , . . . , ud ) ≤ min(u1 , . . . , ud ) = M (u),(16)

où les bornes sont appelées bornes de Fréchet W, M . Les inégalités (16) donnent des bornes
inférieure et supérieure quelque soit la structure de dépendance considérée. Notons que M
est toujours une copule quelque soit la dimension d, tandis que W ne l’est qu’en dimensiond = 2. Une dernière copule particulière, qui a toute son importance, est la copule d’indépendance Π(u) = u1 × · · · × ud . Cette copule sera une copule limite de la plupart des copules
paramétriques.
Un théorème fondamental liant un vecteur aléatoire avec des fonctions de répartition marginales données à une copule est le théorème de Sklar (1959).
Théorème (Sklar (1959)). Soit F : Rd 7→ [0, 1] une fonction de répartition multivariée avec
des marginales F1 , . . . , Fd , alors il existe une copule C telle que pour tout (x1 , . . . , xd ) ∈ Rd ,F (x1 , . . . , xd ) = C(F1 (x1 ), . . . , Fd (xd )). C est unique sur l’ensemble S1 × · · · × Sd où Si est lesupport de la ième marginale.
Notons que si les variables aléatoires marginales sont des variables continues, alors lacopule est unique sur Rd . Sinon elle n’est unique qu’en certains points. En supposant que C
est différentiable sur [0, 1]d , la densité de la loi jointe d’un vecteur aléatoire avec des marginales
de densité fi est donnée parc(x1 , . . . , xd ) =∂ d C(F1 (x1 ), . . . , Fd (xd ))f1 (x1 ) . . . fd (xd ).
∂u1 . . . ∂udNous nous arrêtons ici pour la présentation des copules et renvoyons le lecteur vers les ouvragesde référence : Marshall (1996),Joe (1997),Nelsen (2006).
Retournons à notre problème de ruine. Albrecher et Teugels (2006) montrent le théorèmesuivant à l’aide de la marche aléatoire sous-jacente au processus de risques.
33

Introduction
Théorème (Albrecher et Teugels (2006)). Pour le processus de risque donné en équation(12), avec des montants Xi et des temps d’attente de sinistre Ti identiquement distribués etdes queues de distribution légères. Notons γ le coefficient d’ajustement solution positive del’équation E(exp r(X − cT )) = 1, existant si la condition de profit net E(X) < cE(T ) est
vérifiée. On a alorsψ(u) ∼ Ce−γu ,u→+∞où C = e−B /γE(SN exp(RSN )), N = inf(n > 0, Sn > 0), Sn la somme cumulée des incréments et B une constante positive.

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Albrecher et Teugels (2006) étudient ensuite l’équation de Lundberg pour différentes copules : copules de Spearman, copule EFGM ou les copules Archimédiennes. Nous détaillons
ces dernières, qui vont être utilisées dans cette introduction et au chapitre 4.
Les copules archimédiennes sont caractérisées par un générateur φ : R+ 7→ [0, 1], qui
est une fonction infinement différentiable et complètement monotone, c’est à dire, pour toutk ∈ N, (−1)k φ(k) (t) ≥ 0. Une copule archimédienne est caractérisée de la manière suivanteC(u1 , . . . , ud ) = φ−1 (φ(u1 ) + · · · + φ(ud )) .
Les exemples les plus classiques sont la copule de Clayton φ(t) = (t−α − 1)/α, la copule deGumbel φ(t) = (− log t)α ou encore la copule de Frank φ(t) = log(e−α − 1) − log(e−αt − 1),voir le chapitre 4 de Nelsen (2006).
Dépendance par mélange
Enfin, nous présentons un dernier type de dépendance introduit par Albrecher et al. (2011).
Cela consiste à introduire une dépendance dans la suite des sinistres, soit sur les montants(X1 , X2 , . . . ) ou les temps d’attente de sinistres (T1 , T2 , . . . ) à l’aide d’une variable latente.
Soit Θ une variable aléatoire positive représentant une certaine hétérogénéité sur le portefeuilled’assurance. Tout d’abord, nous supposons que les montants de sinistre Xi sont indépendants
et identiquement distribués conditionnellement à Θ = θ de loi exponentielle E(θ). De plus, lestemps d’attente sont eux-aussi de loi exponentielle λ et indépendents des montants de sinistre.
Conditionnellement à Θ = θ, c’est le modèle de Cramér-Lundberg. Ainsi, on a
λ −u(θ− λ )cψ(u, θ) = min
e
,1 ,
θc
où le minimum utilisé ci-dessus est équivalent à la condition de profit net θ > λ/c. En intégrantpar rapport à la variable θ, on obtientψ(u) = FΘ (θ0 ) + I(u, θ0 ),où θ0 = λ/c etZ +∞I(u, θ0 ) =θ0θ0 −u(θ−θ0 )edFΘ (θ).
θ

Notons dès à présent que la probabilité de ruine est strictement positive quelque soit le niveau
de capital initial u.
34

Dans un tel modèle, une copule archimédienne se cache entre les montants des sinistres.
En effet, la probabilité de survie est donnée par
P (X1 > x1 , . . . , Xn > xn |Θ = θ) =nYe−θxi .

i=1

Par conséquent,
P (X1 > F̄X−1 (u1 ), . . . , Xn > F̄X−1 (un )) =

Z +∞
e

−θ

−1
i=1 F̄X (ui )PndFΘ (θ) = LΘ

0

n
X

!
F̄X−1 (ui ),i=1

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où LΘ est la transformée de Laplace de la variable aléatoire Θ et F̄X la fonction de survie
de X. Il est facile d’identifier une structure archimédienne avec un générateur φ(t) = L−1
Θ (t).
Néanmoins, la dépendance a lieu sur la copule de survie, c’est à dire sur la fonction C(u)définie par
P (U1 > u1 , . . . , Un > un ) = C(u1 , . . . , un ).
Albrecher et al. (2011) donnent trois résultats explicites de probabilités de ruine pour trois
lois. Par exemple, lorsque Θ est une loi gamma Ga(α, λ), alors on aψ(u) =

γ(α − 1, θ0 λ) λα θ0 θ0 u Γ(α − 1, θ0 (λ + u))
+
e ×
,
Γ(α)
Γ(α)
(λ + u)α−1où Γ(., .) est la fonction gamma incomplète supérieure. En utilisant un développement asymptotique de cette fonction, voir par exemple Olver et al. (2010), on obtient γ(α − 1, θ0 λ) λα θ0 λθ01
1
ψ(u) =
+
e
+o
.
Γ(α)
Γ(α)
λ+u
u
Cette formule laisse entrevoir une nouvelle forme d’asymptotique du type A + B/u + o(1/u).
En effet, le chapitre 4 va montrer que cette forme asymptotique est valable quelque soit la loi
pour Θ. Ce chapitre traitera en profondeur les asymptotiques de la probabilité de ruine ψ(u)lié à ce modèle de dépendance.

Le modèle de risque en temps discret
Nous considérons une version discrète du processus de risque présenté jusqu’à présent. Le
modèle de ruine en temps discret, introduit par Gerber (1988), suppose que les primes, lessinistres et les arrivées sont à valeurs discrètes. Par changement de l’échelle de temps, nous
supposons générallement que les primes sont normées à 1. Le processus de risque est le suivant
Ut = u + t −

t
X

Xi ,

i=1

où les montants des sinistres sont à valeurs entières et la condition de profit net E(X) < 1.
Le modèle le plus simple considère des montants de sinistres indépendants et identiquement
distribués.
Gerber (1988) considère la ruine comme le premier instant t où le processus (Ut )t atteint
0. C’est à direψG (u) = P (inf(t ∈ N+ , Ut ≤ 0) < +∞|U0 = u).
35

Introduction
A l’opposé, Shiu (1989) considère la ruine comme le premier instant t où le processus (Ut )tdevient strictement négatifψS (u) = P (inf(t ∈ N+ , Ut < 0) < +∞|U0 = u).
Géométriquement parlant, ψG regarde le premier temps d’atteinte de l’axe des abcissses, tandisque ψS considère le premier temps d’atteinte de la droite horizontale y = −1. Nous pouvons
facilement passer d’une définition à l’autre en changeant le capital initial, en utilisant larelation ψG (u) = ψS (u − 1).
Pour le différencier de sa version continue, le processus de risque P
discret est générallement
affichée en deux composantes : les primes u + t et la perte agrégée ti=1 Xi . Sur la figure 3,
les primes sont traçées en pointillées, les sinistres en trait plein et les sinistres non-nuls sontnotés en lettre. Sur cette trajectoire, selon la définition de Shiu, la ruine intervient en t = 14,
tandis que selon la définition de Gerber, elle a lieu en t = 13.
X14

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cumul. premium
cumul. claim
ruins

X13

X12
X11
X10
X9
X8
X6

u+t

X5
X3

X1

X2

t

∑ Xi

i=1

Time tFigure 3 – Une trajectoire du processus (Ut )tDans le cas de sinistre géométrique, P (X = k) = p(1 − p)k pour k ∈ N ∗ , la probabilitéde ruine peut s’obtenir facilement. Nous présentons ici une version améliorée de Sundt et dos
Reis (2007) utilisant la loi géométrique zéro et un modifiés. La loi géométrique zéro modifiée
a pour fonction de masse de probabilité
P (X = k) = qδ0,k + (1 − q)(1 − δ0,k )ρ(1 − ρ)k−1 ,où δi,j est le produit de Kronecker. En prenant q = ρ, nous retrouvons la loi géométrique
simple. Quant à la version zéro et un modifée de la loi géométrique, les probabilités élémentairessont
P (X = 0) = q = 1 − p et P (X = k/X > 0) = ρδk,1 + (1 − ρ)(1 − α)αk−2 (1 − δk,1 ).
∗. Certains auteurs présentent une version zéro tronquée p(1 − p)k−1 pour k ≥ 1.

36

En prenant α = 1 − ρ, on retrouve la loi zéro modifiée. Les moments sont donnés par




1−ρ
3−α
1−ρ 2
2
E(X) = p 1 +et V ar(X) = qρ + (1 − q)(1 − ρ)
−p 1+
.
1−α
(1 − α)21−αPour cette dernière loi, le paramètre α contrôle la queue de distribution, tandis que les paramètres q, ρ fixent la probabilité en 0 et 1 respectivement. Sur la figure 4, nous avons tracé un
exemple pour chacune des lois.
Probability mass function

0.20
0.15
0.05
0.00

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0.10

P(X=k)

0.25

0.30

G(1/3)
G(1/6, 1/3)
G(1/7, 1/5, 1/3)

0

5

10

15

k

Figure 4 – Loi géométrique simple et modifiée
Le raisonnement de Sundt et dos Reis (2007) consiste à imposer une forme pour la probabilité de ruine et ensuite de déduire la loi de sinistre correspondante. Ils supposent ψS (u) = kw−u
et trouvent que les sinistres doivent être de loi géométrique zéro et un modifés. Ils utilisentl’équation de récurrence suivante obtenue à partir de la définition de la probabilité de ruineen conditionnant par rapport au premier sinistre X1
ψS (u) = P (X1 > u + 1) +u+1
XP (X1 = x)ψS (u + 1 − x),x=0

qui peut être réécrite commeψS (u) = pF̄ (u + 1) + qψS (u + 1) + p

u+1X

f (x)ψS (u + 1 − x),x=1

où q = P (X1 = 0), p = P (X1 > 0), F (u + 1) = P (X1 ≤ u + 1|X1 > 0) et f (x) = P (X1 =x|X1 > 0). Cela fonctionne correctement car en soustrayant astucieusement l’équation derécurrence en u et u + 1, la somme disparait. Si nous choisissions une probabilité de ruine dutype α/(u + β), il est beaucoup plus difficile de retrouver la loi des sinistres. De cette manière,
ils obtiennent le résultat suivant.
Théorème (Sundt et dos Reis (2007)). Dans le modèle discret avec des sinistres géométriquesGe(q, ρ, 1 − α), la probabilité de ruine a pour expressionu (1 − q)(1 − ρ) 1 − q
ψS (u) = min(1 − ρ) + α , 1 .
q(1 − α)q
37

Introduction
Le minimum garantit la condition de profit net (1 − q)(1 − ρ)/q + α < 1.

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Plusieurs extensions ont été proposées au modèle en temps discret, voir, par exemple,
Cossette et Marceau (2000),Cossette et al. (2004),Marceau (2009), où une dépendance entre
les montants de sinistre est introduite. Nous renvoyons le lecteur vers Li et al. (2009) pourune revue complète des modèles en temps discret.
Dans le chapitre 4, nous considérons un modèle à mélange basé sur ce modèle en temps
discret. Les montants des sinistres sont supposés indépendants et identiquement distribués deloi géométrique zéro-modifiée Ge(q, e−θ ) conditionnellement à Θ = θ. Dans un tel modèle, lechapitre 4 donnera des formules explicites de la probabilité de ruine ψS (u) pour certaines lois
Θ et des asymptotiques valable pour toute loi de Θ. Nous renvoyons au prochain chapitre pourplus de détails.

38

Principaux résultats
Cette thèse se décompose en quatre chapitres indépendants dont le fil conducteur est la
modélisation du marché d’assurance non-vie. Chaque chapitre correspond à un article et étudie
une composante du marché de l’assurance.

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Comportement d’un client
Le chapitre 1 est constitué de l’article Dutang (2012b), dans lequel nous cherchons à
modéliser la résiliation des contrats d’assurance par l’assuré. Nous présentons une application
des modèles linéaires généralisés, ainsi que des modèles additifs généralisés, à ce problème. Ces
derniers utilisent des termes non-linéaires dans le prédicteur et peuvent permettre une plus
grand souplesse. Néanmoins, le but de ce chapitre est de mettre en garde contre une utilisation
abusive et simpliste de ces modèles de régression pour prédire les taux de résiliation. Etant
donné leur simplicité d’application, on peut en effet être tenté de les utiliser brutalementsans faire attention à la pertinence des taux prédits. Ce chapitre montre à quel point ces
estimations peuvent être erronées si la régression n’utilise pas le pourcentage de rabais (accordé
par le vendeur d’assurance) et surtout une estimation du prix marché par contrat. D’autrepart, le chapitre propose une méthode simple pour tester l’éventuelle présence d’asymmétrie
d’information reposant sur des hypothèses de forme fonctionnelle.

Compétition et cycles en assurance non-vie
L’article Dutang et al. (2012a) forme le chapitre 2 et a pour but de combler les déficiences del’approche traditionnelle à trois agents : assuré, assureur et marché, où les cycles de marché sontmodélisés d’un côté (voir par exemple, Haley (1993)) et d’un autre côté, un positionnement del’assureur est choisi (voir Taylor (1986) et ses extensions). Nous commençons par l’introductiond’un jeu simple non-coopératif à plusieurs joueurs pour modéliser un marché d’assurance surune période. Dans ce jeu, les assureurs maximisent leur fonction objective Oj (xj , x−j ) souscontrainte de solvabilité gj (xj ) ≥ 0. Les assurés choisissent de résilier ou de renouveler leurcontrat selon une loi multinomiale logit de vecteur de probabilité dépendant du vecteur deprix x. L’existence et l’unicité de l’équilibre de Nash est établie, ainsi que sa sensibilité aux lesparamètres initiaux, voir les propositions 2.2.1 et 2.2.2, respectivement. Un jeu plus complexe
est ensuite proposé en modélisant plus finement la fonction objective Oj (xj , x−j ) et la fonction
contrainte des joueurs gj (xj , x−j ). Bien que l’existence d’équilibre de Nash généralisé soit
toujours garantie, l’unicité est perdue, voir la proposition 2.3.1. Ainsi, cette version améliorée
peut se révéler moins utile en pratique. Une sensibilité aux paramètres est obtenue dans laproposition 2.3.2.
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Principaux résultats

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De plus, une version dynamique du jeu est proposée en répétant le jeu simple sur plusieurs
périodes tout en mettant à jour les paramètres des joueurs et en tenant de compte de lasinistralité observée sur la période t, c’est à dire nous étudions dans ce jeu Oj,t (xj,t , x−j,t ) et
gj,t (xj,t ). En temps infini, la proposition 2.4.1 démontre que le jeu se termine avec au plus
un gagnant. La proposition 2.4.2 donne un ordre stochastique sur le résultat de souscription
par police pour un assureur, permettant de mieux comprendre ce qui favorise la position d’unleader. Enfin, par une approche Monte-Carlo, nous estimons la probabilité pour un joueurd’être ruiné ou de se retrouver leader après un nombre fini de périodes sur un grand nombre
de simulation. Une cyclicité de la prime marché d’environ dix périodes est observée sur laplupart des simulations. L’utilisation de la théorie des jeux non-coopératifs pour modéliser
des problématiques marché est relativement nouvelle : Taksar et Zeng (2011) utilise un jeu
continu à deux joueurs à somme nulle, tandis que Demgne (2010) se sert de modèles standards
de jeux économiques. Par conséquent, ce chapitre apporte une nouvelle preuve de l’utilité dela théorie des jeux non-coopératifs dans la modélisation des marchés de l’assurance.

Calcul d’équilibres de Nash généralisés
Le chapitre 3 se compose de l’article Dutang (2012a). Ce chapitre montre que le calcul effectif d’équilibre de Nash généralisé n’est pas limité aux seuls jeux à deux joueurs, comme c’est
généralement proposé dans les ouvrages de théorie des jeux. Nous nous intéressons aux jeux
généralisés les plus génériques et excluons les jeux généralisés conjointement convexes de notreétude. D’une part, ce chapitre a pour but de faire un panorama des méthodes d’optimisationles plus avancées pour calculer un équilibre de Nash généralisé pour un jeu à plusieurs joueurs.
Ces méthodes se basent sur une reformulation semi-lisse des équations de Karush-Kuhn-Tucker
du problème d’équilibre de Nash. Elles nécessitent l’utilisation du jacobien généralisé, une extension du jacobien classique aux fonctions semi-lisses. D’autre part, nous passons en revue lesprincipaux théorèmes de convergence pour les méthodes de résolution d’équation semi-lisse etétudions leur application dans le contexte des équilibres de Nash généralisé. Une comparaison
numérique de ces méthodes (notamment les méthodes de Newton et Broyden généralisées)
est réalisée sur un jeu test possédant plusieurs équilibres. Le panorama proposé dans ce chapitre est à comparer à Facchinei et Kanzow (2009) étudiant les jeux généralisés (généraux etconjointement convexes).

Asymptotiques de la probabilité de ruine
Enfin, le chapitre 4 basé sur l’article Dutang et al. (2012b) étudie une classe de modèlesde risque avec dépendance introduite par Albrecher et al. (2011). En temps continu, le modèlede risque considéré est basé sur une approche mélange où les montants de sinistre Xi sontconditionnellement indépendants et identiquement distribués de loi exponentielle E(θ) par
rapport la valeur d’une variable latente Θ. Ceci est équivalent à supposer que les montantsde sinistres ont une copule de survie archimédienne. Au sein de ce modèle, nous démontrons
l’existence d’une nouvelle forme d’asymptotique A+B/u pour la probabilité de ruine en temps
infini ψ(u).

40

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Ce nouveau type d’asymptotique en A + B/u est démontré dans le théorème 4.3.1, dont
nous citons ci-dessous l’item 2 : si Θ suit une loi continue de densité fΘ telle que fΘ est presquepartout différentiable et Lebesgue intégrable, alors la probabilité de ruine vérifie fΘ (θ0 )1
ψ(u) = FΘ (θ0 ) +,+ouuoù θ0 = λ/c et pour un capital initial u > 0. Remarquons qu’un résultat similaire peuts’obtenir lorsqu’on ajoute l’hétérogénéité sur les temps d’attente inter-sinistres Ti plutôt queles montants de sinistres Xi . Ce type d’asymptotique est nouveau par rapport à la littérature
actuelle, voir Asmussen et Albrecher (2010).
Dans un second temps, le chapitre 4 analyse une version en temps discret du modèle
de ruine. Ainsi, nous considérons des montants de sinistres de loi géométrique zéro-modiféeGe(q, e−θ ) conditionnellement à l’événement Θ = θ. De nouveau, nous pouvons montrer qu’une
formule asymptotique pour la probabilité de ruine en A + B/u prévaut, voir le théorème 4.3.4.
Nous donnons ci-dessous l’item 2 de ce dernier : si Θ suit une loi continue de densité fΘ telleque fΘ est presque partout différentiable avec une dérivée fΘ0 bornée, alors la probabilité deruine vérifie1qfΘ (θ0 )1
ψ(u) = F̄Θ (θ0 ) +
×
+o
,
u+2
1−q
u+2
où θ0 = − log(1 − q) et pour un capital initial u ≥ 0. Le chapitre se termine par une analysede la dépendance induite par l’approche mélange sur les montants de sinistres dans le modèle
en temps discret. Le cas discret pose des problèmes d’identifiabilité de la copule, qui sontabordés dans la section 4.4. La proposition 4.4.6 quantifie la distance maximale en termesde fonctions de répartition jointe des sinistres entre la version continue et la version discrète.
Des applications numériques sont proposées. Pour les modèles discrets, ce type d’approche parmélange est là nouveau et permet d’obtenir de nouvelles formules fermées pour la probabilitéde ruine.

41

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Principaux résultats

42

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Modèles de régression

43

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Chapitre 1

Sur la nécessité d’un modèle demarché
—The customer, the insurer and themarketIf you’re playing within your capability, what’s the point ?
If you’re not pushing your own technique to its own
limits with the risk that it might just crumble
at any moment, then you’re not really doing your job.
Nigel Kennedy

Ce chapitre se base sur l’article Dutang (2012) soumis au Bulletin Français d’Actuariat.

45

Chapitre 1. Sur la nécessité d’un modèle de marchétel-00703797, version 2 - 7 Jun 2012

1.1

Introduction

In price elasticity studies, one analyzes how customers react to price changes. In thispaper, we focus on its effect on the renewal of non-life insurance contracts. The methodologies
developed can also be applied to new business. Every year insurers face the recurring questionof adjusting premiums. Where is the trade-off between increasing premium to favour higher
projected profit margins and decreasing premiums to obtain a greater market share? We must
strike a compromise between these contradictory objectives. The price elasticity is therefore
a factor to contend with in actuarial and marketing departments of every insurance company.
In order to target new market shares or to retain customers in the portfolio, it is essential
to assess the impact of pricing on the whole portfolio. To avoid a portfolio-based approach, wemust take into account the individual policy features. Moreover, the methodology to estimate
the price elasticity of an insurance portofolio must be sufficiently refined enough to identify
customer segments. Consequently the aim of this paper is to determine the price sensitivityof non life insurance portfolios with respect to individual policy characteristics constituting
the portfolio.
We define the price elasticity as the customer’s sensitivity to price changes relative to theircurrent price. In mathematical terms, the price elasticity is defined as the normed derivativeper (p) = dr(p)dp × r(p) , where r(p) denotes lapse rate as a function of the price p. However, in
this paper, we focus on the additional lapse rate ∆p (dp) = r(p + dp) − r(p) rather er (p) since
the results are more robust and easier to interpret. In the following, we abusively refer to∆p (dp) as the price elasticity of demand.
Price elasticity is not a new topic in actuarial literature. Two ASTIN ∗ workshops (seeBland et al. (1997); Kelsey et al. (1998)) were held in the 90’s to analyze customer retention
and price/demand elasticity topics. Shapiro and Jain (2003) also devote two chapters of theirbook to price elasticity: Guillen et al. (2003) use logistic regressions, whereas Yeo and Smith(2003) consider neural networks.
In the context of life insurance, the topic is more complex as the lapse can occur at
any time, whereas for non-life policies, most lapses occur at renewal dates. There are sometrigger effects due to contractual constraints: penalties are enforced when lapses occur at the
beginning of the policy duration, while after that period, penalties no longer applied. Anotherinfluencial feature is the profit benefit option of some life insurance policies allowing insurers
to distribute part of benefits to customers in a given year. This benefit option stimulates
customers to shop around for policies with higher profit benefits.
In terms of models, Kagraoka (2005); Atkins and Gallop (2007) use counting process tomodel surrenders of life insurance, while Kim (2005) uses a logistic regression to predict thelapse. Milhaud et al. (2011) point out relevant customer segments when using Classification
And Regression Trees models (CART) and logistic regression. Finally, Loisel and Milhaud(2011) study the copycat behavior of insureds during correlation crises.
In non-life insurance, generalized linear models have been the main tool to analyze pricesensitivity, e.g., Dreyer (2000); Sergent (2004); Rabehi (2007). However, generalized linear
model outputs might underestimate the true price sensitivity. This could lead to irrelevant
conclusions, and therefore gross premium optimization based on such results may lead tobiased and sometimes irrelevant pricing decisions, see, e.g., (Hamel, 2007, Part 5), (Bella andBarone, 2004, Sect. 3).
∗. ASTIN stands for Actuarial STudies In Non-Life insurance.

46

1.2. GLMs, a brief introduction

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What makes the present paper different from previous research on the topic is the factthat we tackle the issue of price elasticity from various points of view. Our contribution is
to focus on price elasticity of different markets, to check the impact of distribution channels,
to investigate the use of market proxies and to test for evidence of adverse selection. We
have furthermore given ourselves the dual objective of comparing regression models as well as
identifying the key variables needed.
In this paper, we only exploit private motor datasets, but the methodologies can be applied
to other personal non-life insurance lines of business. After a brief introduction of generalized
linear models in Section 1.2, Section 1.3 presents a naive application. Based on the dubious
empirical results of Section 1.3, the Section 1.4 tries to correct the price-sensitivity predictions
by including new variables. Section 1.5 looks for empirical evidence of asymmetry of information on our datasets. Section 1.6 discusses the use of other regression models, and Section
1.7 concludes. Unless otherwise specified, all numerical applications are carried out with the
R statistical software, R Core Team (2012).

1.2

GLMs, a brief introduction

The Generalized Linear Models (GLM ∗ ) were introduced by Nelder and Wedderburn(1972) to deal with discrete and/or bounded response variables. A response variable on thewhole space of real numbers R is too retrictive, while with GLMs the response variable spacecan be restricted to a discrete and/or bounded sets. They became widely popular with thebook of McCullagh and Nelder, cf. McCullagh and Nelder (1989).
GLMs are well known and well understood tools in statistics and especially in actuarial
science. The pricing and the customer segmentation could not have been as efficient in non-life
insurance as it is today, without an intensive use of GLMs by actuaries. There are even books
dedicated to this topic, see, e.g., Ohlsson and Johansson (2010). Hence, GLMs seem to be thevery first choice of models we can use to model price elasticity. This section is divided into
three parts: (i) theoretical description of GLMs, (ii) a clear focus on binary models and (iii)
explanations on estimation and variable selection within the GLM framework.

1.2.1

Theoretical presentation

In this section, we only consider fixed-effect models, i.e. statistical models where explanatory variables have deterministic values, unlike random-effect or mixed models. GLMs are
an extension of classic linear models, so that linear models form a suitable starting point for
discussion. Therefore, the first subsection shortly describes linear models. Then, we introduce
GLMs in the second subsection.
Starting from the linear model
Let X ∈ Mnp (R) be the matrix where each row contains the value of the explanatory
variables for a given individual and Y ∈ Rk the vector of responses. The linear model assumes
the following relationship between X and Y :Y = XΘ + E,∗. Note that in this document, the term GLM will never be used for general linear model.

47

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Chapitre 1. Sur la nécessité d’un modèle de marché
where Θ denotes the (unknown) parameter vector and E the (random) noise vector. The
linear model assumptions are: (i) white noise: E(Ei ) = 0, (ii) homoskedasticity: V ar(Ei ) =σ 2 , (iii) normality: Ei ∼ N (0, σ 2 ), (iv) independence: Ei is independent of Ej for i 6= j,(v) parameter identification: rank(X) = p < n. Then, the Gauss-Markov theorem gives
us the following results: (i) the least square estimator Θ̂ of Θ is Θ̂ = (X T X)−1 X T Y andσ̂ 2 = ||Y − XΘ||2 /(n − p) for σ 2 , (ii) Θ̂ is a Gaussian vector independent of the randomvariable σ̂ 2 ∼ χ2n−p , (iii) Θ̂ is an unbiased estimator with minimum variance of Θ, such that
V ar(Θ̂) = σ 2 (X T X)−1 and σ̂ 2 is an unbiased estimator of σ 2 .
Let us note that first four assumptions can be expressed into one single assumption
E ∼ N (0, σ 2 In ). But splitting the normality assumption will help us to identify the strong
differences between linear models and GLMs. The term XΘ is generally referred to the linear
predictor of Y .
Linear models include a wide range of statistical models, e.g. the simple linear regression
yi = a + bxi + i is obtained with a 2-column matrix X having 1 in first column and (xi )iin second column. Many properties can be derived for linear models, notably hypothesis
tests, confidence intervals for parameter estimates as well as estimator convergence, see, e.g.,
Chapter 6 of Venables and Ripley (2002).
We now focus on the limitations of linear models resulting from the above assumptions.
The following problems have been identified. When X contains near-colinear variables, theb will be numerically unstable. This would lead to an increase
computation of the estimator Θ∗in the variance estimator . Working with a constrained linear model is not an appropriate
answer. In pratice, a solution is to test models with omitting one explanatory variable after
another to check for near colinearity. Another stronger limitation lies in the fact that the
response variance is assumed to be the same (σ 2 ) for all individuals. One way to deal with thisissue is to transform the response variable by the nonlinear Box-Cox transformation. However,this response transformation can still be unsatifactory in certain cases. Finally, the strongest
limitation is the assumed support of the response variable. By the normal assumption, Y
must lies in the whole set R, which excludes count variable (e.g. Poisson distribution) or
positive variable (e.g. exponential distribution). To address this problem, we have to use amore general model than linear models.
In this paper, Y represents the lapse indicator of customers, i.e. Y follows a Bernoulli
variable with 1 indicating a lapse. For Bernoulli variables, there are two main pitfalls. Since
the value of E(Y ) is contained within the interval [0, 1], it seems natural the expected values Ŷb may fall out of this range for sufficiently
should also lie in [0, 1]. However, predicted values θXlarge or small values of X. Furthermore, the normality hypothesis of the residuals is clearly
not met: Y − E(Y ) will only take two different values, −E(Y ) and 1 − E(Y ). Therefore, themodelling of E(Y ) as a function of X needs to be changed as well as the error distribution.
This motivates to use an extended model that can deal with discrete-valued variables.
Toward generalized linear models
A Generalized Linear Model is characterized by three components:
1. a random component: Yi follows a specific distribution of the exponential family Fexp (θi , φi , a, b, c) † ,∗. This would be one way to detect such isssue.
†. See Appendix 1.8.1.

48

1.2. GLMs, a brief introduction
2. a systematic component: the covariate vector Xi provides a linear predictor ∗ ηi = XiT β,
3. a link function: g : R 7→ S which is monotone, differentiable and invertible, such that
E(Yi ) = g −1 (ηi ),for all individuals i ∈ {1, . . . , n}, where θi is the shape parameter, φi the dispersion parameter,a, b, c three functions and S a set of possible values of the expectation E(Yi ). Let us notethat we get back to linear models with a Gaussian distribution and an identity link function(g(x) = x). However, there are many other distributions and link functions. We say a linkfunction to be canonical if θi = ηi .
Distribution

Canonical link

Mean

PurposeNormal N (µ, σ )identity: ηi = µi

µ = Xβstandard linear regression

Bernoulli B(µ)µ
logit: ηi = log( 1−µ)µ = 1+e1−Xβrate modelling

Poisson P(µ)

log: ηi = log(µi )µ = eXβclaim frequency

Gamma G(α, β)inverse: ηi = µ1iµ = (Xβ)−1

claim severity

Inverse Normal I(µ, λ)squared inverse: ηi = − µ12µ = (Xβ)−2claim severity

tel-00703797, version 2 - 7 Jun 2012

2i

Table 1.1: Family and link functions
There are many applications of GLM in actuarial science. Table 1.1 below lists the mostcommon distributions with their canonical link and standard applications. Apart from theidentity link function, the log link function is the most commonly used link function in actuarial
applications. In fact, with this link function, the explanatory variables have multiplicativeQ
effects on the observed variable and the observed variable stays positive, since E(Y ) = i eβi xi .
For example, the effect of being a young driver and owning an expensive car on average loss
could be the product of the two separate effects: the effect of being a young driver and theeffect of owning an expensive car. The logarithm link function is a key element in mostactuarial pricing models and is used for modelling the frequency and the severity of claims.
It makes possible to have a standard premium and multiplicative individual factors to adjust
the premium.

1.2.2

Binary regression

Since the insurer choice is a Bernoulli variable, we give further details on binary regression
in this subsection.
Base model assumption
In binary regression, the response variable is either 1 or 0 for success and failure, respectively. We cannot parametrize two outcomes with more than one parameter. So, a Bernoulli
distribution B(πi ) is assumed, i.e. P (Yi = 1) = πi = 1 − P (Yi = 0), with πi the parameter.
The mass probability function can be expressed as
fYi (y) = πiy (1 − πi )1−y ,∗. For GLMs, the name ‘linear predictor’ is kept, despite ηi is not a linear predictor of Yi .

49

Chapitre 1. Sur la nécessité d’un modèle de marché
which emphasizes the exponential family characteristic. Let us recall the first two moments
are E(Yi ) = πi and V ar(Yi ) = πi (1 − πi ) = V (πi ). Assuming Yi is a Bernoulli distribution
B(πi ) implies that πi is both the parameter and the mean value of Yi . So, the link function
for a Bernoulli model is expressed as follows
πi = g −1 (xTi β).

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Let us note that if some individuals have identical covariates, then we can group the dataand consider Yi follows a binomial distribution B(ni , πi ). However, this is only possible if allcovariates are categorical. As indicated in McCullagh and Nelder (1989), the link function andthe response variable can be reformulated in term of a latent variable approach πi = P (Yi =1) = P (xTi β − i > 0). If i follows a normal distribution (resp. a logistic distribution), we
have πi = Φ(xTi β) (πi = Flogistic (xTi β)).
Now, the log-likelihood is derived asln(L(π1 , . . . , πn , y1 , . . . , yn )) =nX[yi ln(πi ) + (1 − yi ) ln(1 − πi )] ,i=1

plus an omitted term not involving πi . Further details can be found in Appendix 1.8.1.
Link functions
Generally, the following three functions are considered as link functions for the binary
variableπ
1. logit link: g(π) = ln 1−πwith g −1 being the standard logistic distribution function,
2. probit link: g(π) = Φ−1 (π) with g −1 being the standard normal distribution function,
3. complementary log-log link: g(π) = ln(− ln(1−π)) with g −1 being the standard Gumbel
II distribution function ∗ .
On Figure 1.4 in Appendix 1.8.1, we plot these three link functions and their inverse functions.
All these three functions are the inverses of a distribution function, so other link functions canbe obtained using inverses of other distribution function. Let us note that the first two links
are symmetrical, while the last one is not.
In addition to being the canonical link function for which the fitting procedure is simplified,cf. Appendix 1.8.1, the logit link is generally preferred because of its simple interpretation as
the logarithm of the odds ratio. Indeed, assume there is one explanatory variable X, the logit
link model is p/(1 − p) = eµ+αX . If α̂ = 2, increasing X by 1 will lead to increase the odds
by e2 ≈ 7.389.

1.2.3

Variable selection and model adequacy

As fitting a GLM is quick in most standard software, then a relevant question is to checkfor its validity on the dataset used.
∗. A Gumbel of second kind is the distribution of −X when X follows a Gumbel distribution of first kind.

50

1.2. GLMs, a brief introduction
Model adequacy
The deviance, which is one way to measure the model adequacy with the data and generalizes the R2 measure of linear models, is defined by
D(y, π̂) = 2(ln(L(y1 , . . . , yn , y1 , . . . , yn )) − ln(L(π̂1 , . . . , π̂n , y1 , . . . , yn ))),
where π̂ is the estimate of the beta vector. The “best” model is the one having the lowest
deviance. However, if all responses are binary data, the first term can be infinite. So inpractice, we consider the deviance simply as

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D(y, π̂) = −2 ln(L(π̂1 , . . . , π̂n , y1 , . . . , yn )).
Furthermore, the deviance is used as a relative measure to compare two models. In most softwares, in particular in R, the GLM fitting function provides two deviances: the null devianceand the deviance. The null deviance is the deviance for the model with only an intercept or if
not offset only, i.e. when p = 1 and X is only an intercept full of 1 ∗ . The (second) devianceis the deviance for the model D(y, π̂) with the p explanatory variables. Note that if there are
as many parameters as there are observations, then the deviance will be the best possible, but
the model does not explain anything.
Another criterion introduced by Akaike in the 70’s is the Akaike Information Criterion(AIC), which is also an adequacy measure of statistical models. Unlike the deviance, AIC
aims to penalized overfitted models, i.e. models with too many parameters (compared to thelength of the dataset). AIC is defined by
AIC(y, π̂) = 2k − ln(L(π̂1 , . . . , π̂n , y1 , . . . , yn )),where k the number of parameters, i.e. the length of β. This criterion is a trade-off between
further improvement in terms of log-likelihood with additional variables and the additionalmodel cost of including new variables. To compare two models with different parameter
numbers, we look for the one having the lowest AIC.
In a linear model, the analysis of residuals (which are assumed to be identical and independent Gaussian variables) may reveal that the model is unappropriate. Typically we canplot the fitted values against the fitted residuals. For GLMs, the analysis of residuals is much
more complex, because we loose the normality assumption. Furthermore, for binary data, i.e.
not binomial data, the plot of residuals exhibits straight lines, which are hard to interpret, see
Appendix 1.8.2. We believe that the residual analysis is not appropriate for binary regressions.
Variable selection
From the normal asymptotic distribution of the maximum likelihood estimator, we canderive confidence intervals as well as hypothesis tests for coefficents. Therefore, a p-value isavailable for each coefficient of the regression, which help us to keep only the most significant
variable. However, as removing one variable impacts the significance of other variables, it canbe hard to find the optimal set of explanatory variables.
There are two approaches: either a forward selection, i.e. starting from the null model, we
add the most significant variable at each step, or a backward elimination, i.e. starting fromthe full model, we remove the least significant variable at each step.
∗. It means all the heterogeneity of data comes from the random component.

51

Chapitre 1. Sur la nécessité d’un modèle de marché
Another way to select significant explanatory variables is to use the analysis of deviance.
It consists in looking at the difference of deviance ln L between two models, i.e. ratios of
likelihood. Using an asymptotic distribution, either chi-square or Fisher-Snedecor distributions, a p-value can be used to remove or to keep an explanatory variable. Based on this fact,
statistical softwares generally provide a function for the backward and the forward selection
using an automatic deviance analysis
In conclusion, GLM is a well-known statistical method in actuarial science. This fact
motivates its use to model lapse rate. Since it is a classic among statistical models, fitting
method and variable selection use state-of-art algorithms providing robust estimator. So, there
is absolutely no problem in applying GLMs for a daily use. In the following section, we apply
GLMs to explain the customer price-sensitivity.

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1.3

Simplistic applications and biased business conclusions

This section is intended to present quite naive GLM applications and to show how they
can lead to inconclusive or even biased findings. First, we use a dataset with poor and limited
data, and then a larger dataset with more comprehensive data. Finally, we summarize theissues encountered. It may seem obvious, but to study customer price-sensitivity, insurers
need to collect the premium proposed to customers when renewing policy, especially for those
who lapse.
For confidential reasons, the country names are not revealed, but we study two continental
European insurance markets. In this part of the world, the insurance penetration rate isconsidered high, e.g., 8.6% in France, 7% in Germany, 7.6% in Italy, according to Cummins
and Venard (2007). Thus, the insurance markets studied are mature as well as competitionlevel is intense. Furthermore, data outputs presented in this paper have been perturbed, but
original conclusions have been preserved.

1.3.1

An example of poor data

In this subsection, we work with a (representative) subset of a 1-year lapse history databasein 2003. Each line of the dataset represents a policy for a given vehicle. The dataset suffers amajor problem because only few variables are available.
Descriptive analysis
To better understand interactions between lapses, the premium and other explanatory
variables, we start with a short descriptive analysis. As a general comment, all variables inthe dataset are dependent to the lapse variable according to a Chi-square test. At our disposal,
we have the last year premium and the proposed premium. Computing the premium ratio,
we observe that most of the portfolio experienced a price decrease, probably due to the ageing
and the market conditions. We expect to slightly underestimate the true price sensitivity ofclients, since customers attention will be released.
Turning to customer variables, we focus on gender and driver age variables, reported inTable 1.2. As the age of the customer increases, the lapse rate decreases. So, the most sensitive
clients seem to be the youngest clients. The gender ∗ does not have any particular impact of∗. In a near future, insurers will no longer have the right to discreminate premium against the gender ofthe policyholder according to the directive 2004/113/CE from the European comission.

52

1.3. Simplistic applications and biased business conclusions
the lapse. However the GLM analysis may reveal some links between the gender and lapses.

Lapse rate (%)
Prop. of total (%)(30,47.5](47.5,62.5](62.5,77.5](77.5,92.5]

FEMALE

MALE

20
38

17
42

14
17

14.6
3

18
20

19
80

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Table 1.2: Driver age and Gender
We also have a categoric variable containing a lapse type with three possible values: lapse
by insured, lapse by company and payment default. We observe a total lapse rate of 18%,of which 11% is a payment default, 6% a lapse by the customer, only 1% a lapse by thecompany. The lapse by company has to be removed, because those lapses generally result
from the pruning strategy of insurers. However, default of payment must be taken with caresince it might represent a hidden insured decision. It may result from a too high premiumthat the customer can’t afford. Thus, we choose to keep those policies in our study. Note thatthe lapse motive cannot be used in the regression because its value is not known in advance,
i.e. the lapse motive is endogeneous.
The last variables to explore are policy age and vehicle age. According to Table 1.3, somefirst conclusions can be derived. As the policy age increases, the remaining customers are more
and more loyal, i.e. lapse rates decrease. Unlike the policy age, when the vehicle age increase,the lapse rate increases. One explanation may be that the customer may shop around for a
new insurer when changing the vehicle.

Lapse rate (%)
Prop. of total (%)(1, 5](5,9](9,13](13,17](1,8](8,14](14,20](20,26]

21
38

17
33

18
22

16.9
7

17.6
36

19.4
37

21
22

39
4

Table 1.3: Policy age and vehicle age

GLM analysis
For the GLM analysis of this dataset, we use a backward selection. The explanatory
variables are driver age, gender, policy age, vehicle age, the last year premium and the priceratio, i.e. ratio of the premium proposed and the premium paid last year. In order to havebetter fit and predictive power, all explanatory variables are crossed with the price ratio:
crossing variable xj with price ratio p consists in creating a dummy variable xji × pi for allobservations 1 ≤ i ≤ n.
Note that variable xj might be categorical, i.e. valued in {0, . . . , d}, which allows to zoom
in on some particular features of individuals. The linear predictor for observation i is thus
given byηi = µ × 1 + (x1i , . . . , xki )T β−p + (z1i , . . . , zki )T β+p × pi ,where µ is the intercept, β−p (resp. β+p ) the coefficient for price-noncross variables (resp.
price-cross), xi price-noncross variables, zi price-cross variables and pi the price ratio.
Yet not reported here, we test two models: (i) a GLM with original (continuous) variableand (ii) a GLM with categorized variables. We expect the second model with categorized data
to be better. Using continuous variables limits the number of parameters: 1 parameter for a
53

Chapitre 1. Sur la nécessité d’un modèle de marchétel-00703797, version 2 - 7 Jun 2012

continuous variable and d − 1 parameters for a categorical variable with d categories. Cutting
the driver age, for example, into three values ]18, 35], ]35, 60] and ]60, 99] enables to test forthe significance of the different age classes.
The numerical application reveals that a GLM with categorical data is better in terms ofdeviance and AIC. Hence, we only report this model in Appendix 1.8.2, first column is thecoefficient estimates µ̂, β̂−p and β̂+p . The GLM with continuous variables also has businessinconsistent fitted coefficients, e.g. the coefficient for the price ratio was negative. This also
argues in favor of the GLM with categorized variables. We also analyze (but do not report)
different link functions to compare with the (default) logit link function. But the fit gives
similar estimate for the coefficients µ̂, β̂−p and β̂+p , as well as similar predictions.
To test our model, we want to make lapse rate predictions and to compare against observed
lapse rates. From a GLM fit, we get the fitted probabilities π̂i for 1 ≤ i ≤ n. Plotting those
probabilities against the observed price ratios does not help to understand the link between
a premium increase/decrease and the predicted lapse rate. Recall that we are interested inderiving a portfolio elasticity based on individual policy features, we choose to use an average
lapse probability function defined asn
1 X −1 π̂n (p) =µ̂ + xi (p)T β̂−p + zi (p)T β̂+p × p ,
g
n

(1.1)

i=1

where (µ̂, β̂−p , β̂+p ) are the fitted parameters, xi price-noncross explanatory variables, zi pricecross explanatory variables ∗ and g the logit link function, i.e. g −1 (x) = 1/(1 + e−x ). Notethat this function applies a price ratio constant to all policies. For example, π̂n (1) the average
lapse rate, called central lapse rate, if the premium remains constant compared to last year
for all our customers.
Computing this sum for different values of price ratio is quite heavy. We could have use aprediction for a new obsveration (x̃, ỹ, p̃),


T
T
−1
µ̂ + x̃ β̂−p + ỹ β̂+p × p̃ ,gwhere the covariate (x̃, ỹ, p̃) corresponds to the average individual. But in our datasets, theideal average individual is not the best representative of the average behavior. Equation (1.1)
has the advantage to really take into account portfolio specificities, as well as the summation
can be done over a subset of the overall data.
In Table 1.4, we put the predicted lapse rates, i.e. π̂n (1). We also present a measure ofprice sensitivity, the delta lapse rate defined as∆1− (δ) = π̂n (1 − δ) − π̂n (1) and ∆1+ (δ) = π̂n (1 + δ) − π̂n (1),(1.2)

where δ represents a premium change, for example 5%.
As mentioned in the introduction, this measure has many advantages compared to the pricep‡elasticity † (er (p) = dr(p)dp × r(p) ): it is easier to compute, more robust , easier to interpret.
∗. Both xi and yi may depend on the price ratio, e.g. if xi represents the difference between the proposedpremium and a technical premium.
†. It is the customer’s sensitivity to price changes relative to their current price. A price elasticity of emeans that an increase by 1% of p increase the lapse rate by e%.
‡. Price elasticity interpretation is based on a serie expansion around the point of computation. So, priceelasticity is not adapted for large δ.

54

1.3. Simplistic applications and biased business conclusions∆1− (5%)π̂n (1)∆1+ (5%)All-0.74514.7140.772

Old drivers
Young pol., working male
Young drivers-0.324-0.585-1.166

9.44
15.20819.784

0.333
0.601
1.211

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Table 1.4: Central lapse rates (%) and deltas (pts)

In Table 1.4, we report the predicted lapse rates and deltas for the whole dataset (firstline) as well as for three subsets: old drivers, young policies and working male, young drivers.
This first result exhibits the wide range of behaviors among a portfolio: young vs. old drivers.
However, delta values seem unrealistic: a 5% premium increase will increase the lapse rate
only by 0.772 pts. Based only on such predictions, one will certainly not hesitate to increase
premium.
As this small dataset only provides the driver age, GLM outputs lead to inconclusive or
dubious results. The old versus young segmentation alone cannot in itself substantiate thelapse reasons. We conclude that the number of explanatory variables are too few to get reliable
findings with GLMs, and probably with any statistical models.

1.3.2

A larger databaseIn this subsection, we study another dataset from a different country in continental Europe
in 2004. As for the other dataset, a record is a policy purchased by an individual, so an
individual may have different records for the different covers he bought.

Descriptive analysis
This dataset is very rich and contains much more variables than the previous set. The
full list is available in Appendix 1.8.2. In Table 1.5, we present some explanatory variables.
The dataset contains policies sold through different distribution channels, namely tied-agents,
brokers and direct platforms, cf. first line of Table 1.5. Obviously, the way we sell insurance
products plays a major role in the customer decision to renew or to lapse. The coverage types(Full Comprehensive, Partial Comprehensive and Third-Part Liability) have a lesser influenceon the lapse according to the first table.
The dataset also contains some information on claim history, e.g. the bonus/malus or theclaim number. In Table 1.5, we present a dummy variable for the bonus evolution (compared
to last year). From this table, we observe that a non-stable bonus seems to increase thecustomer propency to lapse. This could be explained by the fact that decreasing or increasing
bonus implies the biggest premium difference compared to last year premium, raising thecustomer attention.
At this stage, the claim number does not seem to influence the lapse. The policy age has
the same impact as in the previous dataset (cf. Table 1.3). The older is the policy the lower
the customer lapses. However, the opposite effect is observed for the vehicle age compared toprevious dataset.
55

Chapitre 1. Sur la nécessité d’un modèle de marché
Coverage

FC

PC

TPL

ChannelAgentBrokerDirectprop. size
lapse rate

36.16
14.26

37.61
12.64

26.23
12.79

prop. size
lapse rate

65.1
7.4

20.1
10.3

6.1
12.1

Claim nb.

0

1

2

3

(3 - 13]Bonus evol.

downstableup

prop. size
lapse rate

70.59
13.75

25.29
13.37

3.60
16.03

0.44
12.82

0.09
35.16

prop. size
lapse rate

33.32
16.69

62.92
11.53

3.76
12.02

Policy age

(0,1]

(1,2]

(2,7]

(7,34]

Vehicle age(0,6](6,10](10,13](13,18]prop. size
lapse rate

24.97
17.43

16.79
15.27

34.38
11.26

23.86
8.78

prop. size
lapse rate

26.06
15.50

31.01
13.56

21.85
12.72

21.08
10.67

Table 1.5: Impact on lapse rates (%)

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GLM analysis
Now, we go to the GLM analysis. We apply a backward selection to select statisticallysignificant variables. The regression summary is put in Appendix 1.8.2. The sign of coefficient
β+p are positive for the two categories of last year premium level ∗ , thus this is businessconsistent. The most significant variables † are the region code, the distribution channel andthe dummy variable indicating the relative difference between the technical premium and the
proposed premium and the dummy variable checking whether the policyholder is also the car
driver.
In terms of prediction, the results presented in Table 1.6 are similar to the previous subsection. As for the “poor” dataset, we use the average lapse function π̂n (p) and delta lapse rate
∆1+ (δ) defined in Equations (1.1) and (1.2), respectively. The overall central lapse rate is low
compared to the previous set but the customers on that market seems more price sensitive,
with bigger deltas for a 5% decrease or increase. Taken into account the distribution channel,
the differences are huge: around 8.7% vs. 11.6% for agent and direct, respectively. Despite
observing higher deltas, we think these estimates still underestimate the true price sensitivity.
∆1− (5%)π̂n (1)∆1+ (5%)

All-0.833

8.966

1.187

Channel agent
Channel broker
Channel direct-0.759-1.255
-1.18

7.732
9.422
11.597

0.75
1.299
1.268

Coverage Full Comp.
Coverage Part. Comp.
Coverage TPL-0.622-0.714-0.8997.723
9.244
10.179

0.97
1.063
1.178

Table 1.6: Central lapse rates (%) and deltas (pts)
Looking at the bottom part, the impact of cover type on central lapse rates is considerably
lower. Central rates are between 8% and 10%, regardless of the product purchased. Delta∗. See lastpremgroup2(0,500] and lastpremgroup2(500, 5e+3].
†. See diff2tech, region2, channel, diffdriverPH7.

56

1.4. Incorporating new variables in the regression
lapse rates ∆1+ are again surprisingly low around 1 pt. In Appendix 1.8.2, we also compare
the observed lapse rate by channel and coverage type against the fitted lapse rate, see Table
1.17. The results are unsatisfactory.

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1.3.3

IssuesThe price-sensitivity assessment appears to be difficult. Getting outputs is easy but having
reliable estimates is harder. We are not confident on the lapse prediction as well as the
additional lapse rates ∆1+ . A first answer is shown in Table 1.18 of Appendix 1.8.2, where we
present the predicted results when the dataset is split according to the distribution channelor the coverage type. This split provides more realistic lapse rates, each fit better catches the
specificity of the distribution channel. Thus we choose to fit nine regressions in the following
in order to catch the full characteristics of the distribution channel and the coverage type.
However, this section reveals major issues of a rapid and simplistic application of GLMs.
We miss something as it does not really make sense that a 5% premium increase on the whole
portfolio leads to a lapse rate increase less than 1pt. In such situation, the insurer has all
reasons to increase premium by 5% and to get a higher gross written premium. The market
competition level drives the level of customer price-sensitivity that we can estimate. Therefore,high cautions is needed when using GLMs predictions with few variables.

1.4

Incorporating new variables in the regression

This section focuses on identifying new key variables needed in the GLM regression inorder to get more reliable results. Attentive readers have probably noticed that some variables
have been forgotten in this first analysis. As we will see, they have a major impact on the
GLM outputs. Furthermore, taking into account previous conclusions on the large dataset of
Subsection 1.3.2, all results presented in this section are obtained by nine different regressions,
one for each channel and each coverage type.

1.4.1

Rebate levels

Firstly, we add to all regressions the rebate level variable, specifying the amount of rebate
granted by the agent, the broker or the client manager to the customer. As reported in Table
1.7, the number of customers having rebates is considerably high. The broker channel grants
a rebate to a majority of customers. Then comes the tied-agent channel and finally the directchannel.

AgentBrokerDirectFull Comp.

Part. Comp.

TPL

56.62
62.25
23.05

36.84
52.5
22.89

22.26
36.24
10.37

Table 1.7: Proportion of granted rebates (%)
It seems logical that the direct channel does not grant rebates since the premium is generally lower through the direct channel than with other distribution channels. The influence57

Chapitre 1. Sur la nécessité d’un modèle de marchéof the coverage type is also substantial: it is harder to get a rebate for a third-part liability(TPL) product than a full comprehensive coverage product.
In order to catch the most meaningful features of the rebate on the lapse decision, therebate variable has been categorized. Despite the dataset is subdivided into 9 parts, thisvariable is always statistically significant. For example in the TPL broker subgroup, theestimated coefficients β̂ for the rebate variable are β̂10−20 = −0.368879, β̂25+ = −0.789049.
In that case, the variable has three categories (0, 10-20 and 25+), thus two coefficients fortwo categories plus the baseline integrated in the intercept. The negative sign means thatthe rebate level has a negative impact on the lapse, i.e. a rebate of 15 decreases the linear
predictor (hence the predicted lapse rate). This is perfectly natural.
Furthermore, when predicting lapse rate with the average lapse function π̂n , we force therebate level to zero. That is to say, in the equationn
1 X −1 π̂n (p) =gµ̂ + xi (p)T β̂−p + zi (p)T β̂+p × p ,ntel-00703797, version 2 - 7 Jun 2012i=1

the explanatory variables xi (p), zi (p) are updated depending on the price ratio p. The rebate
variable appearing in the vector (xi (p), zi (p)) is set to zero when predictions are carried out.
So that a 5% increase really means such premium increase, and not 5% minus the rebate thatthe customer got last year.

Agent
BrokerDirectπ̂n (1)∆1+ (5%)π̂n (1)∆1+ (5%)π̂n (1)∆1+ (5%)7.278
10.98712.922

0.482
2.888
1.154

8.486
9.754
11.303

0.896
2.776
1.263

8.549
10.972
11.893

0.918
3.437
1.490

Full Comp.

Part. Comp.

TPL

Table 1.8: Central lapse rates (%) and deltas (pts)Table 1.8 presents GLM predictions for the nine subgroups. We can observe the majordifferences compared to the situation where the rebate level was not taken into account, cf.
Table 1.6. Notably for the broker channel, the delta lapse rates are high and represent thebroker’s work for the customer to find the cheapest premium. The central lapse rates alsoslightly increase in most cases compared to the previous fit. This subsection shows how
important the rebate variable is when studying customer price-sensitivity.

1.4.2

Market proxyIn this subsection, we add another variable to regressions, a market premium proxy bypolicy. The proxy is computed as the tenth lowest premium among competitor premiums ofa standard third-part liabibility ∗ coverage product. Such computation is carried out on amarket premium database which is filled by all insurers of the market. However, we don’t
have the choice of the market proxy. It would have been a good study to see the influence ofthe market proxy choice, e.g., the fifth, the first lowest or the mean premium, in the GLM fit.
Unfortunately, the market proxy information is only available on two subsets of thedatabase, namely TPL agent and TPL direct subsets. As for the technical premium, we∗. There is no deductible with this product.

58

1.4. Incorporating new variables in the regression
choose to insert that variable in the regression via the relative difference compared to the
proposed premium. We consider
mi =marketi − premiumi,
premiumi

where marketi and premiumi denote the market premium and the proposed premium for theith policy, respectively. In Table 1.9, we give a basic cross-table of lapse and relative market
premium variables. Among the lapsed policies, 65% of them have a higher premium than themarket proxy, whereas for renewed policies it drops to 57%.
m
Renew
Lapse

(-0.75,-0.5](-0.5,-0.25](-0.25,0](0,0.25](0.25,0.5](0.5,0.75](0.75,1]

0.69
0.079

18.4841.326

33.2484.158

28.254
2.637

9.735
0.327

0.571
0.032

0.066
0.006

tel-00703797, version 2 - 7 Jun 2012

Table 1.9: Percentage of policies (%)
However, we cannot conclude that lapses result from a higher premium compared to themarket, just based on this table. In fact, the market proxy is just a proxy for the third-part
liability coverage, computed as the tenth lowest premium. Moreover, the market proxy is atheoretical premium based on the risk characteristics. If a client goes to another company, it
may have a lower or a higher premium depending if he get a rebate or choose an add-on cover.
As the indemnification procedure also varies between two insurers, the market proxy should
be seen as a probe of the market level rather than a true premium.
Now, that we have described the new explanatory variable, we turn our attention to the
GLM regression. The residual deviance and Akaike Information Criterion (AIC) have slightly
decreased with the addition of the market proxy (8866 to 8728 and 8873 to 8735, respectively).
Regression summary for the GLM with market variable is available on request to the author.
The most instructive results are the average lapse prediction. Comparing the Table 1.10
with Table 1.8 reveals that the addition of the market proxy has a major impact of the delta
lapse rate ∆1+ (5%), cf. bolded figures. For the TPL agent subset, it goes from 0.918 to 1.652
pts, while for the TPL direct subset, from 1.490 to 2.738. Central lapse rates before and after
the market proxy inclusion are consistent.

Agent
BrokerDirectπ̂n (1)∆1+ (5%)π̂n (1)∆1+ (5%)π̂n (1)∆1+ (5%)7.278
10.98712.922

0.482
2.888
1.154

8.486
9.754
11.303

0.896
2.776
1.263

8.548
10.97211.958

1.652
3.437
2.738

Full Comp.

Part. Comp.

TPL

Table 1.10: Central lapse rates (%) and deltas (pts)The predicted results are plotted on Figure 1.1, where the x-axis represents central lapse
rates (π̂n (1)), the y-axis delta lapse rates for a 5% premium increase (∆1+ (5%)). The bubble
radius are determined by the proportion of the subet in the whole dataset. The text order inthe legends is the decreasing order of bubble radius.
On Figure 1.1, we clearly observe the difference between those two channels both in termsof central lapse rates and delta lapse rates. These two differences can be explained again by59

Chapitre 1. Sur la nécessité d’un modèle de marché

1

2

3

FC agent
PC agent
FC broker
TPL agent
PC broker

PC direct
TPL broker
FC direct
TPL direct0

delta lapse rate (pts)4

Customer behaviors

4

6

8

10

12

tel-00703797, version 2 - 7 Jun 2012

central lapse rates (%)

Figure 1.1: Comparison of distribution channels and cover types

the fact the brokers are paid to find the cheapest premium. The direct channel shows higher
central lapse rates π̂n (1), but the estimated delta lapse rates are lower than those for Broker
channel. Direct channel are designed for customers shopping around on internet, so it seems
logical that their propency to lapse should be higher. We would have expected the same tohold for delta lapse rates ∆1+ (5%). The estimated delta rate of the direct channel might still
be underestimated. In addition to the absence of market proxies in the TPL direct database,the direct channel is also small in size. Hence, higher uncertainty on those estimates might
explain low delta lapse rates for FC/PC direct subsets.

1.4.3

Backtesting

In this subsection, we present backtesting results for the fitted GLMs. We start by looking
only at an aggregate level: channel and coverage. The results are given in Table 1.11, reporting
observed and fitted lapse rates. The observed lapse rate rj for the jth group is computed as
the average lapse rate variable over the jth group, whereas fitted lapse rate is the average ofthe fitted probabilities
π̂i over the jth group given the observed explanatory variables for eachPnj
individual n1j i=1
π̂i .
The fitted results are good, since for each subgroup, the deviation is below one percentage
point. Compared to the previous backfit table, the improvements with rebate level, market
proxy and datasplit are amazing. The two subgroups for which we use market proxy, theresults are even better (deviation < 0.1pt), see TPL agent and direct. However, we must
recognize that observed price ratio are relatively small: for 85% of the portfolio, the differenceis below 5%. Hence, the model appropriately catches the lapse phenomenon when the variationin premium remains reasonable.
To further assess the predictive power of our GLM fits, we focus on the TPL coverage
product. We consider three subpopulations representing three different behaviors: (i) old
drivers with at least two contracts in the household, (ii) working class with a decreasing60

1.4. Incorporating new variables in the regression
Observed

FittedObserved

FittedObserved

Fitted

7.361
8.123
8.579

7.124
8.084
8.569

10.1679.971
10.867

10.46810.09
11.04212.95811.25811.15312.881
11.193
11.171

Full Comp.
Part. Comp.
TPL

Agent

BrokerDirectTable 1.11: Central lapse rates (%) and deltas (pts)bonus-malus and an old vehicle, (iii) young drivers. We expect the population 3 to be themost price-sensitive.

tel-00703797, version 2 - 7 Jun 2012

Pop. 1
Pop. 2
Pop. 3

Prop.

Obs.

Fit.

Std.Err.

Prop.

Obs.

Fit.

Std.Err.

Prop.

Obs.

Fit.

Std.Err.

13
13
10

4.98
8.45
10.01

5.16
8.65
9.91

0.22
0.32
0.42

5
16
14

7.99
11.59
13.25

8.24
12.36
12.45

0.49
0.50
0.62

3
17
13

6.98
12.44
14.91

8.187
13.02
14.184

0.65
0.61
0.74

Agent

BrokerDirectTable 1.12: Lapse rates and proportions (%)
In Table 1.12, we report the backfit results for the three selected populations separatingeach distribution channel. Each block presents the proportion of population i in the total
subset, the observed lapse rate for population i, the mean of fitted lapse rates and standard
deviations. As expected the difference between the three populations is high whatever thechannel. Population 1 can be tagged as a sluggish behavior, Population 2 a kind of medium
behavior, while Population 3 represents highly sensitive customers.

1.4.4

Market scenarios

Having a market variable in the database allows us to perform market scenarios. In thissubsection, we briefly present this topic particularly interesting for business line managers. We
perform two basic scenarios: a 5% increase of market premium and a 5% decrease of market
premium.

Market -5%
Market 0%
Market +5%-5%Insurer
0%
+5%

-5%

Insurer
0%

+5%

8.007
7.801
7.645

8.763
8.548
8.359

12.5389.604
8.638

14.14311.95810.94317.58914.696
13.589Agent

10.481
10.152
9.916

DirectTable 1.13: Market scenarios (%)
The results are summarized in Table 1.13. It is surprising to see how the tied-agent
customers react very slowly when premium fluctuates. In particular when market decrease of
5% and the proposed premium increases by 5%, then the lapse rate goes only from 8.548%
to 10.481%. While for the direct channel, the lapse rate rockets from 11.958% to 17.589%.
Actually for any difference in premium, the lapse rate fluctuates largely for the direct channel
61Chapitre 1. Sur la nécessité d’un modèle de marché
The two previous sections demonstrate that GLMs are easy to implement, but care on thevariable selection and appropriate data are needed to ensure reliable outputs. In this section,
we show how incorporating new key variables in the GLM regression substantially improves thelapse rate predictions in the different premium scenarios. The rebate level partially reveals
the agent or the broker actions on the customer decisions, while the use of market proxies
illustrates how decisive the competition level is when studying customer price-sensitivity.
In conclusion, the GLM methodology, when used on appropriate data, fulfills the initial
objective to derive average lapse rate prediction taking into account individual features. Furthermore, using the predicted lapse rate values of GLMs, it has been easy to identify customer
segments, which react differently to premium changes. The back-fit of the GLMs on the identified populations is correct. At a customer segment level, GLMs provide a fair estimate oflapse rate and price sensitivity for reasonable premium changes. But at a policy level, we
think lapse predictions should be treated carefully.

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1.5

Testing asymmetry of information

Asymmetry of information occurs when two agents (say a buyer and a seller of insurance
policies) do not have access to the same amount of information. In such situations, one of
the two agents might take advantage of his additional information in the deal. Typically, two
problems can result from this asymmetry of information : adverse selection and moral hazard.
In insurance context, moral hazard can be observed when individuals behave in risker ways,
when they are insured. Insurers cannot control the policyholder’s actions to prevent risk.
Adverse selection depicts a different situation where the buyer of insurance coverage has abetter understanding and knowledge of the risk he will transfer to the insurer than the insurer
himself. Generally, the buyer will choose a deductible in his favor based on its own risk
assessment. Hence, high-risk individuals will have the tendency to choose lower deductibles.
Adverse selection is caused by hidden information, whereas moral hazard is caused by hidden
actions.
Joseph Stiglitz was awarded the Nobel price in economics in 2001 for his pioneer workin asymmetric information modelling. In insurance context, Rothschild and Stiglitz (1976)
models the insurance market where individuals choose a “menu” (a couple of premium anddeductible) from the insurer offer set. Within this model, they show that high-risk individuals
choose contracts with more comprehensive coverage, whereas low-risk individuals will choose
higher deductibles.

1.5.1

Testing adverse selection

The topic is of interest when modelling customer behaviors, since a premium increasein hard market cycle phase, i.e. an increasing premium trend, may lead to a higher loss
ratio. Indeed if we brutally increase the price for all the policies by 10%, most of high-risk
individuals will renew their contracts (in this extreme case), while the low-risk will just run
away. Therefore the claim cost will increase per unit of sold insurance cover.
In this paper, we follow the framework of Dionne et al. (2001), which uses GLMs to
test for the evidence of adverse selection ∗ . Let X be an exogenenous variable vector, Y an∗. Similar works on this topic also consider the GLMs, see Chiappori and Salanié (2000) and Dardanoni
and Donni (2008).

62

1.5. Testing asymmetry of information
endogeneous variable and Z a decision variable. The absence of adverse selection is equivalent
to the prediction of Z based on the joint distribution of X and Y coincides with prediction
with X alone. This indirect characterization leads tol(Z|X, Y ) = l(Z|X),(1.3)

tel-00703797, version 2 - 7 Jun 2012

where l(.|., .) denotes the conditional probability density function.
One way to test for the conditionnal independence of Z with respect to Y is to regress thevariable Z on X and Y and see whether the coefficient for Y is significant. The regression
model is l(Z|X, Y ) = l(Z; aX + bY ). However, to avoid spurious conclusions, Dionne et al.
(2001) recommend to use the following econometric modelb |X) ,
l(Z|X, Y ) = l Z|aX + bY + cE(Y(1.4)
b |X), the conditionnal expectation of Y given the variable X, will be estimated bywhere E(Yb |X) allows toa regression model initially. The introduction of the estimated expectation E(Ytake into account nonlinear effects between X and Y , but not nonlinear effects between Z and
X, Y ∗ .
b |X).
Summarizing the testing procedure, we have first a regression Y on X to get E(YbSecondly, we regress the decision variable Z on X, Y , and E(Y |X). If the coefficient for Y issignificant in the second regression, then risk adverse selection is detected. The relevant choicefor Z is the insured deductible choice, with X rating factors and Y the observed number ofb |X) will be estimated with a Poisson or more sophisticated models, see below.
claims. E(Y

1.5.2

A deductible model

The deductible choice takes values in the discrete set {d0 , d1 , . . . , dK }. The more generalmodel is a multinomial model M(1, p0 , . . . , pK ), where each probability parameter pj depends
on covariates through a link function. If we assume that variables Zi are independent andidentically distributed random variables from a multinomial distribution M(1, p0 , . . . , pK ) and
we use a logit link function, then the multinomial regression is defined byT

exi βj
,
K
P
Tβ
x
1+
e i l

P (Zi = dj ) =l=1

for j = 1, . . . , K where 0 is the baseline category and xi covariate for ith individual, see, e.g.,
McFadden (1981), Faraway (2006) for a comprehensive study of discrete choice modelling.
When reponses (d0 < d1 < · · · < dK ) are ordered (as it is for deductibles), one can also
use ordered logistic models for which
TT

P (Zi = dj ) =eθj −xi βT

1 + eθj −xi β−eθj−1 −xi βT

1 + eθj−1 −xi β.

Note that the number of parameters substantially decreases since the linear predictor for
multinomial logit regression, we have ηij = xTi βj , whereas for the ordered logit, ηij = θj −xTi β.
∗. See Su and White (2003) for a recent procedure of conditional independence testing.

63

Chapitre 1. Sur la nécessité d’un modèle de marché
The parameters θ, called thresholds, have a special interpretation since they link the
response variable Z with a latent variable U by the equation Z = dk ⇔ θk−1 < U ≤ θk .
Hence, the trick to go from a Bernoulli model to a polytomous model is to have different
ordered intercept coefficients θk ’s for the different categorical values.
As in Dionne et al. (2001), our choice goes to the ordered logit model for its simplicity.
So, Z is modelled by the following equationb |Xi )δ ,P (Zi ≤ j/Xi , Yi ) = g −1 θj + XiT β + Yi γ + E(Yfor individual i and deductible j, with g −1 the logistic distribution function ∗ and Xi exogeneous explanatory variables as opposed to endegeneous variables Yi . The parameters of thismodel equation are the regression coefficients β and γ and the threshold parameter θk ’s.

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1.5.3

Application on the large dataset of Subsection 1.3.2

We want to test for evidence of adverse selection on the full comprehensive (FC) coverage
product. So, we study in this subsection only the three datasets relative to that coverage.
First, we model the claim number, and then we test for the asymmetry of information.
Modelling the claim number
Modelling count data in the generalized linear model framework can be done by choosing
an appropriate distribution: the Poisson and overdispersed Poisson distribution, cf. Table
1.1, where the canonical link function is the logarithm. Since for a Poisson distribution P(λ),
P (Y = 0) = e−λ , the GLM Poisson consists in assumingT

E(Y / xi ) = exi β ⇔ − log P (Y = 0/ xi ) = xTi β.
where xi denotes the covariates. In practice, this models suffers a subparametrization of the
Poisson distribution, one single parameter.
One could think that the Negative binomial in an extended GLM † framework will tackle
this issue, but in practice the mass in zero is so high, that both Poisson and negative binomial
distributions are inappropriate. As presented in Table 1.14, the high number of zero-claim
will compromise the good fit of regular discrete distributions.
Claim number
Frequency

0

1

2

3

4

5

5<

43687

5308

667

94

17

2

38

Table 1.14: Claim number for Full Comp. agent subset
As presented in Zeileis et al. (2008) and the references therein, the issue is solved by usinga zero-inflated distribution, e.g., a zero-inflated Poisson distribution. The mass probability∗. Note that in this form, it is easy to see that g −1 can be any distribution functions (e.g. normal orextreme value distributions).
†. The negative binomial distribution does not belong to the exponential family, except if the shape parameter is known. So, the trick is to use a maximum likelihood procedure for that shape parameter at outer
iteration whereas each inner iteration use a GLM fit given the current value of the shape parameter.

641.5. Testing asymmetry of information
function is given by(
P (Y = k) =π
k
(1 − π) λk! e−λif k = 0,otherwise.

Note that Y is a mixture of a Bernoulli distribution B(π) with a Poisson distribution P(λ).
The mean of the zero-inflated Poisson distribution is (1 − π)λ. Using the GLM frameworkand the canonical link functions, a zero-inflated GLM Poisson model is defined as
1

E(Y / xi ) =tel-00703797, version 2 - 7 Jun 2012

1+e

2T

T
x1i γ

e xi β ,where the covariate vectors x1i , x2i are parts of the vector xi . Now there are two (vector)
coefficients to estimate β and γ. The GLM is implemented in R base by the glm function.
For the zero-inflated model, we need to use the pscl package, cf. Jackman (2011).
Still studying the FC agent dataset, we fit three distributions on the claim number: Poisson, zero-inflated Poisson and Negative binomial distributions. As shown in Table 1.19 inAppendix 1.8.2, the three models are similar in terms of log-likelihood or AIC. But, differences appear at the predictions.
Despite being equivalent for first probabilities P (X = 0, 1, 2), classic and zero-inflated
Poisson distributions decrease too sharply compared to the observed number of claims. The
negative Binomial distribution (fourth line) is far better. In Appendix 1.8.2, we give theregression summary for zero-inflated negative binomial distribution on the FC agent subset.
We obtain the same conclusion for other FC subsets.
Claim number

0

1

2

3

4

5

6

Observed

43687

5308

667

94

17

2

2

Poissonzeroinfl. Poisson
zeroinfl. NB

43337.943677.643704.6

5896.05267.75252.6

500.9
745.0
704.7

39.8
80.2
98.8

3.7
7.5
14.9

0.417
0.665
2.457

0.054
0.058
0.442

Table 1.15: Claim number prediction for Full Comp. agent subset

Testing adverse selection
Now that we have modelled the claim frequency, we turn to the modelling of the deductible
choice as described in the previous section: an ordered logistic model. We test for evidence ofadverse selection on three datasets: agent, broker and direct with Full. Comp. products. Let
us note that we cannot test adverse selection on TPL covers, since there is no deductible for thiscover. As reported in Subsection 1.5.1, adverse selection testing is done by a fit of a GLM to
explain the deductible choice Zi . In addition to the exogeneous variables Xi for ith individual,
the regression will use the observed claim number Yi (endogeneous) and its expected valueb |Xi ) (exogeneous).
coming from the zero-inflated negative binomial regression E(YThe numerical illustrations reveal that it is more relevant to cluster some deductible values
which are too few in the dataset. Actually, the deductible is valued in {0, 150, 300, 500, 600,
1000, 2000, 2500}. As 300 euros is the standard deductible, very high deductibles are rarely
chosen. So, we choose to regroup deductible values greater than 500 together. In Table 1.16,
65

Chapitre 1. Sur la nécessité d’un modèle de marché
we report the proportion of customers by deductible value for the first two datasets. Small
deductible values might reveal high-risk individuals, so we decide to keep those values.
Deductible (e)

0

150

300

500+

0

150

300

500+

Proportion (%)

5.17

10.29

70.85

13.68

4.78

7.85

68.21

17.46

Agent channel

Broker channel

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Table 1.16: Frequency table for Full Comp. deductibles values
As shown in Appendix 1.8.2 for FC agent subset, the endogeneous variable Yi is notstatistically significant despite being negative, i.e. the higher the loss number, the lower theb |Xi ) is significant. For the two other FC datasets,
deductible. But the expected value E(Yb |Xi ) are not significant, but these datasets are also smaller inboth coefficients for Yi and E(Ysize. We conclude that there is no adverse selection for FC datasets.
After removing insignificant variables in the deductible regression, we integrate the deductible choice predicted probabilities to the lapse regression. Let Zi denotes the deductible
for the ith individual, we incorporate fitted probabilities P̂ (Zi = 0), P̂ (Zi = 150) andP̂ (Zi = 500+). We choose to consider 300 euro as the baseline category, as 300-euro deductible is the standard “unchosen” deductible. For the FC agent dataset, the three probabilities, P̂ (Zi = 0), P̂ (Zi = 150) and P̂ (Zi = 500+), are significant, see Appendix 1.8.2, whereas
for the two other FC datasets some probabilities are not significant. We perform the usual
predictions for the lapse rate (-5%, 0% and +5% for the proposed premium). But we do notpresent here the lapse rate predictions since predictions are almost unchanged ∗ .
This section shows how to use GLM modelling to test for evidence of adverse selection.
In our dataset, no adverse selection is detected. The inclusion of deductible choice probability neither improves the lapse predictions nor helps in understanding the lapse decision at
aggregate level. But we believe that the deductible choice (especially non standard ones) bya customer plays a major role in the propensy of lapse when renewing its policy. Low-risk
invididuals, i.e. with high deductibles, are likely to be the most sensitive customers, unlike tohigh-risk individuals.

1.6

Other regression models

This section presents other regression models. There are mainly two (static) extensions
to GLMs in two directions: (i) additive models where the linear predictor is composed ofsmooth terms and (ii) mixed models where we add a random term (as opposed to fixed term,
i.e. deterministic). These two extensions are available for the exponential family distribution,
leading to generalized additive models and generalized linear mixed models, respectively. In
this paper, we discard mixed models as they are inefficient in our context. The first subsection introduces generalized additive models, and then the second subsection is devoted toan application. The last subsection details other regression models than generalized additivemodels.
∗. difference less than 0.1% pt.

66

1.6. Other regression models

1.6.1

Model presentation

The Generalized Additive Models (GAM) were introduced by Hastie and Tibshirani (1990)
by unifying generalized linear models and additive models. So, GAMs combine two flexible
and powerful methods: (i) the exponential family which can deal with many distribution for
the response variable and (ii) additive models which relax the linearity assumption of thepredictor.
Theoretical presentation

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In this subsection, we present Generalized Additive Models in two steps: from linear to
additive models and then from additive to generalized additive models. Fitting algorithms arethen briefly presented, whereas smoothing techniques are detailed in Appendix 1.8.1. Finally,
we apply GAMs on the large dataset of Subsection 1.3.2.
Assuming observations Xi and response variables Yi are identically and independently
distributed random variables having the same distribution of generic random variables X andY , respectively. In a linear model, the model equation is
Y = XΘ + E
where Y as always stands for the response variable, X the design matrix and E the random
noise. Linear models assume by definition a linear relationship motivated by mathematicaltractability rather than empirical evidence. One candidate to extend the linear model is the
additive model defined by
p
X
Y =α+
fj (Xj ) + E,
j=1

with fj smooth function of the jth explanatory variable Xj and E is assumed to be a centered
random variable with variance σ 2 .
A GAM is characterized by three components:
1. a random component: Yi follows a distribution of the exponential family Fexp (θi , φi , a, b, c),
2. a systematiccomponent: the covariate vector Xi provides a smooth predictor ηi =Pp
α + j=1 fj (Xij ),3. a link function g : R 7→ S which is monotone, differentiable and invertible, such that
E(Yi ) = g −1 (ηi ),for i ∈ {1, . . . , n}, where θi is the shape parameter, φi the dispersion parameter, a, b, c threefunctions (characterizing the distribution), fj ’s smooth functions and S a set of possible valuesof the expectation E(Yi ). Note that linear models (and GLMs) are special cases of additivemodels (and GAMs) with fj (x) = βj x.
We present here only the main idea of fitting algorithms and we do not go into details, see
Appendix 1.8.1 for a list of smoothing procedures. All smoothers have a smoothing parameterλ, (the polynom degree, the bandwidth or the span). A first concern is how to choose acriterion on which to optimize λ (hence to have an automatic selection). Then, a second
concern is to find a reliable estimate of the parameters α and smooths coefficients given asmoothing value λ.
We present the procedure in the reverse way. Assuming a value of λ, we present an algorithm to fit the model. Hastie and Tibshirani (1990) propose a local averaging generalized
67

Chapitre 1. Sur la nécessité d’un modèle de marché
Fisher scoring method. However, Wood (2008) proposes a recent and reliable method: the Penalized Iteratively Reweighted Least Square method (PIRLS). The PIRLS is (unsurprisingly)
an iterative method aiming to minimize the penalized deviancee = D(f1 , . . . , fp ) +
D

p
X

Z
λj

fj00 (xj )2 dxj ,j=1

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where the second term penalizes the wiggly behavior of smooth functions.
Given a set of basis functions (bjk )jk , we can express the smooth function fj as fj (x) =PKje
k=1 βjk bjk (x). So, in the end, the GAM can be represented as a GLM with ηi = Xi β
ei containing the basis functions evaluated at the covariate values and β containing
with X
linear parameter α and coefficients βjk ’s. Thus, the first term is fully determined. Hence, thepenalized deviance is given byeD(β)= D(β) +X

λj β T Sj β,

jwhere Sj contains known coefficients and zero’s and D(β) the GLM version of the deviancefor the fixed-basis GAM model. In Appendix 1.8.1, we present in details the PIRLS algorithmeto solve the problem min D(β).
The PIRLS algorithm gives for any λ the corresponding fitted coefficient β̂(λ), i.e. smoothfunctions fˆj . Now, we must find a criterion to select the appropriate vector λ. We cannot
choose the smoothing parameter λ as the parameter minimizing the deviance, because themodel will overfit the data. In the literature, there are many criteria to select the smoothing parameter: likelihood measures such as Restricted Maximum Likelihood (REML), Maximum Likelihood (ML) and cross validation measures such as Generalized Cross Validation(GCV), Generalized Approximate Cross Validation (GACV). These methods differ whether
the smoothing parameter is treated as a random effect or not. So, we either maximize aquantity linked to the likelihood (ML/REML) or minimize a prediction error (GCV/GACV).
Expressions of log-likelihood criterion (ML and REML) use the deviance of the model,
the satured deviance and a third-term penalizing the wiggliness of the smooth function fj .
The optimization procedure consists in using a Newton method for the optimization of theparameter λ where in each iteration a PIRLS is used (to find β(λ)). So, this is a nested
optimization where outer iteration optimizes over λ and the inner iterations optimized overβ, see Wood (2010) for details.
An alternative approach seeks in minimizing the prediction error. The predictive error
may seem difficult to assess, but the trick is to use a leave-one-out procedure. It consistsin computing n deviances D−i where D−i is the deviance without the ith observation. The
deviance cross validation is just a sum of the D−i ’s. In practice we do not fit n times themodel (clearly too expensive!) but an approximation is used to compute the GCV or GACV.
Then again, a nested optimization procedure using the PIRLS scheme is used.
In Appendix 1.8.1, we report an example of GAM fit, showing the criterion and the choiceof polynomial basis have few impact on the final model. Thus, in the following, we use the
REML criterion and thin plate regression: the default in the R function gam.
68

1.6. Other regression models
Binary regression and model selection
As for GLMs, the binary regression means we assume that Yi follows a Bernoulli distribution B(πi ), πi being linked to explanatory variables. So, the model equation is
πi = g −1 (ηi ),where g is the link function and ηi the predictor. Unlike the GLM where the predictor was
linear, for GAMs the predictor is a sum of smooth functions:
α0 +

p
X

fj (Xj ) or α0 +tel-00703797, version 2 - 7 Jun 2012

j=1

p1
X
i=1

αi Xi +p2
X

fj (Xj ),j=1

the latter being a semi-parametric approach. As suggested in Hastie and Tibshirani (1995),
the purpose to use linear terms can be motivated to avoid too much smooth terms and are
longer to compute (than linear terms). For instance, if a covariate represents the date or thetime of events, it is “often” better to consider the effect as an increasing or decreasing trendwith a single parameter αi .
As for GLMs, we are able to compute confidence intervals using the Gaussian asymptotic
distribution of the estimators. The variable selection for GAMs is similar to those of GLMs.
The true improvement is a higher degree of flexibility to model the effect of one explanatory
variables on the response.
The procedure for variable selection is similar to the backward approach of GLMs, but aterm is dropped only if no smooth function and no linear function with this term is relevant.
That is to say, a poor significance of a variable modelled by a smooth function might besignificant when modelled by a single linear term. We will use the following acceptance rulesof Wood (2001) to drop an explanatory variable:(a) Is the estimated degrees of freedom for the term close to 1?
(b) Does the plotted confidence interval band for the term include zero everywhere?
(c) Does the GCV score drop (or the REML score jump) when the term is dropped?
If the answer is “yes” to all questions (a, b, c), then we should drop the term. If only question(a) answer is “yes”, then we should try a linear term. Otherwise there is no general rule toapply. For all the computation of GAMs, we use the recommended R package mgcv written
by S. Wood.

1.6.2

Application to the large dataset

In Section 1.3.2, the GLM analysis of this large dataset reveals that the channel distribution
strongly impacts the GLM outputs. Especially, the lapse gap between tied-agent and otherchannels is far stronger than what we could expect. Moreover, the price sensitivity gap
measured by the lapse deltas is also high. Let us see this it still holds with GAM results.
On each channel and cover, we first estimate a GAM by modelling all the terms by asmooth function. And then we apply the Wood’s rules to remove, to linearize or to categorize
the explanatory variables. In Appendix 1.8.2, we provide the regression summary for one of
the nine subsets.
Comments on regression summary
In this subsection, we briefly comment on the nine regression summaries. Let us start with
the Third-Part Liability cover. For the agent subset, for which we have a market proxy, we69tel-00703797, version 2 - 7 Jun 2012

Chapitre 1. Sur la nécessité d’un modèle de marché
keep four non linear terms (premium difference variables and car class) all modelled jointlywith the price ratio. We try to model these terms independently of price ratio, but this wasworse in terms of REML scores. On the broker subset, we keep two non linear terms (difference
to technical premium and vehicle age). Only the first term is modelled jointly with the priceratio, because the second term has a linear effect with the price ratio. Due to a small size,
the direct subset was hard to handle with a GAM. We restrict the price ratio to be a smoothterm of small order. This dataset also shows some strange results with a negative elasticity
for small premium increase.
Studying Partial Comprehensive coverage is also challenging. For the agent subset, despite
many attempts, only the price ratio (alone) has a real benefit to be modelled non linearly.
This dataset is sufficiently big to make a lot of explanatory variables significant. And so, we
believe a big part of price sensitivity is explained by linear terms. As for the TPL covers,
the same variables are modelled non linearly for the broker subset, jointly with the priceratio. The high estimated degrees of freedoms emphasize this non linearity. Turning to thedirect channel, only the difference to technical premium variable is modelled through a smoothfunction, jointly with the price ratio.
Finally, we study the Full Comprehensive coverage product. As always, the agent subset
has many nonlinear terms. Three terms (driver age, difference to technical premium and carclass) are smoothed together with the price ratio. Again, the estimated degrees of freedom
are high, especially for the difference to technical premium variable. Regarding the broker
subset, four terms (driver age, vehicle age, difference to technical premium and car class) aremodelled non linearly. We retrieve the difference with technical premium and the vehicle age
as non linear terms. There might be a process made by brokers to target old vehicles and/or
to detect a strong difference with technical premium. So, the brokers have a major impact on
the lapse decision. Ending with the direct subset, only two terms are modelled non linearly(the driver age, difference to technical premium): the estimated degree of freedom for thepolicyholder age variable is high. This may be linked to the close relationship between the
motor (technical) premium and the policyholder age.
Examples of fitted smooth functions
In the preceding analysis, we observe some trends between channel distributions. Notably,
the broker channel results are more sensitive to the difference with technical premium and thevehicle age variables than the other two channels. There is also a data size effect, since thedata sets gradually increase in size from TPL and PC to FC covers. Of course, the more wehave data, the more the regression is reliable.
On Figure 1.2, we plot two fitted smooth functions from two different GAM regressions ∗ .
Figure 1.2a represents the smooth function for the price ratio variable of the PC-agent regression. We observe that the smooth function is highly non linear, i.e. a high degree of freedomof 6.35. The smooth function features a very sharp increase of the price ratio around 1: suchsteep increase is not possible with a linear predictor.
Figure 1.2b is the plot of the bivariate smooth function of the price ratio and the difference
to technical premium variable for FC broker dataset. There is a small hollow in the curvearound the point (1, 0), a price ratio of 1 and a zero difference with technical premium. Locally,∗. The grey area represents the standard error bandwidth around the smooth function. It is standard touse an area rather than two simples curves for the confidence interval: this suggests smooth functions lies insuch area.

701.6. Other regression modelss(pric2.0

eratio1.5,diff2te

1.0
0.5

ch,18

0.0

.04)

-0.5
-0.4
-0.2

1.5
ice

pr
ra

0.0 chtef2
0.2 diftio

1.0

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0.4(a) PC agent - price ratio smooth function(b) FC broker - bivariate smooth function

Figure 1.2: GAM smooth functions

the price elasticity of the lapse decision is negative. Fortunately, this business inconsistency issmall and located. If we had market variables for this dataset, it could be of interest to check
whether this anomaly vanishes.
Discussion on predictions
As for the GLM analysis, we turn to the analysis of the distribution channel and thecoverage type by looking at the lapse rate predictions. We also consider an average lapse ratefunction defined as


p
n
X
X
1
π̂n (p) =g −1 µ̂ + xi (p)T β̂−p + zi (p)T β̂+p × p +fˆj (z̃i (p), p) ,
(1.5)
n
i=1

j=1

where (µ̂, β̂−p , β̂+p ) are the fitted parameters, fˆj are the fitted smooth functions, (xi , zi , z̃i )are parts of explanatory variables of the ith individual and g is the logit link function. Whatdifferentiates Equation (1.5) with Equation (1.1) is the inclusion of additive terms in thepredictor.
On Figure 1.3, we plot the usual bubble plot to compare GAMs and GLMs. We observethat GAM delta lapse rate predictions are higher than GLM ones in most cases. This isespecially true for PC agent or FC broker: there is a high jump upward. Only two channelcovers have a lower delta lapse rate ∆1+ (5%) with GAMs: the FC direct case, a case where
the dataset is small (so the GAM model selection was hard) and the FC agent case where thedifference is limited.
In terms of central lapse rates, most of predictions π̂n (1) are higher, i.e. shift to the righton Figure 1.3. It means that the customers in the portfolio are more price-sensitive even if we
propose exactly the same premium as last year. On a private motor insurance, most people
expect a better bonus-malus from year to another, hence a premium decrease.
71

Chapitre 1. Sur la nécessité d’un modèle de marché

Customer behaviors (GAM)

Customer behaviors (GLM)
4
3
2

delta lapse rate (pts)4

PC direct
TPL broker
FC direct
TPL direct00

PC direct
TPL broker
FC direct
TPL direct2

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FC agent
PC agent
FC broker
TPL agent
PC broker

112

delta lapse rate (pts)34

FC agent
PC agent
FC broker
TPL agent
PC broker

6

8

10

12

2

4

6

8

central lapse rates (%)

central lapse rates (%)(a) GAM(b) GLM

10

12

Figure 1.3: GAM vs. GLM - comparison of distribution channels and cover types

Now, we stop the GAM analysis and conclude on the pros and cons of GAMs. GAMs
are less known tools than GLMs in actuarial science. But since their introduction in the
90’s, GAMs are well studied and use state-of-the-art fitting procedures. There are two ways
to perform model selections: prediction errors vs. likelihoods. In this paper, we follow the
Wood’s rule to select variables based on the restricted maximum likelihood. We tested otherstatistical quantities, but the impact remains limited.
As for GLMs, GAMs allow us to assess an overall estimated price elasticity (via π̂n (1)and ∆1+ (5%)) taking into account the individual features of each policy. The additional
complexity coming with additive modelling compared to GLMs permit to really fit the data.
Especially for broker lines, we get a more cautious view of customer price sensitivity. For small
datasets, GAM predictions may lead to irrelevant results. Furthermore, as already noticed for
GLMs, GAMs predictions are reliable for with a small range of price change: extrapolating
outside observed price ratio range leads to doubtful results.
Finally, GAMs need a longer time to fit than GLMs and require a better computing power.
This is a limitation for GAMs to be used easily by everyone. In addition, some user judgement
is needed to select, to linearize or to reject explanatory variables in order to get the final modelfor GAMs. Even with Wood’s rules, newcomers may find it hard to choose between two GAM
models with the same “score”, i.e. with the same likelihood or prediction errors.

1.6.3

Other regression models

GLMs and GAMs are static models. One option to take into account for dynamics would
be to use time serie models on regression coefficients of GLMs. But this was impossible with
our datasets due to a limited number of years and it is rather a trick than an appropriate
solution. Generalized Linear Mixed Models (GLMM), where the linear predictor becomes thesum of a (unknown deterministic) fixed term and a random term, are a natural extension of
GLMs to deal with heterogeneity across time.
Among many others, Frees (2004) presents GLMMs in the context of longitudinal and72tel-00703797, version 2 - 7 Jun 2012

1.7. Conclusion
panel data. Since a panel data model cannot deal with right-censoring (that occurs when apolicy is terminated), they are not appropriate to our policy termination problem, i.e. lapse.
Despite discarding GLMMs for dynamic lapse modelling, we try to use the GLMMs on one
period in order to model endogeneous effects such as dropping coverage with a random term.
Unfortunately, this reveals inefficient.
The Survival Regression Model of Cox (1972) allow to remove the inherent limits of thestatic regression models previously presented. By nature, they take into account the dynamic
aspects of the response variable. by a lifetime variable. In our context, we model the lifetimeof a policy. As GLMs and GAMs demonstrate, renewing a policy for the first time is not
motivated by the same factors as renewing one for the tenth time. An application of suchmodels may be found in Brockett et al. (2008) and Chapter 4 of Dutang (2011).
The full power of survival models is not only to model one lapse reason. Other policy
termination factors can be integrated so as to model the complete life cycle of a policy. Witha full picture integrating other cash flows such as claims, and premiums, insurance risk could
also be better assessed. Further advanced models than the Cox model regression exists, such as
state-space models, e.g., Fahrmeir (1994) or stochastic counting processes, see, e.g., Andersen
et al. (1995); Aalen et al. (2008). Some attempts have been done to use Fahrmeir (1994)’s
state space model, but the fitting process was too heavy to be quickly used.

1.7

Conclusion

Fitting price-sensitivity is a complex topic. Being dependent on the market’s environment,
price elasticity forecasts require rigorous attention to details to prevent the risk of erroneous
conclusions. Not surprisingly, a data cleaning process is essential prior to any regression fitting.
In short, some supplied explanatory variables substantially affect the results. Omitting thesevariables in the data can, in itself, lead to unreliable findings.
These must-have variables include distribution channels, market premium proxies, rebate
levels, coverage types, driver age, and cross-selling indicators. In Section 1.3, the small dataset
only provides the driver age: this example leads to inconclusive results. On the large dataset,the coverage type, and the cross-selling indicators were added to the regression fit. This
enabled us to refine our analysis. Having or not having a household policy with the same
insurer was thus proven to be a driving factor in renewing or allowing a contract to lapse.
However, fully reliable predictions are only achieved when the rebate level and market
premium proxies are used. In Section 1.4, the price sensitivity fit was considerably enhanced,
along with our ability to fine tune the results, thanks to the inclusion of distribution channels,a market proxy, and a rebate level. With the gradual addition of explanatory variables, wehave seen an increased accuracy of the lapse rate predictions. Disposing of market variables
proved to make testing market scenarios possible (e.g. -5%, +5%). Being able to provide suchforecasts is highly valuable in taking pricing actions. If those market proxies are no longer
available, we are likely to get back to less meaningful results.
Adverse selection resulting from an asymmetry of information is a widely known risk ininsurance. Section 1.5 investigates for empirical evidence of adverse selection and studies its
relationship to the lapse decision of customers. On our large dataset, no adverse selection isdetected. At aggregate level, adverse selection does not have a big influence. Nevertheless,at individual level, choosing a non-standard deductible when underwriting a new policy willcertainly have consequences on the termination of this policy.
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Chapitre 1. Sur la nécessité d’un modèle de marché
Generalized Linear Models are widely known and respected methods in non-life insurance.
However, they have some inherent constraints with GLMs. Thus, in Section 1.6, we test
Generalized Additive Models, which allow for non linear terms in the predictor. Like GLMs,
the quality of the findings attained is directly related to the data provided. Using limited
variables will produce approximate results, whereas, dealing with an extensive set of variables
lead to proven results.
Applying GAMs, despite their additional complexity, can be justified in cases where GLMs
fail to provide realistic lapse predictions and we have substantial datasets. Note that GAMs
can model interactions between explanatory variables. Not restricted to linear terms, theyconsequently provide us with a more adaptive tool. Caution should however be exercised, as
they may overfit the data when applied to limited datasets. This could then imply businessinconsistency.
In this paper, we have explored the price elasticity topic from various viewpoints. Onceagain, our research has further demonstrated that the quality of data used in actuarial studies
unequivocally affects the findings reached. In addition, the key role of the market proxies
in estimating price sensitivity has been established. Market competition modelling, see, e.g.,
Demgne (2010), Dutang et al. (2012), is therefore relevant.
The conclusions drawn from customer price sensitivity studies should in any respect beweighed carefully. Charging higher premiums to loyal customers could seem unfair in light of
the fact that those same customers usually have a better claims history. By the same token,
relying on the market context with its inherent uncertainty to predict price sensitivity could
be misleading. In summary, insurers must have a well informed overview of the market, thecustomer base, and a keen awareness of the pros and cons of potential pricing adjustments.
The models presented herein serve as decision-making support tools and reinforce businessacumen.

1.8

Appendix

1.8.1

Generalized linear and additive models

Univariate exponential family
Clark and Thayer (2004) defines the exponential family by the following density or massprobability function
f (x) = ed(θ)e(x)+g(θ)+h(x) ,where d, e, g and h are known functions and θ the vector of paremeters. Let us note that the
support of the distribution can be R or R+ or N. This form for the exponential family is called
the natural form. When we deal with generalized linear models, we use the natural form of
the exponential family, which isθx−b(θ)f (x, θ, φ) = e a(φ)+c(x,φ),where a, b, c are known functions and θ, φ ∗ denote the parameters. This form is derived
from the previous by setting d(θ) = θ, e(x) = x and adding a dispersion parameter φ. The
exponential family of distributions in fact contains the most frequently used distributions.
∗. the canonic and the dispersion parameters.

74

1.8. Appendix
For example, the normal distribution N (µ, σ 2 ) with θ = µ and φ = σ 2 , see Clark and Thayer(2004) for details.
Fitting procedure
To determine the parameter vector β, we use the maximum likelihood estimation. For nobservations, the log-likelihood of the model given a distribution from the exponential family
is written as follows:n Xyi θi − b(θi )ln(L(θ1 , . . . , θn , φ, y1 , . . . , yn )) =+ c(yi , φ) .
(1.6)
a(φ)
i=1

tel-00703797, version 2 - 7 Jun 2012

Let us define µi = E(Yi ) and ηi = g(µi ) = Xi β, the linear prediction where i is the number ofthe observation, n the total number of observations.
For all i and j,∂ ln(Li ) ∂µi
yi − µi∂ ln(Li )=×= (g −1 )0 (g(µi )) ×
Xij .
∂βj
∂µi
∂βj
V ar(Yi )P ∂ ln(Li )Pi −µi
Maximum likelihood equations are then:= i (g −1 )0 (g(µi )) × Vyar(YXij = 0, fori ∂βji)
all j. Therefore, we get the equations, as a function of the βi ’s:
X ∂ ln(Li ) X
yi − g −1 (Xi β)=
Xij = 0.
(1.7)
(g −1 )0 (Xi β) × 0 −1 −1∂βj(b ) (g (Xi β))iiThese equations are not linear with respect to the βi s, and cannot be solved easily. As always
for complex equation, we use an iterative algorithm to find the solution. Most softwares, such
as R, use an iterative weighted least-squares method, see Section 2.5 of McCullagh and Nelder(1989).
Link functions for binary regression
Log-likelihood for canonical link Using the expression of the variance function and thecanonical logit function (g −1 (x) = 1+e1−x and (b0 )−1 (x) = x(1 − x)), Equation (1.7) becomes
0=

X
i

X
yi − 1+e1−ηi
e−ηi
×
X
=
(yi (1 + e−ηi ) − 1)Xij ,
ij
1
e−ηi
1 + e−ηi
−η
−η
1+e

i 1+e

i

i

for j = 1, . . . , p. These equations are called the likelihood equations. If we put it in a matrix
version, we get the so-called score equation
X T (Y − µ(β)) = 0.
Thus, the Fisher information matrix for β in the case of logit link is 2
∂ ln L
4
I(π) = −E= diag(πi (1 − πi )).
∂βj ∂βk
Since we use the maximum likelihood estimator, the estimator β̂ has the good property of
being asymptotically unbiased and Gaussian with variance matrix approximated by Fisher
information I(π(β̂)) ∗ .
∗. see subSection 4.4.4 of McCullagh and Nelder (1989).

75

Chapitre 1. Sur la nécessité d’un modèle de marché
Inverse link functions
1.0

Link functions

0.6
0.0

0.2

0.4

π = g(η)(−1)

50-5η = g(π)

inv. logit
inv. probit
inv. cloglog

0.8

logit
probitcompl. log-log

0.0

0.2

0.4

0.6

0.8

1.0

-4

-2

0

2

4

η

π

tel-00703797, version 2 - 7 Jun 2012

Figure 1.4: Link functions for binary regression

Univariate smoothing
In this paragraph, we present gradually some classic smoothing procedures, from the simplest to more complex methods. Probably the simplest method to get a smooth function is toregress a polynom on the whole data. Assuming observations are denoted by x1 , . . . , xn andy1 , . . . , yn , a multiple regression model is appropriate with
Y = α0 + α1 X + · · · + αp X p .
P
Using f (x) = i αi xi is clearly not flexible and a better tool has to be found. One way
to be more flexible in the smoothing is to subdivide the interval [min(x), max(x)] into K
segments. And then we can compute the average of the response variable Y on each segment[ck , ck+1 [. This is called the bin smoother in the literature. As shown on Hastie and Tibshirani(1990) Figure 2.1, this smoother is rather unsmooth.
Another way to find a smooth value at x, we can use points about x, in a symmetricneighborhood NS (x). Typically, we use the k nearest point at the left and k nearest at theright of x to compute the average of yi ’s. We haves(y|x) =1
CardNS (x)

X

yi ,

i∈NS (x)where the cardinal CardNS (x) does not necessarily equal to 2k + 1 if x is near the boundaries.
Again we do not show the result and refers the reader to Hastie and Tibshirani (1990) Figure
2.1. This method, called the running mean, takes better into account the variability of thedata. However we lose the smoothness of previous approches.
An extension of this approach is to fit the linear model y = µ + αx on the points (xi , yi )in the neighborhood (for i ∈ NS (x)). That is to say we have a serie of intercepts µ and slopes
α for all observations. We called this method the running line, which generalizes the runningmean, where α is forced to 0.
76

1.8. Appendix
Another enhancement is to weight the points in the regression (for xi ) inversely relative
to the distance to xi . Generally we use the tricube weight function
w(z) = (1 − |z|3 )3 11|z|<1 .
|x −x |

So the weight for xj when computing the smooth value of xi is w(zj ) with zj = i b j and b
the bandwith. Introduced by Cleveland (1979), this method is known as LOcally WEigthed
Smoothing Scatterplots (LOWESS). Other weight function can be used as long as it is asymmetric, decreasing from 0, strictly positive on ] − 1, 1[ and null elsewhere.
Loess - span = 0.5
120

120

Loess - span = 0.25

80

100

weight. running mean
weight. running line
weight. running square

0

20

40

60

distance60
200

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distance

80

100

weight. running mean
weight. running line
weight. running square

5

10

15

20

25

5

speed

10

15

20

25

speed

Figure 1.5: LOWESS

Summarising the LOWESS approach, we have w a weight function, d the degree of polynoms (1 for running line, 0 for running mean), f the span defined as the proportion of points
to use in the regression ∗ . On Figure 1.5, we plot different LOWESS methods for 2 spans.
Clearly the righthand side plot is smoother than the lefthand one. For a given span, as one
can expected the running mean method is more robust, unlike the running square which goesup and down easily.
A popular method for smoothing is the Kernel smoothing. Choosing a Kernel function(i.e. a symmetric density function) k, the associated smoother is

Pn
x−xi
yi
i=1 k
b
 ,
s(y|x) = Pn
x−xi
i=1 k
b
where b denoting the bandwidth is used to scale the data (x − xi )/b.
As suggested in Venables and Ripley (2002), Kernel smoothing can be seen as a local
weighted running mean approach. However, the power of the Kernel approach relies on theuse of more complex Kernel functions. Common Kernel functions are the standard normal
density function or the Epanechnikov function (a bisquare function). With Figure 1.6a, notsurprisingly, increasing the bandwidth increases the smoothness of the fitted curve.
The last and recent tool to fit a smooth curve is to use spline functions. The approachconsists in splitting the interval in K knots (t1 , . . . , tK ) and fit a polynom on each segment,
while imposing smooth conditions at the knots. One intuitive spline function is the polynomof third degree, since the smooth conditions (f 0 = f 00 = 0) are easily written done.
∗. closely linked to the bandwidth.

77

Chapitre 1. Sur la nécessité d’un modèle de marché

B-splines smoothing
120

120

Gaussian Kernel smoothing60

distance

80

100

degree=1
degree=2
degree=4

0

20

40

60
0

20

40

distance

80

100

bandwidth=1
bandwidth=2
bandwidth=3

5

10

15

20

25

speed

(a) Kernel smoothing

5

10

15

20

25

speed

(b) B-spline smoothing

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Figure 1.6: Kernel and B-spline smoothing

A crucial fact is that using K polynomials can be represented by a banded matrix, a bandfor each segment. Using the matrix representation emphasizes that we use a basis of functions
to approximate the function f . Many polynomial basis can be used (e.g. Bernstein polynoms
for Bézier curves).
One popular basis is the B-spline basis. They are defined recursively starting polynoms ofdegree 0 defined by Bi,0 (t) = 11ti ≤t<ti+1 ∗ and higher order Bi,d obtained by convex combination(with ti ’s increasing knots) of (Bi,d−1 )i ’s. To fit the data, we then minimize a penalized leastsquare and use some quantiles as the knots (for instance quartiles).
The thin plate regression uses a basis of thin plate (also known as polyharmonic functions)2m−d if d is odd. The smoothfunctions φmd (r) = αmd r2m−d log(r)Pn if d is even and αmd rfunctions is computed as s(y|x) = i=1 δi φmd (||x − xi ||2 ). A low-rank approximation of thissmooth function is then used to decrease the computational burden, see Wood (2003). Thismethod avoids the knot placement problems of traditional regression spline modelling.
On Figure 1.7, we plot smooth functions for the cars data with different smoothing
selection and basis functions (where default is the bottom-left figure). Both for the REML
and GCV criterion, the function basis has no influence on the estimation of the smooth function
f . Let us note that the estimated degrees of freedom are also very closed whatever the method
is.
On Figure 1.6b, we use the quantiles 20%, 40%, 60% and 80% as interior knots and threedifferent degrees. As one can expect, B-splines with high degrees better fit the data. To
conclude with this smoother presentation, all smoothers presented here are linear smoothers,
since they can be written by ŷ = Sy with S the smoother matrix (depending on observationsx).
PIRLS algorithm
The Penalized Iteratively Reweighted Least Square (PIRLS) has the following scheme
1. Initiate µ0i typically with yi .
e k)
2. Iterate while no change in deviance D(β∗. See theorem 1.5 of Steihaug (2007).

78

1.8. Appendix
Cubic regression splines (GCV)

15

20

25

40
20
0

s(speed,1.48)
5

10

15

20

25

5

10

15

20

speed

speed

Thin plate regression splines (REML)

Cubic regression splines (REML)

B-basis regression (REML)

10

15

20

25

40
20
0

s(speed,1.52)

-40

-40
5

25

-20

0
-20

-20

0

s(speed,1.53)

20

20

40

40

speed

-40

s(speed,1.53)

-40

-40
10

-20

-20

0

s(speed,1.49)

20

20
0

s(speed,1.49)

-20
-40

5

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B-basis regression (GCV)40

40Thin plate regression splines (GCV)

5

10

speed
(default method)

15

20

25

5

speed

10

15

20

25

speed

Figure 1.7: Additive model tests(a) compute the weight wik =1
g 0 (µk−1
)
i

q

ωi
,
V (µk−1)i(b) evaluate the pseudo datazik = g 0 (µk−1)(yi − µk−1
) + ηik−1 ,iiwith ηik−1 = g(µk−1),i(c) minimize with respect to β the least square objective2e||W (z − Xβ)||+X

λj β T Sj β,jwith W = diag(w1 , . . . , wn ). We get β k .
e k and µi = g −1 (η k ).
(d) prepare next estimate with η k = Xβ
i

1.8.2

R outputs

Bronchitis dataset
Let us study the example of Bronchitis data of Turner (2008). The data consists of 212
patients, on which we measure the presence/absence of bronchitis B for bron, the air pollution
level in the locality of residence P for poll and the number of cigarettes smoked per day C
for cigs, see Appendix 1.8.2.
79

Chapitre 1. Sur la nécessité d’un modèle de marché
Let us first regress the bronchitis indicator on all variables 
1 P 1 C1
B1 ..  ,
Y =  ...  and X =  ... ...
. 
1 P n Cn

Bn

with a logit link function. The regression summary is given below
Call: glm(formula = bron ~ 1 + cigs + poll, family = binomial)
Deviance Residuals:
Min
1Q
Median-2.4023 -0.5606 -0.4260

3Q-0.3155

Max
2.3594

tel-00703797, version 2 - 7 Jun 2012

Coefficients:
Estimate Std. Error z value Pr(>|z|)(Intercept) -10.08491
2.95100 -3.417 0.000632 ***cigs
0.21169
0.038135.552 2.83e-08 ***poll
0.13176
0.04895
2.692 0.007113 **--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Null deviance: 221.78
Residual deviance: 174.21

on 211on 209degrees of freedom
degrees of freedom - AIC: 180.21

So the GLM fit seems good because all variables (including intercept) are significant witha very low p-value. However the plot of residuals ∗ (see Figure 1.8a) against fitted values † isquite puzzling. Two distinct curves are shown: one for ill patients and the other for healthy
ones.
When categorizing the P variable, we lose information but we transform binary data into
binomial data. This makes the fit better on this aspect, see Figure 1.8b. So for the same data,with the same (significant) variables, the two analyses of residuals lead to different conclusions.
Hence, conclusions of residual analysis must be taken with great care.
GLM outputs of Section 1.3.1
See below the summary table with coefficients values, standard errors, z-statistics and pvalue. For confidentiality reason, all the deviance and AIC statistics shown in this paper havebeen scaled by the same positive coefficient.
Here follows the regression summary when variables are categorical.
Call: glm(formula = did_lapse ~ agepolgroup2 + priceratio:agegroup4 +priceratio * (gender + agevehgroup2 + prembeforegroup2),
family = binomial(), data = workdata)
Deviance Residuals:
Min
1Q
Median-3.1587 -0.6633 -0.6060

3Q-0.5193Max
2.8747

Coefficients:(Intercept)

Estimate Std. Error z value Pr(>|z|)-2.522477
0.120852 -20.873 < 2e-16 ***∗. Working residuals are ˆi = Yi − π̂i . Note that using other residual types, Pearson, Studentized, do notchange this behavior.
†. Fitted values are π̂i .

80

1.8. Appendix
Residuals vs. Fitted

0.5
0.0
-1.5

-2

-1.0

-0.5

(working) residuals

0-1(working) residuals

1

1.0

2

1.5

Residuals vs. Fitted

0.2

0.4

0.6

0.8

1.0

0.0

0.2

(a) Binary data

tel-00703797, version 2 - 7 Jun 2012

0.4

0.6

0.8

fitted values

fitted values(b) Binomial data

Figure 1.8: Analysis of residuals for binary regression

agepolgroup2(4,49]-0.153793
0.007270 -21.154 < 2e-16 ***genderMALE
0.681454
0.117045
5.822 5.81e-09 ***agevehgroup2(5,10]-0.684290
0.106741 -6.411 1.45e-10 ***agevehgroup2(10,99]-0.262674
0.101038 -2.600 0.00933 **prembeforegroup2(500,1e+03]-0.295837
0.137011 -2.159 0.03083 *prembeforegroup2(1e+03,1e+04]-0.923435
0.283603 -3.256 0.00113 **priceratio
1.018771
0.120903
8.426 < 2e-16 ***priceratio:agegroup4(35,60]-0.352247
0.008083 -43.579 < 2e-16 ***priceratio:agegroup4(60,99]-0.674209
0.011248 -59.938 < 2e-16 ***priceratio:genderMALE-0.607070
0.116885 -5.194 2.06e-07 ***priceratio:agevehgroup2(5,10]
0.956935
0.106426
8.992 < 2e-16 ***priceratio:agevehgroup2(10,99]
0.766736
0.100552
7.625 2.44e-14 ***priceratio:prembeforegroup2(500,1e+03]
0.569856
0.138151
4.125 3.71e-05 ***priceratio:prembeforegroup2(1e+03,1e+04] 1.340304
0.285123
4.701 2.59e-06 ***--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Null deviance: 53978
Residual deviance: 53258

on 56026
on 56012

degrees of freedom
degrees of freedom - AIC: 53261

Variable list for Subsection 1.3.2
The dataset is quite rich, therefore we have the detailed features of each policy. We write
below a subset of the available variables:– Policy: a dummy variable indicating the lapse, the policy age, the cover type (TPL, PC
or FC) and the product, the bonus class for PC and FC covers and the bonus evolution,– Policyholder: the policyholder age and the gender, the marital status and the job group,– Premium: the last year premium, the technical premium and the proposed premium,the payment frequency, the market premium, i.e. the tenth lowest NB premium for aparticular category,– Car: the mileage, the vehicle age, the car usage, the car class,– Cross-selling: the number of AXA contract in household, a dummy variable on household
81Chapitre 1. Sur la nécessité d’un modèle de marché
policy,– Claims: the claim amount, the claim number per year,– Agent: the cumulative rebate, the technical rebate, the age difference between the agent
and the policyholder.
GLM outputs for Subsection 1.3.2
The regression summary is given below
Call: glm(formula = lapse ~ lastprem_group2 + diff2tech + directdebit +product + nbclaim0708percust + vehiclage + householdNbPol +polholderage + maritalstatus2 + jobgroup2 + gender + polage +bonusevol2 + cover + priceratio:(lastprem_group2 + diff2tech +paymentfreq + glasscover + region2 + nbclaim08percust + householdNbPol +diffdriverPH7 + channel + typeclassTPL + bonusevol2), family = binomial("logit"),
data = idata)

tel-00703797, version 2 - 7 Jun 2012

Deviance Residuals:
Min
1Q
Median-3.1241 -0.4366 -0.3427

3Q-0.2402Max
3.3497

Coefficients:
Estimate Std. Error z value Pr(>|z|)(Intercept)-2.6456876 0.1822517 -14.517 < 2e-16 ***lastprem_group2(500,5e+03]
0.2008839 0.0952157
2.110 0.034878 *
diff2tech
6.9600797 0.7949370
8.756 < 2e-16 ***directdebit-0.0422104 0.0097823 -4.315 1.60e-05 ***productT1-0.1060909 0.0185019 -5.734 9.80e-09 ***productT2-1.0107703 0.0336376 -30.049 < 2e-16 ***productT3-0.3869057 0.0193135 -20.033 < 2e-16 ***nbclaim0708percust
0.0802148 0.0061759 12.988 < 2e-16 ***vehiclage-0.0172387 0.0010180 -16.934 < 2e-16 ***householdNbPol-0.1638354 0.0156899 -10.442 < 2e-16 ***polholderage-0.0106258 0.0003000 -35.417 < 2e-16 ***maritalstatus2b-0.1455813 0.0266586 -5.461 4.74e-08 ***maritalstatus2d-0.1088016 0.0119736 -9.087 < 2e-16 ***jobgroup2public-0.1529926 0.0079183 -19.321 < 2e-16 ***gender
-0.0739520 0.0077666 -9.522 < 2e-16 ***polage-0.0245842 0.0006806 -36.123 < 2e-16 ***bonusevol2up-down
1.9010618 0.1746998 10.882 < 2e-16 ***coverpartial compr.
0.0244814 0.0099107
2.470 0.013504 *coverTPL-0.0349025 0.0131839 -2.647 0.008112 **priceratio:lastprem_group2(0,500]
1.0418939 0.1840274
5.662 1.50e-08 ***priceratio:lastprem_group2(500,5e+03] 1.0246974 0.2000580
5.122 3.02e-07 ***priceratio:diff2tech-8.7933934 0.7867136 -11.177 < 2e-16 ***priceratio:paymentfreq-0.0136538 0.0010577 -12.909 < 2e-16 ***priceratio:glasscover-0.0865708 0.0139001 -6.228 4.72e-10 ***priceratio:region2_02-04-05-11
0.3608514 0.0207136 17.421 < 2e-16 ***priceratio:region2_03-09-10
0.1368317 0.0109978 12.442 < 2e-16 ***priceratio:region2_04-05-06-07
0.0935641 0.0103280
9.059 < 2e-16 ***priceratio:region2_12-13
0.3938396 0.0166819 23.609 < 2e-16 ***priceratio:region2_14-15-16
0.4424354 0.0160587 27.551 < 2e-16 ***priceratio:region2_17_
0.4812002 0.0243385 19.771 < 2e-16 ***priceratio:nbclaim08percust-0.0374916 0.0102707 -3.650 0.000262 ***priceratio:householdNbPol
0.0794544 0.0157004
5.061 4.18e-07 ***priceratio:diffdriverPH7learner 17
0.2768748 0.0578518
4.786 1.70e-06 ***priceratio:diffdriverPH7only partner
0.0976821 0.0077879 12.543 < 2e-16 ***priceratio:diffdriverPH7young drivers 0.1684370 0.0148135 11.371 < 2e-16 ***priceratio:channelbroker
0.3954067 0.0089064 44.396 < 2e-16 ***priceratio:channeldirect
0.3715832 0.0132034 28.143 < 2e-16 ***priceratio:typeclassTPL
0.0108773 0.0016963
6.412 1.43e-10 ***bonusevol2up-down:priceratio-1.8295464 0.1740807 -10.510 < 2e-16 ***--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 182

1.8. Appendix
Null deviance: 62279
Residual deviance: 58739

Group jObserved rj

AgentBrokerDirect8.840
9.245
11.837

on 121813
on 121809

Fitted n1j

degrees of freedom
degrees of freedom - AIC: 58747

Pnj

i=1 π̂i (pi )

7.714
8.896
9.005

Group jFC
PC
TPL

Observed rj

Fitted n1j

8.962
9.464
10.222Pnj

i=1 π̂i (pi )

7.492
8.846
12.522

Table 1.17: Lapse rates (%)∆1− (5%)π̂n (1)∆1+ (5%)∆1− (5%)π̂n (1)∆1+ (5%)-0.983-1.344-1.2468.652
9.123
12.341

1.23
1.841
1.143

-0.759-1.255
-1.18

8.732
9.422
11.597

0.75
1.299
1.268

tel-00703797, version 2 - 7 Jun 2012

Channel agent
Channel broker
Channel direct
ChannelOne fit by channel

One fit for all channels∆1− (5%)π̂n (1)∆1+ (5%)∆1− (5%)π̂n (1)∆1+ (5%)-0.926-0.635-0.9738.297
9.347
12.011

1.01
1.195
1.876

-0.622-0.714-0.8998.723
9.244
10.179

0.97
1.063
1.178

Coverage FC
Coverage PC
Coverage TPL
Coverage

One fit by coverage

One fit for all coverages

Table 1.18: Predicted lapse rates by channel and coverage

GLM outputs for Subsection 1.4.2
The regression summary without using the market proxy is given below.
Call: glm(formula = lapse ~ diff2tech + product2 + region2 + cumulrebate3 +nbclaim0608percust + isinsuredinhealth + isinsuredinlife +
vehiclage + householdNbPol + polholderage + maritalstatus2 +jobgroup2 + gender + typeclassTPL + bonusevol2 + priceratio:(diff2tech +paymentfreq + nbclaim08percust + nbclaim0608percust + nbclaim0708percust +isinsuredinaccident + householdNbPol + gender + typeclassTPL +bonusevol2), family = binomial("logit"), data = idata)
Deviance Residuals:
Min
1Q
Median-1.2613 -0.4104 -0.3482

3Q-0.2792

Max
3.1127

Coefficients:(Intercept)
diff2techproduct2T1
product2T2
region2_02-04-11
region2_05
region2_08-09
region2_10
region2_12-13
region2_14-15-16

Estimate Std. Error z value Pr(>|z|)-1.3513224 0.1034727 -13.060 < 2e-16 ***7.8972018 1.4461272
5.461 4.74e-08 ***-0.1275087 0.0321359 -3.968 7.25e-05 ***-0.2762145 0.0348857 -7.918 2.42e-15 ***0.2886433 0.0427885
6.746 1.52e-11 ***0.1878357 0.0277600
6.766 1.32e-11 ***0.0661201 0.0259573
2.547 0.010857 *
0.4506006 0.0906820
4.969 6.73e-07 ***0.3729663 0.0404406
9.223 < 2e-16 ***0.4591227 0.0406760 11.287 < 2e-16 ***83

tel-00703797, version 2 - 7 Jun 2012

Chapitre 1. Sur la nécessité d’un modèle de marché
region2_17
0.4469127 0.0609890
7.328 2.34e-13 ***cumulrebate3
0.0131512 0.0220328
0.597 0.550581
nbclaim0608percust
0.2538161 0.0861386
2.947 0.003213 **isinsuredinhealth-0.2117021 0.0737189 -2.872 0.004082 **isinsuredinlife-0.0904838 0.0403864 -2.240 0.025061 *
vehiclage-0.0418472 0.0024594 -17.015 < 2e-16 ***householdNbPol-0.1608386 0.0347312 -4.631 3.64e-06 ***polholderage-0.0142367 0.0007987 -17.824 < 2e-16 ***maritalstatus2b-0.2473493 0.0756033 -3.272 0.001069 **maritalstatus2d-0.1026557 0.0339761 -3.021 0.002516 **jobgroup2public-0.1564253 0.0212887 -7.348 2.01e-13 ***gender
-0.8573031 0.1748974 -4.902 9.50e-07 ***typeclassTPL-0.1127455 0.0320514 -3.518 0.000435 ***bonusevol2up-down
3.5129944 0.6064173
5.793 6.91e-09 ***priceratio:diff2tech-8.7833478 1.4474939 -6.068 1.30e-09 ***priceratio:paymentfreq-0.0314041 0.0025894 -12.128 < 2e-16 ***priceratio:nbclaim08percust-0.1047064 0.0383473 -2.730 0.006324 **priceratio:nbclaim0608percust -0.2269052 0.0913726 -2.483 0.013017 *priceratio:nbclaim0708percust
0.1429228 0.0365854
3.907 9.36e-05 ***priceratio:isinsuredinaccident -0.1395317 0.0505194 -2.762 0.005746 **priceratio:householdNbPol
0.0817417 0.0347087
2.355 0.018519 *priceratio:gender
0.7813407 0.1758044
4.444 8.81e-06 ***priceratio:typeclassTPL
0.1300911 0.0320887
4.054 5.03e-05 ***priceratio:bonusevol2up-down-3.3300573 0.6048578 -5.506 3.68e-08 ***--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Null deviance: 9151
Residual deviance: 8866

on 18893
on 18860

degrees of freedom
degrees of freedom - AIC: 8873

GLM outputs for Subsection 1.5.3

log L
AIC
Deg. of free.

Poissonzeroinfl. Poisson

zeroinfl. NB-2757145197
27-2837246797
26-2810546258
26

Table 1.19: Model adequacy for claim frequency of FC agent
Here follows the regression summary for zero-inflated NB distribution fit.
Call: zeroinfl(formula = nbclaim08FC ~ bonuspercentnew + bonusevol2 + lastprem_group2+ isinsuredinhealth + isinsuredinlife + isinsuredinaccident + polage + vehiclage + polholderage+ typeclassFC + diffdriverPH2 + gender | lastprem_group2 + diff2tech+ isinsuredinaccident + polage + polholderage, data = subdata, dist = "negbin")Pearson residuals:
Min
1Q Median
3QMax-0.6907 -0.3701 -0.3263 -0.2836 27.6615
Count model coefficients (negbin with log link):
Estimate Std. Error z value Pr(>|z|)(Intercept)-2.5053555 0.0463173 -54.091 < 2e-16 ***bonuspercentnew-0.0045481 0.0004473 -10.168 < 2e-16 ***bonusevol2up-down
0.2814031 0.0108215 26.004 < 2e-16 ***lastprem_group2(500,5e+03]
0.2867385 0.0125864 22.782 < 2e-16 ***isinsuredinhealth0.2536512 0.0129962 19.517 < 2e-16 ***isinsuredinlife
0.1500995 0.0101994 14.716 < 2e-16 ***isinsuredinaccident
0.1545091 0.0132603 11.652 < 2e-16 ***polage-0.0045662 0.0008071 -5.657 1.54e-08 ***vehiclage-0.0116381 0.0012641 -9.207 < 2e-16 ***polholderage
0.0052154 0.0006398
8.152 3.59e-16 ***84

1.8. Appendix
typeclassFC
0.0259947
diffdriverPH2all drivers > 24 0.1603390
diffdriverPH2commercial
0.5143316
diffdriverPH2learner 170.2501158
diffdriverPH2same-0.1661160
diffdriverPH2young drivers
0.2524112
gender
-0.0593577
Log(theta)
0.2848294

0.0012908 20.139 < 2e-16 ***0.0110572 14.501 < 2e-16 ***0.0338102 15.212 < 2e-16 ***0.0642750
3.891 9.97e-05 ***0.0111876 -14.848 < 2e-16 ***0.0158128 15.962 < 2e-16 ***0.0088454 -6.711 1.94e-11 ***0.0330418
8.620 < 2e-16 ***tel-00703797, version 2 - 7 Jun 2012

Zero-inflation model coefficients (binomial with logit link):
Estimate Std. Error z value Pr(>|z|)(Intercept)-7.299505
0.367536 -19.861 < 2e-16 ***lastprem_group2(500,5e+03] -0.484487
0.081025 -5.979 2.24e-09 ***diff2tech-7.214606
0.562964 -12.815 < 2e-16 ***isinsuredinaccident-0.256634
0.098848 -2.596 0.00942 **polage-0.011704
0.004260 -2.747 0.00601 **polholderage
0.094674
0.004658 20.326 < 2e-16 ***--Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1
Theta = 1.3295
Number of iterations in BFGS optimization: 77
Log-likelihood: -2.81e+04 on 24 Df

GLM outputs for Subsection 1.5.3
It follows the regression summary for ordered logistic regression for FC agent subset.
Call: polr(formula = deductibleFC3 ~ nbclaim08FC + ClaimNBhat + bonuspercentnew +lastprem_group2 + diff2tech + isinsuredinaccident + polage +vehiclage + polholderage + typeclassFC, data = subdata, Hess = TRUE,
method = "logistic")
Coefficients:
Value Std. Errort value pvalue
nbclaim08FC-2.900e-02 8.425e-03 -3.442e+00 0.180
ClaimNBhat
1.656e+00 9.401e-02 1.762e+01 0.036
bonuspercentnew
1.391e-02 3.357e-04 4.143e+01 0.015
lastprem_group2(500,5e+03] -3.026e-01 1.129e-02 -2.679e+01 0.024
diff2tech-1.720e+00 6.900e-02 -2.493e+01 0.026
isinsuredinaccident-2.964e-01 9.988e-03 -2.968e+01 0.021
polage-2.789e-02 3.594e-04 -7.759e+01 0.008
vehiclage
4.625e-02 1.056e-03 4.381e+01 0.015
polholderage-9.538e-03 2.921e-04 -3.266e+01 0.019
typeclassFC
1.169e-01 1.154e-03 1.013e+02 0.006
Intercepts:
Value
Std. Error t value
0|150-2.3565
0.0354-66.5322
150|300 -0.4060
0.0334-12.1655
300|500
4.17640.0341
122.4217
Residual Deviance: 664289.21
AIC: 664315.21

The GLM regression summary for lapse on the FC agent subset including deductible choiceprobabilities is available on request to the author.
GAM outputs for Subsection 1.6.2
Below we give the regression summary for the TPL agent dataset. Other summaries areavailable on request to the author.
85

Chapitre 1. Sur la nécessité d’un modèle de marché
Family: binomial - Link function: logit
Formula: lapse ~ product2 + region2 + cumulrebate3 + nbclaim0608percust +isinsuredinhealth + isinsuredinlife + vehiclage + householdNbPol +polholderage + maritalstatus2 + jobgroup2 + gender + bonusevol2 +priceratio:(paymentfreq + nbclaim08percust + nbclaim0608percust +nbclaim0708percust + isinsuredinaccident + bonusevol2) +s(priceratio, diff2tech) + s(priceratio, diff2top10agent) +s(priceratio, diff2top10direct) + s(priceratio, typeclassTPL)

tel-00703797, version 2 - 7 Jun 2012

Parametric coefficients:
Estimate Std. Error z value Pr(>|z|)(Intercept)-0.9881832 0.0744176 -13.279 < 2e-16 ***product2T1-0.2957239 0.0365839 -8.083 6.30e-16 ***product2T2-0.5888125 0.0439784 -13.389 < 2e-16 ***region2_02-04-11
0.2474500 0.0432128
5.726 1.03e-08 ***region2_05
0.1820856 0.0279436
6.516 7.21e-11 ***region2_08-09
0.0627676 0.0260959
2.405 0.016161 *region2_10
0.4597820 0.0908178
5.063 4.13e-07 ***region2_12-13
0.3600178 0.0408722
8.808 < 2e-16 ***region2_14-15-16
0.4440049 0.0377465 11.763 < 2e-16 ***cumulrebate3
0.1287561 0.0241245
5.337 9.44e-08 ***nbclaim0608percust
0.2144964 0.0968126
2.216 0.026720 *isinsuredinhealth-0.2018414 0.0739308 -2.730 0.006331 **isinsuredinlife-0.0978298 0.0405763 -2.411 0.015908 *
vehiclage-0.0367641 0.0025963 -14.160 < 2e-16 ***householdNbPol-0.0783881 0.0048668 -16.107 < 2e-16 ***polholderage-0.0150938 0.0008334 -18.111 < 2e-16 ***maritalstatus2b-0.2629597 0.0760885 -3.456 0.000548 ***maritalstatus2d-0.1017553 0.0341228 -2.982 0.002863 **jobgroup2public-0.1161175 0.0217312 -5.343 9.12e-08 ***gender
-0.0790535 0.0209269 -3.778 0.000158 ***bonusevol2up-down
7.4827223 1.0625789
7.042 1.89e-12 ***priceratio:paymentfreq-0.0343715 0.0026481 -12.980 < 2e-16 ***priceratio:nbclaim08percust-0.0893319 0.0393116 -2.272 0.023062 *priceratio:nbclaim0608percust -0.2010502 0.1016136 -1.979 0.047864 *priceratio:nbclaim0708percust
0.1538349 0.0369590
4.162 3.15e-05 ***priceratio:isinsuredinaccident -0.1409923 0.0508941 -2.770 0.005600 **priceratio:bonusevol2up-down-7.2677291 1.0573222 -6.874 6.26e-12 ***--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Approximate significance of smooth terms:
edf Ref.df Chi.sq p-values(priceratio,diff2tech)
12.440 16.687 113.56 < 2e-16 ***s(priceratio,diff2top10agent)
8.901 12.069 29.36 0.00361 **s(priceratio,diff2top10direct) 8.177 11.277 18.63 0.07569 .
s(priceratio,typeclassTPL)
4.160 5.687 43.91 5.43e-08 ***--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R-sq.(adj) =REML score =0.0176
Deviance explained = 3.46%
44028 Scale est. = 1
n = 187733

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Théorie des jeux

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Chapitre 2

Theorie des jeux et cycles de marché—
A game-theoretic approach to non-life
insurance market cycles

Little by little, one travels far.
J.R.R. Tolkien (1892–1973)

Ce chapitre se base sur l’article Dutang et al. (2012) dont une partie est déjà soumise àl’European Journal of Operation Research.

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2.1

Introduction

Insurance market cycles and the study of their causes have been puzzling actuaries formany years. Feldblum (2001) discusses four main causes that could explain the presence ofunderwriting through their aggregate effect. These causes are (i) actuarial pricing procedure,(ii) underwriting philosophy, (iii) interest rate fluctuations and (iv) competitive strategies. He
compares contributions through out the 20th century on the topic, see also Markham (2007)
for an overview.
Actuarial pricing procedures are subject to claim cost uncertainty, information lag (dueto accouting, regulatory and legal standards). Such effects are likely to generate fluctuations
around an equilibrium price, when extrapolating premiums, see, e.g., Venezian (1985); Cummins and Outreville (1987). In addition, a hard behavior of underwriters combined with alack of coordination is an extra recipe for underwriting cycles. In particular, price policies
cannot be sold independently of the market premium, but neither can the market premium be
driven by one’s individual actions. This is called underwriting philosophy by Feldblum (2001),
and is also noticed in Jablonowski (1985), who assumes (i) insurers do not make decisions in
isolation from other firms in the market, and (ii) profit maximization is not the exclusive,or even the most important, motivation of insurers. Interest rate deviations further increase
the frequency and the amplitude of market cycles, as they have an impact on the investment
result and (indirectly) on the maximum rebate that underwriters can afford to keep presumably low-risk customers. Fields and Venezian (1989) were among the first to demonstrate thiseffect. Finally, the competition level on most mature insurance markets are such that any
increase in market share can only be carried out by price decrease ∗ (due to very little productdifferentiation). Coupled with capital constraints (e.g. Gron (1994)) and price inelasticity,
insurers are forced not to deviate too much from market trends.
On a different level, basic economic models suggest that the equilibrium premium is themarginal cost, as any upward deviation from this equilibrium will result in losing all the policies
in the next period. This theory would imply that all insurers price at the market premium
However, in practice customers do not move from a insurer to a cheaper insurer as swiftly
as economic models anticipate. There is an inertia of the insurance demand, preventing all
insured to shop arround for the cheapest insurer when their premium is slightly higher than themarket premium. So customer behavior is much more complicated. In addition to customer
loyalty, Feldblum (2001) points out that it is difficult for a new insurer to enter successfully
into the non-life insurance market.
More refined economic models focus on moral hazard and adverse selection. The celebrated
model of Rothschild and Stiglitz (see Rothschild and Stiglitz (1976)) deals with a utility-based
agent framework where insureds have private information on their own risk. Insurers provide
a menu of contracts (a pair of premium and deductible) and high-risk individuals choose
full coverage, whereas low-risk individuals are more attracted by partial coverage. Note thatthe equilibrium price may not exist if all insurers offer just one type of contract. Picard(2009) considers an extension by allowing insurers to offer participating contracts (such as
mutual-type contracts). This feature guarantees the existence of an equilibrium, which forces(rational) insureds to reveal their risk level. An important source of applications of suchmodels is health insurance where moral hazard and adverse selection play a major role, see,∗. The hunger for market share is driven by the resulting reduction of claim uncertainty is increasing thepolicy number, which is motivated by the law of large numbers.

942.1. Introduction
e.g., Geoffard et al. (1998), Wambach (2000); Mimra and Wambach (2010) and Picard (2009).

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However, the economic models mentioned above can not address the insurance market
cycle dynamics, so that one has to look for further alternatives. Taylor (1986, 1987) deals
with discrete-time underwriting strategies of insurers and provides first attempts to model
strategic responses to the market, see also, Kliger and Levikson (1998); Emms et al. (2007);Moreno-Codina and Gomez-Alvado (2008). The main pitfall of the optimal control approachis that it focuses on one single insurer and thus implicitly assumes that insurers are playing agame against an impersonal market player and the market price is independent of their own
actions.
In this paper, we want to investigate the suitability of game theory for insurance market
modelling. The use of game theory in actuarial science has a long history dating back to K.
Borch and J. Lemaire, who mainly used cooperative games to model risk transfer between
insurer and reinsurer, see, e.g., Borch (1960, 1975), Lemaire and Quairière (1986). Bühlmann(1984) and Golubin (2006) also studied risk transfer with cooperative tools. Among the articles
using noncooperative game theory to model the non-life insurance market, Bertrand oligopoly
models are studied by Polborn (1998), Rees et al. (1999), Hardelin and de Forge (2009). Powers
and Shubik (1998, 2006) also study scale effects of the number of insurers and the optimal
number of reinsurers in a market model having a central clearing house. More recently, Taksar
and Zeng (2011) study non-proportional reinsurance with stochastic continuous-time games.
Demgne (2010) seems to be the first study from a game theory point of view of (re)insurance
market cycles. She uses well known economic models: pure monopoly, Cournot’s oligopoly(i.e. war of quantity), Bertrand’s oligopoly (i.e. war of price) and Stackelberg (leader/followergame). For all these models, she tests various scenarios and checks the consistency of model
outputs with reinsurance reality.
Finally, in many ruin theory models, one assumes that the portfolio size remains constantover time (see, e.g., Asmussen and Albrecher (2010) for a recent survey). Non-homogeneous
claim arrival processes have usually been studied in the context of modelling catastrophe
events. More recently, non-constant portfolio size has been considered, see, e.g., Trufin et al.
(2009) and the references therein. Malinovskii (2010) uses a ruin framework to analyze different
situations for an insurer in its behavior against the market.
This paper aims to model competition and market cycles in non-life insurance markets
with noncooperative game theory in order to extend the player-vs-market reasoning of Taylor(1986, 1987)’s models. A main contribution is to show that incorporating competition whensetting premiums leads to a significant deviation from the actuarial premium and from a
one-player optimized premium. Furthermore, the repeated game models a rational behavior
of insurers in setting premium in a competitive environment, although the resulting market
premium is cyclical. The rest of the paper is organized as follows. Section 2.2 introduces
a one-period noncooperative game. Existence and uniqueness of premium equilibrium areestablished. Section 2.3 relaxes assumptions on objective and constraint components of the
one-period model. The existence of premium equilibrium is still guaranteed, but uniquenessmay not hold. A reasonable choice of an equilibrium is proposed in this situation. Section 2.4
then works out with the repeated version of the one-period model of Section 2.3. A conclusionand perspectives are given in Section 2.5.
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2.2

A one-period model

In a first attempt to model the non-life insurance market cycle, we ignore for simplicity
investment results, although they play a key role for third-part liability insurance product for
which interest rate fluctuations have a big impact, as well as loss reserving. So, our framework
is consistent only for short-tail business.
Consider I insurers competing in a market of n policyholders with one-year contracts(where n is considered constant). The “game” for insurers is to sell policies to this insured
market by setting the premium. Let (x1 , . . . , xI ) ∈ RI be a price vector, with xj representing
premium of insurer j. Once the premium is set by all insurers, the insureds choose to renew
or to lapse from their current insurer. Then, insurers pay claims, according to their portfolio
size, during the coverage year. At the end of the year, underwriting results are determined,
and insurer capital is updated: some insurer may be bankrupt.
In the next subsections, we present the four components of the game: a lapse model, a loss
model, an objective function and a solvency constraint function. In the sequel, a subscriptj ∈ {1, . . . , I} will always denote a player index whereas a subscript i ∈ {1, . . . , n} denotes aninsured index.

2.2.1

Lapse model

Being with current insurer j, the insurer choice Ci of insured i for the next period follows an I-dimensional multinomial distribution MI (1, pj→ ) with probability vector pj→ =(pj→1 , . . . , pj→I ) summing to 1. The probability mass function is given by P (Ci = k | j) =pj→k . It seems natural and it has been verified empirically that the probability to choose an
insurer is highly influenced by the previous period choice. In other words, the probability tolapse pj→k with k 6= j is generally much lower than the probability to renew pj→j . To ourknowledge, only the UK market shows lapse rates above 50%. Those probabilities have todepend on the premium xj , xk proposed by insurer j and k, respectively.
Assume at the beginning of the game that the insurer portfolio sizes are nj (such thatPI
j=1 nj = n). The portfolio size Nj (x) of insurer j for the next period is a random variable
determined by the sum of renewed policies and businesses coming from other insurers. Hence,
Nj (x) = Bjj (x) +IX

Bkj (x).

k=1,k6=jNj (x) is a sum of I independent binomial variables (Bkj )k where Bkj has parameters B(nk , pk→j (x)).
In the economics literature, pj→k is considered in the framework of discrete choice models. In the random utility maximization setting, McFadden (1981) or Anderson et al. (1989)
propose multinomial logit and probit probability choice models. In this paper, we choose amultinomial logit model, since the probit link function does not really enhance the choicemodel despite its additional complexity. Working with unordered choices, we arbitrarily set
the insurer reference category for pj→k to j, the current insurer. We define the probability
for a customer to go from insurer j to k given the price vector x by the following multinomial
logit model
1
if j = k,
P f (x ,x )
 1+ e j j ll6=jpj→k = lgkj (x) =(2.1)
efj (xj ,xk )
if j 6= k, 1+ P efj (xj ,xl )
l6=j

96

2.2. A one-period model
where the sum is taken over the set {1, . . . , I} and fj is a price sensitivity function. In thefollowing, we consider two types of price functionsxj
f¯j (xj , xl ) = µj + αjand f˜j (xj , xl ) = µ̃j + α̃j (xj − xl ).
xl

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The first function f¯j assumes a price sensitivity with the ratio of the proposed premium xj andcompetitor premium xl , whereas f˜j works with the premium difference xj − xl . Parametersµj , αj represent a base lapse level and price sensitivity. We assume that
Pinsurance products
display positive price-elasiticity of demand αj > 0. One can check that k lgkj (x) = 1.
The above expression can be rewritten as
lgkj (x) = lgjj (x) δjk + (1 − δjk )efj (xj ,xk ) ,with δij denoting the Kronecker product. It is difficult to derive general properties of the
distribution of a sum of binomial variables with different probabilities, except when the size
parameters nj are reasonably large, in which case the normal approximation is appropriate.
With this insurer choice probability, the expected portfolio size of insurer j reduces toXN̂j (x) = nj × lgjj (x) +nl × lgjl (x),l6=jwhere nj denotes the last year portfolio size of insurer j.

2.2.2

Loss model

Let Yi be the aggregate loss of policy i during the coverage period. We assume no adverse
selection among insured of any insurers, i.e. Yi are independent and identically distributed(i.i.d.) random variables, ∀i = 1, . . . , n. As already mentioned, we focus on short-tail business.
Thus, we assume a simple frequency – average severity loss model
Yi =

Mi
X

Zi,l ,

l=1

where the claim number Mi is independent from the claim severities Zi,l . Therefore, theaggregate claim amount for insurer j is
Nj (x)

Sj (x) =X

Nj (x) Mi

Yi =

i=1

X X

Zi,l ,

i=1 l=1

where Nj (x) is the portfolio size of insurer j given the price vector x. We consider two maini.i.d.

i.i.d.

cases of the loss models: (i) Poisson-lognormal model: Mi ∼ P(λ) and Zi,l ∼ LN (µ1 , σ12 ),i.i.d.

i.i.d.

(ii) negative binomial-lognormal model: Mi ∼ N B(r, p) and Zi,l ∼ LN (µ2 , σ22 ). We choosea different parameter set for the claim severity distribution, because if we want a significant
difference between the two loss models, changing only the claim number distribution does notreveal sufficient. These two instances of the frequency-average severity model are such the
PNj (x)aggregate claim amount Sj (x) = i=1
Yi is still a compound distribution of the same kind,since Yi are assumed i.i.d. random variables.
PMf (x)Hence, the insurer aggreggate claim amount Sj (x) is a compound distribution l=1j Zl
such that the claim number Mj and claim severity Zl follow
97Chapitre 2. Theorie des jeux et cycles de marché
fj (x) ∼ P(Nj (x)λ) and Zl i.i.d.
– a Poisson-lognormal with M∼ LN (µ1 , σ12 ),fj (x) ∼ N B(Nj (x)r, p) and Zl i.i.d.
– a negative binomial-lognormal with M∼ LN (µ2 , σ22 ).
In the numerical applications, these two loss models are denoted PLN and NBLN, respectively.
In addition to these two base loss models, we will also test a variation of the Negative
binomial model, in which the claim numbers are correlated among insurers. Concretely,– draw u from a uniform distribution U(0, 1).
– set λj = Qj (u) where Qj is the quantile function a random Gamma variable with shapepparameter Nj (x)r and rate parameter 1−p.
– draw a compound variable Sj with claim frequency P(λj ) and claim frequency LN (µ2 , σ22 ).
Since a Poisson-Gamma mixture follows a negative binomial distribution, the resulting margingal claim number distribution for insurer j is a Negative Binomial distribution with parameters
BN (Nj (x)r, p). However, now the loss frequency among insurers is comonotonic. We will denote this model by PGLN in the numerical applications.

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2.2.3

Objective function

In the two previous subsections, we presented two components of the insurance markets:the lapse model (how insureds react to premium changes) and the loss model (how insureds
face claims). We now turn our attention to the underwriting strategy of insurers, i.e. on how
they set premiums.
In Subsection 2.2.1, we assumed that price elasticity of demand for the insurance productis positive. Thus, if the whole market underwrites at a loss, any actions of a particular insurer
to get back to profitability will result in a reduction of his business volume. This has twoconsequences for possible choice of objective functions: (i) it should use a decreasing demand
function of price xj given the competitors price x−j = (x1 , . . . , xj−1 , xj+1 , . . . , xI ) and (ii) it
should depend on an assessment of the insurer break-even premium per unit of exposure πj .
We suppose that insurer j maximizes the expected profit of renewing policies defined as



nj
xj
Oj (x) =1 − βj−1(xj − πj ) ,(2.2)
n
mj (x)
where πj is the break-even premium j and mj (x) is a market premium proxy. The objectivefunction Oj defined as the product of a demand function and an expected profit per policy
represent a company-wide expected profit. Oj targets renewal business and does not take
into account new business explicitly. In addition to focusing on renewal business only, theobjective function locally approximates the true insurer choice probability lgjj presented in
Subsection 2.2.1. However, since the demand function Dj (x) = nj /n(1 − βj (xj /mj (x) − 1)) is
not restricted to [0,1], demand Dj can exceed the current market share nj /n, but profit per
policy will decline when the premium decreases. Thus, maximising the objective function Oj
leads to a trade-off between increasing premium to favour higher projected profit margins anddecreasing premium to defend the current market share. Note that Oj has the nice property
to be infinitely differentiable.
Parameter πj corresponds to the estimated mean loss of insurer j and is expressed as
πj = ωj aj,0 + (1 − ωj )m0
where aj,0 is the actuarial premium defined as empirical average loss per policy over a certain
number of past years, the market premium m0 is defined as the past average of the market
982.2. A one-period model
premium weighted by the gross written premium and ωj ∈ [0, 1] is the credibility factor of
insurer j. If insurer j is the market leader, then ωj should be close to 1, whereas when insurer
j is a follower, ωj should be close to 0. Note that πj takes into account expenses implicitly
via the actuarial and the market premiums.
The market proxy used in Equation (2.2) is the mean price of the other competitors
mj (x) =

1 X
xk .
I −1
k6=j

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The market proxy aims to assess other insurer premiums without specifically targeting one
competitor. By excluding the price xj to compute the market proxy mj (x), we suppose
insurer j is not dominant in the market. If, for example, insurer j underwrites 80% of thetotal premium available in the market, mj (x) will not be appropriate, but in such cases themarket competition is low. We could have used the minimum of the competitors’ premium,but then mj (x) would not have been a continuous function of the price vector x. Furthermore,insurer j does not necessarily take into account to be the cheapest insurer.

2.2.4

Solvency constraint function

In addition to maximizing a certain objective function, insurers must satisfy a solvency
constraint imposed by the regulator. Currently, European insurers report their solvency margin in the Solvency I framework, based on the maximum of a percentage of gross written
premium and aggregate claim mean. According to Derien (2010), a non-life insurer computes
its solvency margin as
SM = max(18% × GWP, 26% × AC) × max(50%, AC net of reins/AC gross of reins),
where GWP denotes the gross written premium and AC the aggregate claim mean ∗ . Discarding reinsurance, the Solvency I framework leads to a solvency margin
SM = max(9% × GWP, 13% × AC).
This approach is not really satisfactory, as it does not take into account the risk volalityof underwritten business. Since 2005, actuaries are well busy with the upcoming Solvency
II framework. In this new framework, the quantitative part leads to the computation of
two capital values, both based on the difference between a certain quantile and the mean of
the aggregate loss. The solvency capital requirement (SCR) is based on the 99.5%-quantile,whereas the minimum capital requirement (MCR) is based on the 85%-quantile.
In our game context, we want to avoid the simplistic Solvency I framework, but still wantto keep the tractablity for the SCR computation rule. We recall that the aggregate claim
amount is assumed to be a frequency - average severity model, i.e. Cat-losses are ignored.
A simplification is to approximate a q-quantile Q(n, q) of aggregate claim amount of n i.i.d.
√risks by a bilinear function of n and n√Q(n, q) = E(Y )n + kq σ(Y ) n,

(2.3)∗. The percentages 18% and 26% are replaced respectively by 16% and 23% when the GWP exceeds 57.5
Meur or AC exceeds 40.3 Meur.

99

Chapitre 2. Theorie des jeux et cycles de marchétel-00703797, version 2 - 7 Jun 2012

where the coefficient kq has to be determined and Y is the generic individual claim severity
variable. The first term corresponds to the mean of the aggregate claim amount, while thesecond term is related to standard deviation.
Three methods have been tested to compute the solvency coefficient kq : (i) a normal
approximation kqN = Φ−1 (q), where Φ is the distribution function of the standard normal
distribution, (ii) a simulation procedure with 105 sample size to get kqS as the empiricalquantile and (iii) the Panjer recursion to compute the aggregate claim quantile kqP ∗ .
While the normal approximation is based on the first two moments of the distribution
only, simulation and Panjer methods need to have assumptions on claim frequency and claim
severity distributions: we use the PLN and NBLN models defined in Subsection 2.2.2. We alsoneed a risk number n. In Table 2.1, we report solvency coefficients for n = 1000 risks. Panjer
and simulation methods appear twice since two loss models (PLN and NBLN) are tested.
prob qkqN

kqP -PLN

kqP -NBLN

kqS -PLN

kqS -NBLN

0.75
0.8
0.85
0.9
0.95
0.99
0.995

0.674
0.842
1.036
1.282
1.645
2.326
2.576

1.251
1.431
1.642
1.912
2.321
3.117
3.419

0.913
1.104
1.332
1.627
2.083
2.997
3.352

0.649
0.829
1.029
1.299
1.695
2.475
2.777

0.627
0.812
1.03
1.312
1.759
2.633
2.976

Table 2.1: Solvency coefficient k
Numerical experiments show that the normal approximation is less conservative for highquantiles (i.e. kqN < kqP ) when the claim number follows a negative binomial distribution,and the reverse for the Poisson distribution. Based on this study, we choose to approximate
quantiles at 85% and 99.5% levels with coefficients k85 = 1 and k995 = 3.
Thus, using the approximation (2.3), the solvency capital requirement SCR is deduced as√
SCRq ≈ kq σ(Y ) n,
which is more complex than the Solvency I framework. Numerical investigations show that
the Solvency I requirement corresponds to a 75% quantile. Therefore, we decide to choose theadapted solvency constraint function
gj1 (xj ) =Kj + nj (xj − πj )(1 − ej )− 1,√k995 σ(Y ) nj

(2.4)

where k995 is the solvency coefficient and ej denotes the expense rate as a percentage of grosswritten premium. The numerator corresponds to the sum of current capital Kj and expected
profit on the in-force portfolio (without taking into account new business). It is easy to see√that the constraint gj1 (x) ≥ 0, is equivalent to Kj + nj (xj − πj )(1 − ej ) ≥ k995 σ(Y ) nj , but
gj1 is normalized with respect to capital, providing a better numerical stability.
In addition to the solvency constraint, we need to impose bounds on the possible premium.
A first choice could be simple linear constraints as xj − x ≥ 0 and x − xj ≥ 0, where x∗. See, e.g., Theorem 12.4.3 of Bowers et al. (1997). Panjer recursion requires that the claim distribution is
discrete. So before using Panjer algorithm, we use a lower discretization of the lognormal claim distribution.

100

2.2. A one-period modeland x represent the minimum and the maximum premium, respectively. But the following
reformulation is equivalent and numerically more stable:
gj2 (xj ) = 1 − e−(xj −x) ≥ 0 and gj3 (xj ) = 1 − e−(x−xj ) ≥ 0.
The minimum premium x could be justified by a prudent point of view by regulators while
the maximum premium x could be set, e.g., by a consumer right defense association. In thesequel, we set x = E(Y )/(1 − emin ) < x = 3E(Y ), where emin is the minimum expense rate.
Overall, the constraint function gj (xj ) ≥ 0 is equivalent to

√
(2.5)
{xj , gj (xj ) ≥ 0} = xj ∈ [x, x], Kj + nj (xj − πj )(1 − ej ) ≥ k995 σ(Y ) nj .

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2.2.5

Game sequence

For noncooperative games, there are two main solution concepts: Nash equilibrium andStackelberg equilibrium. The Nash equilibrium assumes player actions are taken simultaneously while for the Stackelberg equilibrium actions take place sequentially, see, e.g., Fudenberg
and Tirole (1991); Osborne and Rubinstein (2006). In our setting, we consider the Nash equilibrium as the most appropriate concept. We give below the definition of a generalized Nashequilibrium extending the Nash equilibrium with constraint functions.
Definition. For a game with I players, with payoff functions Oj and constraint function gj ,a generalized Nash equilibrium is a vector x? = (x?1 , . . . , x?I ) such that for all j = 1, . . . , I, x?jsolves the subproblem
max Oj (xj , x?−j ) s.t. gj (xj , x?−j ) ≥ 0.
xj

where xj and x−j denote action of player j and the other players’ action, respectively.
A (generalized) Nash equilibrium is interpreted as a point at which no player can profitably
deviate, given the actions of the other players. When each player’s strategy set does not dependon the other players’ strategies, a generalized Nash equilibrium reduces to a standard Nashequilibrium. Our game is a Nash equilibrium problem since our constraint functions gj definedin Equation (2.4) depend on the price xj only.
The game sequence is given as follows:(i) Insurers set their premium according to a generalized Nash equilibrium x? , solving for
all j ∈ {1, . . . , I}
x−j 7→ arg max Oj (xj , x−j ).
xj ,gj (xj )≥0(ii) Insureds randomly choose their new insurer according to probabilities pk→j (x? ): we
get Nj (x).
(iii) For the one-year coverage, claims are random according to a frequency-average severity model relative to the portfolio size Nj (x? ).
(iv) Finally the underwriting result is determined by U Wj (x? ) = Nj (x? )x?j (1−ej )−Sj (x? ),where ej denotes the expense rate.
If the solvency requirement is not fullfilled, in Solvency I, the regulator response is immediate: depending on the insolvency severity, regulators can withdraw the authorisation tounderwrite new business or even force the company to go run-off or to sell part of its portfolio.
In Solvency II, this happens only when the MCR level is not met. There is a buffer between
101Chapitre 2. Theorie des jeux et cycles de marché
MCR and SCR where regulators impose some specific actions to help returning to the SCRlevel.
In our game, we choose to remove players which have a capital below MCR and to authorize
players to continue underwriting when capital is between the MCR and the SCR. Note thatthe constraint function will be active when computing the Nash equilibrium, if the capital isbetween the MCR and SCR.

2.2.6

Properties of the premium equilibrium

In this subsection, we investigate the properties of premium equilibrium. We start byshowing existence and uniqueness of a Nash equilibrium. Then, we focus on the sensitivity
analysis on model parameters of such equilibrium.

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Proposition 2.2.1. The I-player insurance game with objective function and solvency constraint function defined in Equations (2.2) and (2.5), respectively, admits a unique (Nash)premium equilibrium.
Proof. The strategy set is R = [x, x]I , which is nonempty, convex and compact. Givenx−j ∈ [x, x], the function xj 7→ Oj (x) is a quadratic function with second-degree term−βj x2j /mj (x) < 0 up to a constant nj /n. Thus, this function is (strictly) concave. Moreover, for all players, the constraint functions gj1 are linear functions, hence also concave. By
Theorem 1 of Rosen (1965), the game admits a Nash equilibrium, i.e. existence is guaranteed.
By Theorem 2 of Rosen (1965), uniqueness is verified if we have the following inequality for
all x, y ∈ R,
I
I
X
X
rj (xj − yj )∇xj Oj (y) +rj (yj − xj )∇xj Oj (x) > 0,
(2.6)
j=1

j=1

for some r ∈ RI with strictly positive components ri > 0. As the function xj 7→ Oj (x)is a strictly concave and differentiable function for all x−j , we have ∇xj Oj (x)(yj − xj ) >Oj (y) − Oj (x) and equivalently ∇xj Oj (y)(xj − yj ) > Oj (x) − Oj (y). Thus,(xj − yj )∇xj Oj (y) + (yj − xj )∇xj Oj (x) > Oj (y) − Oj (x) + Oj (x) − Oj (y) = 0.
Taking r = 1, equation (2.6) is verified.
Proposition 2.2.2. Let x? be the premium equilibrium of the I-player insurance game. For
each player j, if x?j ∈]x, x[, the player equilibrium x?j depends on the parameters in the following
way: it increases with break even premium πj , solvency coefficient k995 , loss volatility σ(Y ),expense rate ej and decreases with sensitivity parameter βj and capital Kj . Otherwise whenx?j = x or x, the premium equilibrium is independent of any parameters.
Proof. The premium equilibrium x?j of insurer j solves the necessary Karush-Kuhn-Tucker
conditions:
X j?
∇xj Oj (x? ) +λl ∇xj gjl (x?j ) = 0,
1≤l≤3
(2.7)
j?
?
? T j?
0 ≤ λ , gj (xj ) ≥ 0, gj (xj ) λ = 0,102

2.2. A one-period model
where λj? ∈ R3 are Lagrange multipliers, see, e.g., Facchinei and Kanzow (2009). In the lastpart of equation (2.7), gj (x?j )T λj? = 0 is the complementarity equation implying that the lconstraint gjl is either active (gjl (x?j ) = 0) or inactive (gjl (x?j ) > 0), but λj?
l = 0.
j?
We suppose that x?j ∈]x, x[. Hence, λj?
2 = λ3 = 0. There are two cases: either the solvency
constraint gj1 is active or not. Let us assume the solvency constraint is inactive. Insurer j’s
premium equilibrium verifies ∇xj Oj (x? ) = 0, i.e.

nj
n



x?j
πj
1 − 2βj+ βj + βj?
mj (x )
mj (x? )


= 0.

(2.8)

Let xjy be the premium vector with the j component being y, i.e. xjy = (x1 , . . . , xj−1 , y, xj+1 , . . . , xI ).
We denote by z a parameter of interest and define the function F as

tel-00703797, version 2 - 7 Jun 2012

Fxj (z, y) =∂Oj j(x , z),∂xj ywhere the objective function depends (also) on the interest parameter z. Equation (2.8) canbe rewritten as Fxj? (z, x?j ) = 0.
By the continuous differentiability of F with respect to z and y and the fact that Fxj (z, y) =0 has at least one solution (z0 , y0 ), we can invoke the implicit function theorem, see Appendix
2.6.1. So there exists a function ϕ defined in a neighborhood of (z0 , y0 ) such that Fxj (z, ϕ(z)) =jx
0 and ϕ(z0 ) = y0 . Furthermore, if ∂F∂y (z0 , y0 ) 6= 0, the derivative of ϕ is given by∂Fxj(z, y).
ϕ (z) = − ∂zj
∂Fx
(z,
y)
∂y
y=ϕ(z)
0In our case, we have
∂ 2 Oj jnj∂Fxj(z, y) =< 0.
(xy , z) = −2αj2∂ynmj (x)∂xj
As a consequence, the sign of ϕ0 is simply
!
j

∂F
x
sign ϕ0 (z) = sign(z, ϕ(z)) .
∂z
Let us consider z = πj . We havenj β j∂Fxj(z, y) => 0.
∂z
nmj (x)Thus, the function πj 7→ x?j (πj ) is increasing.
Let z be the sensitivity coefficient βj . We havenj
∂Fxj(z, y) =
∂z
n


−2βj

πj
y
+1+
mj (x)
mj (x)

.
103

Chapitre 2. Theorie des jeux et cycles de marché
Using Fxj (z, ϕ(z)) = 0, it leads to
nj −1
∂Fxj
(z, ϕ(z)) =
< 0.
∂z
n z

tel-00703797, version 2 - 7 Jun 2012

Thus, the function βj 7→ x?j (βj ) is decreasing. In such a case of an inactive constraint, thepremium equilibrium is independent of the initial portfolio size nj .
When the solvency constraint is active, the premium equilibrium x?j verifies gj1 (x?j ) = 0,i.e.
√k995 σ(Y ) nj − Kj?
xj = πj +.
(2.9)
nj (1 − ej )Here, the implicit function theorem is not necessary since x?j does not depend on x?−j . We
deduce that x?j is an increasing function of πj , k995 , σ(Y ), ej and a decreasing function Kj .
The function nj 7→ x?j (nj ) is not necessarily monotone. Let z be nj . Differentiating
Equation (2.9) with respect to z, we getkσ(Y ) Kj
1−,+√
ϕ0 (z) = 3/22zz (1 − ej )whose sign depends on the value of the other parameters.

2.2.7

Numerical illustration

All numerical applications are carried out with the R software, R Core Team (2012). Pleaserefer to Appendix 2.6.1 for computation details.
Base parameters
We consider a three-player game operating a 10 000-customer insurance market, i.e. n =10000, I = 3. To ensure that insurers already have underwritten business, we provide d-year
history for portfolio size, where d = 3. Although we provide a 3-year history in this section,
we only consider the one-period equilibrium. So only the value at year 0 matters.
Insurer portfolio size nj (t)’s are given in Table 2.2. The portfolio size is chosen such that
player 1 is the leader, player 2 the challenger and player 3 the outsider with 45%, 32% and
23% market shares, respectively.
time

P1

P2

P3

-2
-1
0

4200
4700
4500

3800
3200
3200

2000
2100
2300

Table 2.2: Insurer portfolio sizes
We consider two types of loss model: (i) loss E(Y ) = 1, σ(Y ) = 4.472, Poisson-Lognormal
model, (ii) loss E(Y ) = 1, σ(Y ) = 10, Negative Binomial-Lognormal model. The loss history issuch that the actuarially based premiums āj,0 ’s are given in Table 2.3 and the market premiumm̄0 is 1.190, 1.299 for PLN and NBLN, respectively.
1042.2. A one-period model

PLN
NBLN

P1

P2

P3

1.066
1.079

1.159
1.189

0.972
1.035

Table 2.3: Actuarially based premium āj,0The weight parameters (ωj )j used in the computation of the insurer break-even premiumare ω = (1/3, 1/3, 1/3). Before giving the sensitivity parameters βj ’s, we present the lapse
models. For customer behavior, we have two parameters µj , αj per player given a pricesensitivity function. At first, we consider the price function based on the premium ratio

tel-00703797, version 2 - 7 Jun 2012xj
f¯j (xj , xl ) = µj + αj .
xl
The central lapse rate parameters (i.e. lapse rate when every insurers use the same premium)are set to 10%, 14% and 18% for j = 1, 2 or 3, respectively. In addition to this first constraint,
we also impose that on increase of 5% compared to other players increases the total lapse rateby 5 percentage points.
Let x1 = (1, 1, 1) and x1.05 = (1.05, 1, 1). The two constraints are equivalent tolg21 (x1 ) + lg31 (x1 ) = 10% and lg21 (x1.05 ) + lg31 (x1.05 ) = 15%
for Insurer 1. We get µ1 = −12.14284 and α1 = 9.25247. With this central lapse rate
parameters, the expected numbers of lost policies when all insurers propose the same premium
are 450.1, 448.0 and 414.0.
Secondly, we consider the price function based on the premium differencef˜j (xj , xl ) = µ̃j + α̃j (xj − xl ).
Calibration is done similarly as for fj . In Figure 2.5 in Appendix 2.6.1, we plot the total lapse
ratio function of each player for the two different price function fj (left graph) and f˜j (rightgraph). In a grey dot-dash horizontal line, we highlight the central rates at 10%, 14% and
18% (the premium of other players is set to 1.4). In the central graph, we plot the total lapse
rate function of player 1 with the two different price functions.
parameters βj of objective functions are fitted in the following way 1 −Price sensitivityβjxj
mj (x) − 1≈ lgjj (x). Using x = (1.05, 1, 1), we getβj =1 − lgjj (x)0.05.

Using the premium ratio function f¯j , we have (β1 , β2 , β3 ) = (3.0, 3.8, 4.6).
Last parameters are capital values and the expense rates. Capital values (K1 , K2 , K3 )are set such that the solvency coverage ratio is 133%. Expense rates are (e1 , e2 , e3 ) =(10%, 10%, 10%).
Results and sensitivity analysis
Since we consider two loss models (PLN, NBLN) and two price sensitivity functions f¯j , f˜j(denoted by ‘ratio’ and ‘diff’, respectively), we implicitly define four sets of parameters, which105

Chapitre 2. Theorie des jeux et cycles de marché
differ on loss model and price sensitivity functions. In Table 2.4, we report premium equilibriaof the four models (PLN-ratio, PLN-diff, NBLN-ratio and NBLN-diff), differences between
equilibrium vector x? and actuarial and average market premium, and expected difference inportfolio size (∆N̂1 negative means insurer 1 expects to lose customers).

PLN-ratio
PLN-diff
NBLN-ratio
NBLN-diff

x?1

x?2

x?3

||x? − ā||2||x? − m̄||2

∆N̂1

∆N̂2

∆N̂3

1.612
1.659
1.727
1.777

1.583
1.621
1.697
1.738

1.531
1.566
1.648
1.685

11.06411.19111.99412.152

13.19913.29715.64515.752-258.510-382.879-239.465-385.404-43.479-38.401-35.301-29.645301.989421.280274.766
415.049

tel-00703797, version 2 - 7 Jun 2012

Table 2.4: Base premium equilibrium
The premium equilibrium vector x? is quite similar between the four different tested models. The change between price sensitivity functions f¯j , f˜j from an insurer point of view is achange in sensitivity parameter βj in its objective function. The change between f¯j , f˜j results
in a slight increase of premium equilibrium whereas the change between PLN or NBLN loss
models is significantly higher. Unlike the sensitivity function change, a change in loss models
does not impact the objective function but the constraint function (an increase in σ(Y )).
In Tables 2.5, 2.6, we perform a sensitivity analysis considering the NBLN-ratio modelas the base model. Table 2.5 reports the analysis with respect to capital (K3 decreases) andsensitvity parameter (βj increases). Table 2.6 focuses on actuarially based premiums (āj,0increases), average market premium (m̄0 increases) and credibility factors (ωj increases).

basecapital downbasesensitivity up

x?1

x?2

x?3

||x? − ā||2||x? − m̄||2

∆N̂1

∆N̂2

∆N̂3

1.727
1.797

1.697
1.764

1.648
1.79

11.99412.18515.64515.678-239.465-96.943-35.301
87.126274.766
9.817

x?1

x?2

x?3

||x? − ā||2||x? − m̄||2

∆N̂1

∆N̂2

∆N̂3

1.727
1.643

1.697
1.62

1.648
1.575

11.99411.73615.64515.479-239.465-207.466-35.301-42.836274.766
250.302

Table 2.5: Base premium equilibrium

base
actuarial up
market up

basecredibility up

x?1

x?2

x?3

||x? − ā||2||x? − m̄||2

∆N̂1

∆N̂2

∆N̂3

1.727
1.766
1.91

1.697
1.714
1.874

1.648
1.665
1.822

11.99413.01512.72

15.64515.70620.503-239.465-325.31-240.803-35.301
4.752-29.484274.766
320.558
270.287

x?1

x?2

x?3

||x? − ā||2||x? − m̄||2

∆N̂1

∆N̂2

∆N̂3

1.727
1.68

1.697
1.657

1.648
1.599

11.994
11.83915.64515.545-239.465-232.155-35.301-68.27274.766
300.425

Table 2.6: Base premium equilibrium

1062.3. Refinements of the one-period model

2.3

Refinements of the one-period model

In this section, we propose refinements on the objective and constraint functions of theprevious section.

2.3.1

Objective function

The objective function given in Subsection 2.2.3 is based on an approximation of the true
demand function. For insurer j, the expected portfolio size is given byXN̂j (x) = nj × lgjj (x) +nl × lgjl (x),l6=jtel-00703797, version 2 - 7 Jun 2012

where lglj ’s are lapse functions and lgjj the “renew” function. Note that the expected size N̂j (x)contains both renewal and new businesses. So, a new objective function could beej (x) = N̂j (x) (xj − πj ),Onwhere πj is the break-even premium as defined in Subsection 2.2.3. However, we do not conej (x) does not verify some generalized convexity
sider this function, since the function xj 7→ O
properties, which we will explain in Subsection 2.3.3. And also, the implicit assumption isthat insurer j targets the whole market: this may not be true in most competitive insurance
markets.
Instead, we will test the following objective function
ej (x) =O

nj lgjj (x)n(xj − πj ),(2.10)

taking into account only renewal business. This function has the good property to be infinitely
differentiable. Using the definition lgjj in equation (2.1), one can show that the function
xj 7→ lgjj (x) is a strictly decreasing function, see Appendix 2.6.1. As for the objective function
ej is a trade-off between increasing premium for better expected profit andOj , maximising O
decreasing premium for better market share.

2.3.2

Constraint function

We also change the solvency constraint function xj 7→ gj1 (xj ) defined in equation (2.4),
which is a basic linear function of the premium xj . We also integrate other insurer premiumx−j in the new constraint function, i.e. xj 7→ g̃j1 (x). We could use the following constraintfunction
Kj + N̂j (x)(xj − πj )(1 − ej )qg̃j1 (x) =− 1,
k995 σ(Y ) N̂j (x)the ratio of the expected capital and the required solvency capital. Unfortunately, this function
does not respect a generalized convexity property, that we will define in the Subsection 2.3.3.
So instead, we consider a simpler versiong̃j1 (x) =Kj + nj (xj − πj )(1 − ej )q
− 1,k995 σ(Y ) N̂j (x)(2.11)

107

Chapitre 2. Theorie des jeux et cycles de marché
by removing the expected portfolio size N̂j in the numerator. This function also has the good
property to be infinitely differentiable. The other two constraint functions gj2 , gj3 are identicalas in Subsection 2.2.4.

2.3.3

Properties of premium equilibrium

Conditions on the existence of a generalized Nash equilibrium can be found in Facchinei andKanzow (2009) or Dutang (2012b). In our setting, we need to show (i) the objective function
Oj (x) is quasiconcave with respect to xj , (ii) the constraint function gj (x) is quasiconcavewith respect to xj , (iii) the action set {xj ∈ Xj , gj (xj , x−j ) ≥ 0} is nonempty.
Recall that a function f : X 7→ Y is concave if ∀x, y ∈ X, ∀λ ∈ [0, 1], we have f (λx +(1 − λ)y) ≥ λf (x) + (1 − λ)f (y). Note that a convex and concave function is linear. If
inequalities are strict, we speak about strict concavity. A function f : X 7→ Y is quasiconcave
if ∀x, y ∈ X, ∀λ ∈]0, 1[, we havetel-00703797, version 2 - 7 Jun 2012

f (λx + (1 − λ)y) ≥ min(f (x), f (y)).
Again, if inequalities are strict, we speak about strict quasiconcavity. As for concavity, there
exist special characterizations when f is C 2 .
Proposition. When f is a differentiable function on an open convex O ⊂ Rn , then f isquasiconcave if and only if ∀x, y ∈ O, f (x) ≥ f (y) ⇒ ∇f (y)T (x − y) ≥ 0.
When f is a C 2 function on an open convex O ⊂ Rn , then f is quasiconcave if and only if∀x ∈ O, ∀d ∈ Rn , dT ∇f (x) = 0 ⇒ dT ∇2 f (x)d ≤ 0.
Proof. See Theorems 2 and 5 of Diewert et al. (1981).
From the last proposition, it is easy to see that for a C 2 univariate function, quasiconcavity
implies unimodality. Furthermore, f is pseudoconcave if and only if ∀x, y, we have f (x) >f (y) ⇒ ∇f (y)T (x − y) > 0.
Proposition. When f is a C2 function on an open convex O ⊂ Rn , then if ∀x ∈ O, ∀d ∈Rn , dT ∇f (x) = 0 ⇒ dT ∇2 f (x)d < 0, then f is pseudoconcave, which in turn implies strict
quasiconcavity.
Proof. See Corollary 10.1 of Diewert et al. (1981).
Examples of quasiconcave functions include monotone, concave or log-concave functions.
A univariate quasiconcave function is either monotone or unimodal. More properties can befound in Diewert et al. (1981). Figure 2.6 in Appendix 2.6.1 relates on the different conceptsof convexity.
Proposition 2.3.1. The I-player insurance game with objective function and solvency constraint function defined in Equations (2.10) and (2.11), respectively, admits a generalized Nashpremium equilibrium, if for all j = 1, . . . , I, g̃j1 (x) > 0.
Proof. Properties of the expected portfolio size function have been established in Appendix
2.6.1. The objective function can be rewritten as
ej (x) = lgj (x, f )(xj − πj ),O
j108

2.3. Refinements of the one-period model
ej has been built to be continuous on RI . Note that we stress theup to a constant nj /n. O+
dependence on the price sensitivity function f . Using Appendix 2.6.1, the gradient of theobjective function is proportional toXej (x)∂O0= lgjj (x, f )(1 − Sj (x)(xj − πj )), where Sj (x) =fj1(xj , xl ) lglj (x, f ).
∂xj
l6=j

The gradients cancel at 1 = Sj (xj? )(x?j − πj ), where xj? = (x1 , . . . , xj−1 , x?j , xj+1 , . . . , xI ).
The second-order derivative is given byX
ej (x)∂2O0= lgjj (x, f ) (xj − πj )2Sj2 (x) − 2Sj (x) − (xj − πj )fj1
(xj , xl )2 lglj (x, f )
2
∂xj
l6=j
= lgjj (x, f )2Sj (x) [(xj − πj )Sj (x) − 1] − lgjj (x, f )(xj − πj )

X

0
fj1
(xj , xl )2 lglj (x, f ).

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l6=j

The sign of the second order derivative at xj? is!
Xej (xj? )
∂2O
j j?
?
0
sign
=
−
lg
(x
,
f
)(x
−
π
)
fj1
(x?j , xl )2 lglj (xj? , f ).
j
j
j
∂x2j
l6=j

However, the root of the gradient is such that x?j − πj = 1/Sj (xj? ) > 0. So we have
∂ 2 Oj (xj? )
sign
∂x2j

!
< 0.

Hence, the function xj 7→ Oj (x) is pseudoconcave, and thus strictly quasiconcave.
Functions gj2 , gj3 are strictly concave since second-order derivatives are∂ 2 gj2 (x)∂ 2 gj3 (x)−(xj −x)< 0 and= −e= −e−(x−xj ) < 0.
∂ 2 xj∂ 2 xjWe verify quasiconcavity of the function g̃j1 with respect to xj . The function xj 7→ g̃j1 (x) ismonotone since its gradient!
∂g̃j1 (x)Kj + nj (xj − πj )(1 − ej )∂ N̂j (x)nj (1 − ej )
q
=
−
+
3/2
∂xj
∂xj
2k995 σ(Y )N̂j (x)k995 σ(Y ) N̂j (x)is positive for all x ∈ RI+ . Thus, function xj 7→ g̃j1 (x) is (strictly) quasiconcave.
Let Xj = [x, x]. The constraint set is Cj (x−j ) = {xj ∈ Xj , g̃j1 (xj , x−j ) ≥ 0} wherexj 7→ g̃j1 (x) is strictly increasing, continuous and by assumption, for all j = 1, . . . , I, g̃j1 (x) > 0.
Thus, Cj (x−j ) is a nonempty convex closed set. Furthermore, the point-to-set mapping Cj is
upper semi-continuous by using Example 5.10 of Rockafellar and Wets (1997). Using Theorem
13 of Hogan (1973) and the continuity of g̃j1 , the point-to-set mapping is also lower semicontinuous. By Theorem 4.1 of Facchinei and Kanzow (2009), there exists a generalized Nashequilibrium.
109

Chapitre 2. Theorie des jeux et cycles de marché
Non-uniqueness issues

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Uniqueness of a generalized Nash equilibrium is not guaranteed in general. Furthermore,
there is no particular reason for a player to choose a certain Nash equilibrium rather thananother one. Rosen (1965) studied uniqueness of such an equilibrium in a jointly convex
game (i.e. where objective functions are convex and the constraint function is common andconvex). To deal with non-uniqueness, he studies a subset of generalized Nash equilibrium,
where Lagrange multipliers resulting from the Karush-Kuhn-Tucker (KKT) conditions arenormalized. Such a normalized equilibrium is unique given a scale of the Lagrange multiplierwhen the constraint function verifies additional assumptions. Other authors such as von
Heusinger and Kanzow (2009) or Facchinei et al. (2007) define normalized equilibrium when
Lagrange multipliers are set equal.
Another way is to look for generalized Nash equilibria having some specific properties, suchas Pareto optimality. The selection of the equilibrium is particularly developed for games withfinite action sets. In that setting, one can also use a mixed strategy, by playing ramdomly one
among many equilibrium strategies.
Parameter sensitivity
Proposition 2.3.2. Let x? be a premium equilibrium of the I-player insurance game. For
each player j, if x?j ∈]x, x[, player equilibrium x?j depends on parameter in the following way:
it increases with break-even premium πj , solvency coefficient k995 , loss volatility σ(Y ), expense
rate ej and decreases with lapse parameter µj , αj and capital Kj . Otherwise when x?j = x orx, premium equilibrium is independent of any parameters.
Proof. As explained in Appendix 2.6.1, the KKT conditions at a premium equilibrium x? aresuch there exist Lagrange multipliers λj? ,
ej
∂g̃j1
∂O
(x) − λj?
(x) = 0,1
∂xj
∂xj
when assuming gj2 , gj3 functions are not active. And the complementarity constraint is such1 ?
that λj?
1 × g̃j (x ) = 0.
If the solvency constraint g̃j1 is inactive, then we necessarily have λ?j1 = 0. Let xjy be thepremium vector with the j component being y, i.e. xjy = (x1 , . . . , xj−1 , y, xj+1 , . . . , xI ). We
denote by z a parameter of interest, say for example ej . We define the function F as
Fxj (z, y) =∂Oj j(x , z),∂xj ywhere the objective function depends on the interest parameter z. By the continuous differentiability of F with respect to z and y, we can invoke the implicit function theorem, see
Appendix 2.6.1. So there exists a function ϕ such that Fxj (z, ϕ(z)) = 0, and the derivative is
given by∂Fxj
(z, y)
ϕ (x) = − ∂zj.
∂Fx∂y (z, y) y=ϕ(z)0110

2.3. Refinements of the one-period modelIn our case ∗ , we have
Fxj (z, y) =andnj j jlg (x )[1 − Sj (xjy )(y − πj )],n j y∂ 2 Oj j∂ 2 Oj j∂Fxj∂Fxj(xy , z), and(x , z).
(z, y) =(z, y) =
∂z
∂z∂xj
∂y
∂x2j yThe first-order derivative is given by
X

 nj
nj
∂Fxj
0
(z, y) = 2 lgjj (xjy )Sj (xjy ) (y − πj )Sj (xjy ) − 1 − lgjj (xjy )(y −πj )fj1(y, xl )2 lglj (xjy ).
∂y
n
n
l6=j

Using Fxj (z, ϕ(z)) = 0 whatever z represents, it simplifies to
X
nj
∂Fxj
0
(z, ϕ(z)) = − lgjj (xjϕ(z) )(ϕ(z) − πj )fj1
(ϕ(z), xl )2 lglj (xjϕ(z) ).
∂yn

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l6=j

Let z now be the insurer’s break-even premium z = πj . We have∂Fxj(z, y) = nj lgjj (xjy )Sj (xjy ).
∂zThus, the derivative of ϕ is
Sj xjϕ(z)ϕ0 (z) =(ϕ(z) − z)

P
l6=j

.

0 (ϕ(z), x )2 lgl xj
fj1
l
j
ϕ(z)

By definition, Fxj (z, ϕ(z)) = 0 is equivalent to
1 = Sj xjϕ(z) (ϕ(z) − z).
Thus ϕ(z) − z > 0. We conclude that ϕ0 (z) > 0, i.e. the function πj 7→ x?j (πj ) is increasing.
Let z be the intercept lapse parameter z = µj . By differentiating the lapse probability, wehave∂ lgjj∂z(xjy ) = − lgjj (xjy )X
l6=jlglj (xjy )and∂ lgkj j(xy )∂z= − lgkj (xjy )j6=kXlglj (xjy ) + lgkj (xjy ).

l6=j

We get∂Fxj(z, y) = −nj lgjj (xjy )(1 − lgjj (xjy )) 1 − Sj (xjy )(y − πj ) − nj lgjj (xjy )2 Sj (xjy ).
∂z
Note the first term when y = ϕ(z) since Fxj (z, ϕ(z)) = 0. We finally obtain
Sj xjϕ(z) lgjj xjϕ(z).
ϕ0 (x) = −P 0(ϕ(z) − z)
fj1 (ϕ(z), xl )2 lglj xjϕ(z)l6=j∗. To simplify, we do not stress the dependence of lgkj and Sj on f .

111

Chapitre 2. Theorie des jeux et cycles de marché
Using 1 = Sj xjϕ(z) (ϕ(z) − z), we have2Sj xjϕ(z) lgjj xjϕ(z) < 0.
ϕ0 (x) = − P
0 (ϕ(z), x )2 lgl xj
fj1
l
j
ϕ(z)
l6=j

Thus, the function µj 7→ x?j (µj ) is decreasing.
Let z be the slope lapse parameter z = αj .
∂ lgjj∂z

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and∂ lgkj j(xy )∂z(xjy ) = − lgjj (xjy )X∆j,l (xjy ) lglj (xjy )l6=j= − lgkj (xjy )X∆j,l (xjy ) lglj (xjy ) + lgkj (xjy )∆j,k (xjy ),

l6=j

j6=k

where ∆j,l (xjy ) = xj /xl if we use the premium ratio function fj and xj − xl if we use thepremium difference function f˜j . We get∂Fxj(z, y) = −nj lgjj (xjy )Sj∆ (xjy )[1 − Sj (xjy )(y − πj )]∂zX0− nj lgjj (xjy )2 Sj∆ (xjy ) − nj lgjj (xjy )fj1(y, xl )∆j,l (xjy ) lglj (xjy ),l6=jwhere Sj∆ (xjy ) =we havePjjll6=j ∆j,l (xy ) lgj (xy ).

Again the first term cancels when y = ϕ(z). Hence,


 P




0 (ϕ(z), x )∆ lgl xj
lgjj xjϕ(z) Sj∆ xjϕ(z) + l6=j fj1
l
j,l j
ϕ(z)


ϕ0 (z) = −.
P 0(ϕ(z) − z)
fj1 (ϕ(z), xl )2 lglj xjϕ(z)
l6=j





Using 1 = Sj xjϕ(z) (ϕ(z) − z), we have

ϕ0 (z) = −Sj






 P




j
j
l
0 (ϕ(z), x )∆
 lgjj xjϕ(z) Sj∆ xjϕ(z) + l6=j fj1
x
lg
x
l
j,l
j
ϕ(z)
ϕ(z)


xjϕ(z).
P 0
fj1 (ϕ(z), xl )2 lglj xjϕ(z)l6=j0 (ϕ(z), x ) >If we use the premium ratio function, we have ∆j,l (.) = ϕ(z)/xl > 0 as well as fj1
l
0. It is immediate that ϕ0 (z) < 0. Otherwise when we use the premium difference function(∆j,l (.) = ϕ(z) − xl ), we cannot guarantee that the numerator is positive.
If the solvency constraint g̃j1 is active, then we necessarily have λ?j1 > 0, gj1 (x? ) = 0. Let xjy
be the premium vector with the j component being y as above. We denote by z a parameterof interest, then we define the function G as

Gjx (z, y) = g̃j1 (xjy , z),112

2.3. Refinements of the one-period model
where the objective function depends on the interest parameter z. Again, we apply the implicit
function theorem with a function φ such that Gjx (z, φ(z)) = 0. The first-order derivative is
given by∂gj1
∂Gjx
(z, y) > 0,(z, y) =∂y∂xj
since xj 7→ g̃j1 is a strictly increasing function. Therefore, the sign of φ0 is!
∂Gjx
sign φ (z) = −sign(z, φ(z)) .
∂z
0

Let z = πj be the actuarial premium. We havetel-00703797, version 2 - 7 Jun 2012

∂Gjx
(z, y) = −∂znj (1 − ej )q
< 0,k995 σ(Y ) N̂j (xjy )independently of y or z. So, sign(φ0 (z)) > 0, i.e. the function πj 7→ x?j (πj ) is increasing as in
the previous case.
Let z = Kj be the capital. We have∂Gjx(z, y) =
∂z

1
q
> 0.
k995 σ(Y ) N̂j (xjy )So sign(φ0 (z)) < 0, i.e. the function Kj 7→ x?j (Kj ) is decreasing.
Let z = σ(Y ) be the actuarial premium. We have
Kj + nj (y − πj )(1 − ej )∂Gjx
1q(z, y) = − 2 ×,∂zz
k995 N̂j (xjy )jjxwhich simplifies to ∂G∂z (z, φ(z)) = −1/z < 0 using the definition of G . Thus, the function?
σ(Y ) 7→ xj (σ(Y )) is decreasing. By a similar reasoning, we have for z = k995 , that φ isdecreasing.

2.3.4

Numerical application

We use the same set of parameters as in Subsection 2.2.7. As discussed above, a generalized
premium equilibrium is not necessarily unique: in fact there are many of them. In Tables 2.7
and 2.8, we report generalized Nash equilibria found with different starting points (210 feasible
points randomly drawn in the hypercube [x, x]I ). Premium equilibrium are sorted according
to the difference with average market premium m̄.
In Table 2.8, this computation is done for the Negative Binomial-Lognormal loss model(NBLN), whereas Table 2.7 reports the computation for Poisson-Lognormal model (PLN).
Both tables use the price ratio function f¯j . The last column of those tables reports thenumber of optimization sequences converging to a given equilibrium.
Most of the time, other equilibriums found hit one of the barriers x, x. It may appear
awkward that such points are optimal in a sense, but one must not forget the Lagrange
113

Chapitre 2. Theorie des jeux et cycles de marché
x?1

x?2

x?3

||x? − ā||2||x? − m̄||2

∆N̂1

∆N̂2

∆N̂3

Nb

1
1.30411
11.01851.38561.419
1.04493

1
1.025
1.31831.00011.36941.08441.45411.39311.17381
1.0283
1.00651.3427
1.39931.4501
1.12473
1.5381

0.11320.0497
0.09640.11620.133
0.11440.11213.43793.27670.00840.06450.07540.08960.22150.26960.30043.90754.0418

-19
-2479
1001
722
2646
-1729
-1564
3787
-4490

-16
1264
-1899
701
-1507
2758
-1233
-1490
5412

35
1216
898
-1423
-1139
-1029
2797
-2297
-922

1
13
4
3
114
142
111
1
3

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Table 2.7: Premium equilibria - PLN price ratio function

x?1

x?2

x?3

||x? − a||2||x? − m||2

∆N̂1

∆N̂2

∆N̂3

Nb

1.36441
11.00441.48751.555
1.561
1.73463
3
3
3

1.0574
1.39421.001
1.4216
1.17261.6092
1.25263
1.4664
1.36991.90413

1.06611.02081.42061.45691.57921.25083
1.43483
1.76581.54971.75420.12390.12010.13980.08870.18360.23233.08653.2546
6.226
3.47943.69416.407

0.06110.10030.11920.17810.27810.35983.53943.7733
6.83843.77894.0712
7.0956

-2635
1315
851
3333
-1622
-1369
-1405
-955
-4485
-4482
-4462
-4354

1397
-2258
818
-1923
2696
-1210
3695
-3174
6746
5299
-743
-2970

1239
943
-1670
-1411
-1075
2579
-2291
4129
-2261
-817
5205
7324

10
1
3
109
116
97
4
5
2
4
12
4

Table 2.8: Premium equilibria - NBLN price ratio function

multipliers (not reported here). Those are not zero when a constraint gji is active, (wherei = 1, 2, 3 and j = 1, . . . , I).
Tables 2.12 and 2.13 in Appendix 2.6.1 report the computation when we use the pricedifference function f˜j . The number of different premium equilibria is similar as in the previous
case.
This numerical application reveals that in our refined game, we have many generalized
premium equilibria. In our insurance context, a possible way to deal with multiple equilibria
is to choose as a premium equilibrium, the generalized Nash equilibrium x? that is closest to
the average market premium m̄. This option is motivated by the high level of competition
present in most mature insurance markets (e.g. Europe and North America) where each
insurer sets the premium with a view towards the market premium.
However, this solution has drawbacks: while a single Nash equilibrium may be seen asa self-enforcing solution, multiple generalized Nash equilibria cannot be self-enforcing. We
will not pursue this refined one-shot game further and focus on the simple insurance game ofSection 2.2.
114

2.4. Dynamic framework

2.4

Dynamic framework

In practice, insurers play an insurance game over several years, gather new information on
incurred losses, available capital and competition level. We present in this section the dynamic
framework based on the one-shot game of Subsection 2.2. The first subsection gives a justification for the chosen dynamic model, compared to other possible dynamic game models. The
following subsections present the dynamic game, some properties and numerical illustrations.

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2.4.1

Dynamic game models

Defining dynamic games is quite complex. Basar and Olsder (1999) is a major reference on
noncooperative dynamic game theory. Extending a static game to a dynamic game consists
not only of adding a time dimension t for the control variable. It also requires the definitionof a state equation (xt+1 = f (xt , . . . )) and a state variable xt , “linking” all the information
together, see Definition 5.1 of Basar and Olsder (1999) ∗ .
Depending on which information the players have about the state variable, different classesof games are defined: open-loop (knowing only the first state x1 ), closed-loop (all states xt
up to time t), feedback (only the current state xt ). Computational methods for dynamic
equilibrium generally use backward equations, e.g. Theorem 6.6 of Basar and Olsder (1999)
for feedback strategies and Theorem 6.10 in a stochastic setting. This method does notcorrespond to the insurance market reality: (i) premium is not set backwardly, the claim
uncertainty is a key element in insurance pricing, (ii) the time horizon is infinite rather thanfinite.
A class of discrete-time games, first introduced by Shapley (1953), use a finite state space
where a transition probability models the evolution of the current state depending on player
actions. As the set of possible strategies (a serie of pure or mixed actions) is huge, Shapley(1953) focuses on strategies depending on the current state only. These games are also referred
to Markov games. Despite our game has a Markovian property, we do neither limit our strategy
space to a finite set, nor use a finite state space.
Another kind of dynamic games is evolutionary games, e.g. Sigmund and Hofbauer (1998).
Evolutionary games are different in spirit to the classical game theory since they try to model
non-rational behavior of players meeting randomly. The different types of individuals represent
the different type of strategies. Again a recurrence equation is used to model the average
proportion of individuals of type i at time t. In the actuarial literature, Ania et al. (2002)
use an evolutionary setting to extend the Rotschild-Stiglitz framework on optimal insurance
coverage by individuals. Non-rational behavioral game theory does not seem the appropriate
tool for insurance market cycles.
Finally, repeated games study long-term interactions between players during the repetition of one-shot finite games. The horizon either infinite or finite plays a major role in theanalysis of such games, in particular whether punishment strategies and threats are relevant.
Most of the theory (Folk theorems) focuses on the set of achievable payoffs rather than thecharacterization of the equilibrium. Folk theorems demonstrate that wellfare outcomes canbe attained when players have a long-term horizon, even if it is not possible in the one-shot
game, see, e.g., Osborne and Rubinstein (2006). Our game does not belong to this framework,∗. We deal with discrete-time games, if working continuous-time games, the state equation is replaced bya differential equation. Such games are thus called differential games.

115

Chapitre 2. Theorie des jeux et cycles de marché
since strategic environments (action sets) evolve over a time, the action set is not finite andstochastic pertubations complete the picture.
We choose a repeated game but with infinite action space, such that at each period,
insurers set new premiums depending on past observed losses. A generalized Nash equilibrium
is computed at each period. Our repeated game does not enter the framework of dynamic
games as presented in Basar and Olsder (1999), but it shares some properties of Markov games
and classical repeated games.

2.4.2

Deriving a dynamic model

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In this subsection, we describe the repeated game framework. Now, insurers have a pasthistory: past premium x?j,t gross written premium GWPj,t , portfolio size nj,t , capital Kj,t at
the beginning of year t. Let d be the history depth for which economic variables (e.g. market
premium) will be computed. In this setting, objective Oj,t and constraint functions gj,t arealso time-dependent.
At the beginning of each time period, the average market premium is determined asd1X
m̄t−1 =dPN?
j=1 GWPj,t−u × xj,t−u

u=1 |GWP.,t−u{z,}market premium for year t−uwhich is the mean of last d market premiums. With current portfolio size nj,t−1 and initial
capital Kj,t−1 , each insurer computes its actuarially based premium asdāj,t =

1
1 X sj,t−u,
1 − ej,t dnj,t−uu=1 | {z }avg ind loss

where sj,t denotes the observed aggregate loss of insurer j during year t. Thus, break-evenpremiums are πj,t = ωj āj,t + (1 − ωj )m̄t−1 . Thus, the objective function in the dynamic model
is given by



nj,t
xj
Oj,t (x) =1 − βj,t−1(xj − πj,t ) ,nmj (x)and the solvency constraint function by
1gj,t(xj ) =

Kj,t + nj,t (xj − πj,t )(1 − ej,t )− 1.
√
k995 σ(Y ) nj,t

It is important to note that the characteristics of insurers evolve over time, notably the breakeven premium πj,t , the expense rate ej,t and the sentivity parameter βj,t .
The game sequence for period t is as follows
1. The insurers maximize their objective function subject to the solvency constraint
sup Oj,t (xj,t , x−j,t ) such that gj,t (xj,t ) ≥ 0.
xj,t

2. Once the premium equilibrium vector x?t is determined, customers randomly lapse or
renew, so we get a realization n?j,t of the random variable Nj,t (x? ).
116

2.4. Dynamic framework
3. Aggregate claim amounts Sj,t are randomly drawn according to the chosen loss model(either PLN, NBLN or PGLN) with the frequency shape parameter multiplied by n?j,t .
So we get a new aggregate claim amount sj,t for period t per insurer j.
4. The underwriting result for insurer j is computed by U Wj,t = n?j,t × x?j,t × (1 − ej ) − sj,t .

tel-00703797, version 2 - 7 Jun 2012

5. Finally, we update the capital by the following equation Kj,t+1 = Kj,t + U Wj,t .
This game sequence is repeated over T years, but insurers are pulled out of the market
when they have either a tiny market share (< 0.1%) or a negative capital. Furthermore, weremove players from the game when the capital is below the minimum capital requirement(MCR), whereas we keep them if capital is between MCR and solvency capital requirement(SCR).
Let It ⊂ {1, . . . , I} be the set of insurers at the beginning of year t and Rt ⊂ {1, . . . , I}
the set of removed insurers at the end of year t. If some insurers are removed, i.e. Card(Rt ) >0, then corresponding policyholders randomly move to other insurers according to a It+1 dimensional multinomial distribution. Say from l ∈ Rt to j ∈ It+1 , insured randomly move−?
?
with multinomial distribution MIt+1 (nl,t , p−l→ (xt )), where the probability vector pl→ (xt ) has
jth component given by
lgjl (x?t )
?
(x
)
=
p−
.
P
l→j t
1 − k∈Rt lgkl (x?t )When there are no more insurers, i.e. Card(It+1 ) = 0, the game ends, while if there is asingle insurer, i.e. Card(It+1 ) = 1, the game continues and the survivor insurer set the highest
premium.
In the current framework, we make the following implicit simplifying assumptions: (i)the pricing procedure is done (only) once a year (on January 1), (ii) all policies start at
the beginning of the year, (iii) all premium are collected on January 1, (iv) every claim is(fully) paid on December 31 and (v) there is no inflation and no stock/bond market to invest
premium.
In practice, these assumptions do not hold: (i) pricing by actuarial and marketing departments can be done more frequently, e.g. every 6 months, (ii) policies start and are renewed
all over the year, (iii) premium is collected all over the year, (iv) claims are settled every
day and there are reserves for incurred-but-not-reported claims and (v) there are inflation on
both claims and premiums, and the time between the premium payment and a possible claim
payment is used to invest in stock/bond markets. However, we need the above simplifications
to have a sufficiently simple model.

2.4.3

Properties of premium equilibrium

Proposition 2.4.1. For the repeated I − player insurance game defined in the previous subsection, the probability that there is at least two non-bankrupt insurers at time t decreases
geometrically as t increases.
Proof. As reported in Appendix 2.6.1, insurer choice probability functions xj 7→ lgjl (x) are(strictly) decreasing functions from 1 to 0. Note that lgjl (x) = 0 (respectively lgjl (x) = 1) is
only attained when xj tends to +∞ (−∞). When i 6= j functions xi 7→ lgjl (x) are strictlyincreasing. Let xj− = (x, . . . , x, x, x, . . . , x) and xj− = (x, . . . , x, x, x, . . . , x). We have0 < lgjl (xj− ) < lgjl (x) < lgjl (xj− ) < 1,117

Chapitre 2. Theorie des jeux et cycles de marché
for all x ∈ [x, x]I . Taking supremum and infimum on player j, we get
0 < pl = inf lgjl (xj− ) and sup lgjl (xj− ) = pl < 1.
jj

Using the definition of portfolio size Nj,t (x) given in Subsection 2.2.1 as a sum of binomial
random variables Blj,t (x), we have
P (Nj,t (x) = mj |Nj,t−1 > 0, Card(It−1 ) > 1)


X
=P
Blj,t (x) = mj Nj,t−1 > 0, Card(It−1 ) > 1l∈It−1X=Y

P (Blj,t (x) = m̃l )m̃1 ,...,m̃It−1 ≥0 l∈It−1
s.t.

tel-00703797, version 2 - 7 Jun 2012l m̃l =mj

Y nl,t−1 

X

=

P

m̃l

m̃1 ,...,m̃It−1 ≥0 l∈It−1s.t.

Pnl,t−1 −m̃jlgjl (x)m̃l 1 − lgjl (x)l m̃l =mj

Y nl,t−1 lpm̃
(1 − pl )nl,t−1 −m̃j = ξ > 0
l
m̃l

X

>

m̃1 ,...,m̃It−1 ≥0 l∈It−1s.t.

P

l m̃l =mj

Therefore,
P (Card(It ) = 0|Card(It−1 ) > 1) =Nj,t (x)



X

Yi 

P ∀j ∈ It−1 , Nj,t (x) ≥ 0, Kj,t−1 + Nj,t (x)x?j,t (1 − ej ) <i=1

≥ P ∀j ∈ It−1 , Nj,t (x) > 0, Kj,t−1 + Nj,t (x)x?j,t (1 − ej ) <Nj,t (x)



X

Yi 

i=1

X

=

Y

Kj,t−1 + mj x?j,t (1 − ej ) <Pt (Nj,t (x) = mj ) P

m1 ,...,mIt−1 ≥0 l∈It−1
s.t.

P

P

Yi

i=1

Y

Kj,t−1 + mj x?j,t (1 − ej ) <Pt (Nj,t (x) = mj ) P

m1 ,...,mIt−1 >0 l∈It−1
s.t.

!

l ml =n

X

≥

mj
X

mj
X

!
Yi

i=1

l ml =nX>ξY

m1 ,...,mIt−1 >0 l∈It−1
s.t.

P

Kj,t−1 + mj x?j,t (1 − ej ) <

mj
X

!
Yi

>0

i=1

P
l ml =nThus, we have
P (Card(It ) > 1|Card(It−1 ) > 1) =1 − P (Card(It ) = 0|Card(It−1 ) > 1) − P (Card(It )1|Card(It−1 ) > 1)
≤ 1 − P (Card(It ) = 0|Card(It−1 ) > 1) < 1 − ξ˜ < 1.
118

2.4. Dynamic framework
By successive conditioning, we get
P (Card(It ) > 1) = P (Card(I0 ) > 1)

t
Y


t
P (Card(Is ) > 1|Card(Is−1 ) > 1) < 1 − ξ˜ .

s=1

So, the probability P (Card(It ) > 1) decreases geometrically as t increases.
Proposition 2.4.2. For the repeated I − player insurance game defined in the previous subsection, if for all k 6= j, xj ≤ xk and xj (1 − ej ) ≤ xk (1 − ek ), then the underwriting result by
policy is ordered U Wj ≤icx U Wk where U Wj is the random variable
Nj (x)X
1
U Wj = xj (1 − ej ) −Yi .
Nj (x)

tel-00703797, version 2 - 7 Jun 2012

i=1

Proof. Let us consider a price vector x such that xj < xk for all k 6= j. Since the change
probability pk→j (for k 6= j) is a decreasing function (see Appendix 2.6.1), pk→j (x) > pk→l (x)for l 6= j given the initial portfolio sizes nj ’s are constant.
Below we use the stochastic orders (≤st , ≤cx ) and the majorization order (≤m ) whose
definitions and main properties are recalled in the Appendices 2.6.2 and 2.6.2 respectively.
Using the convolution property of the stochastic order I times, we can show a stochastic orderof the portfolio size
Nk (x) ≤st Nj (x), ∀k 6= j.
Let us consider the underwriting result per policy1
uwj (x, n) =nnxj (1 − ej ) −nX!
= xj (1 − ej ) −

Yi

i=1

n
X
1
i=1

n

Yi ,

for insurer j having n policies, where Yi denotes the total claim amount per policy.
Let n < ñ be two policy numbers and añ , an ∈ Rñ be defined as




1

1
1
1
.
añ =
,...,
and an = 
,
.
.
.
,
,
0,
.
.
.
,
0
n

|
{z
}
ñ
ñ
n
| {z } size ñ−nsize nSince añ ≤m an and (Yi )i ’s are i.i.d. random variables, we have
ñ
X
1
i=1

ñ

Yi ≤cx

n
X
1
i=1

n

P

i añ,i Yi ≤cx

P

i an,i Yi i.e.

Yi .

For all increasing convex functions φ, the function x 7→ φ(x + a) is still increasing andconvex. Thus for all random variables X, Y such that X ≤icx Y and real numbers a, b, a ≤ b,
we have
E(φ(X + a)) ≤ E(φ(X + b)) ≤ E(φ(Y + b)),i.e. a + X ≤icx b + Y .
As xj (1 − ej ) ≤ xk (1 − ek ) and using the fact that X ≤cx Y is equivalent to −X ≤cx −Y ,we haveuwj (x, ñ) ≤icx uwk (x, n), ∀k 6= j.
119

Chapitre 2. Theorie des jeux et cycles de marché
Using Theorem 3.A.23 of Shaked and Shanthikumar (2007), except that for all φ convex,
E(φ(uwj (x, n))) is a decreasing function of n and Nk (x) ≤st Nj (x), we can show
U Wj = uwj (x, Nj (x)) ≤icx uwk (x, Nk (x)) = U Wk .

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2.4.4

Numerical illustration

In this subsection, we present numerical illustrations of the repeated game. As explained at
the beginning of this section, objective and solvency constraint functions depend on parameters
evolving over time: the portfolio size nj,t , the capital Kj,t , the break-even premium πj,t . Doing
this, we want to mimic the real economic actions of insurers on a true market: in private motor
insurance, each year insurers and mutuals update their tariff depending on last year experienceof the whole company and the claim experience of each particular customer through bonusmalus. In our game, we are only able to catch the first aspect.
In addition to this parameter update, we want to take into account the portfolio size
evolution over time. As nj,t will increase or decrease, the insurer j may become a leader orlose leadership. Hence, depending on market share (in terms of gross written premium), we
update the lapse, the expense and the sensitivity parameters αj,t , µj,t , ej,t and βj,t , respectively.
Before the game starts, we define three sets of parameters for the leader, the outsider and thechallenger, respectively. At the beginning of each period t, each insurer has its parameter
updated according to its market share GWPj,t−1 /GWPt−1 . When the market share is above
40% (respectively 25%), insurer j uses the “leader” parameter set (respectively the “outsider”
parameter set), otherwise insurer j uses the “challenger” parameter set. There is only one
parameter not evolving over time: the credibility factor ωj which is set to a common value ofωj = 9/10 in our numerical experiments.
For the following numerical application, the parameters are summarized in Table 2.9. Lapse
parameters ∗ are identical as in Subsection 2.2.7, but we change the expense and sensitivity
parameters to get realistic outputs.

leaderchallenger
outsider

αj

µj

ej

βj

-12.143-9.814
-8.37

9.252
7.306
6.161

0.2
0.18
0.16

3.465
4.099
4.6

Table 2.9: Parameter sets

Random paths
In order to compare the two different loss models (PLN, NBLN) and the sensitivity functions f¯j , f˜j , we fix the seed for losses and insured moves. That’s why we observe similar
patterns for the four situations. On Figures 2.1 and 2.2, we plot the individual premium
x?j,t , the gross written premium GWPj,t , the loss ratio LRj,t and the solvency coverage ratio
Kj,t /SCRj,t .
∗. We give here only the lapse parameter for the price sensitivity function f¯j , but there are also threeparameter sets for f˜j .

120

2.4. Dynamic framework

tel-00703797, version 2 - 7 Jun 2012

An interesting feature of these random paths is that a cyclic pattern is observed for themarket individual premium mt , strong correlations of gross written premiums GWPj,t andloss ratio LRj,t for each insurer j. We fit a basic second-order autoregressive model on market
premium (i.e. Xt − m = a1 (Xt−1 − m) + a2 (Xt−2 − m) + t ) ∗ . Estimation on the serie (mt )tleads to period of 11.01 and 9.82 years, respectively for Figures 2.1 and 2.2.

Figure 2.1: A random path for NBLN loss model and f˜jFurthermore, this numerical application shows that insurers set premium well above thepure premium E(Y ) = 1. Thus, the insurer capitals tend to infinite as we observe on the(bottom right) plot of the solvency coverage ratio. We also do the computation of PLN/NBLN
loss models with the sensitivity function f¯j . Similar comments apply, see Appendix 2.6.2.
Some Monte-Carlo estimates
In this subsection, we run the repeated game a certain of times with the NBLN loss modeland the price sensitivity function f˜j in order to assess certain indicators by a Monte-Carlo
method. We choose a sample size of 214 ≈ 16000 and a time horizon T = 20. Our indicators
are: (i) the ruin probability of insurer j before time Te and (ii) the probability a leader at time
Te, where Te = T /2 or T .
Results are given in Table 2.10. Estimates of ruin probabilities are extremely low, because
the safety loadings of equilibrium premium are very high, see previous Figures. Leadershipprobabilities are more interesting. Recalling that Insurer 1 is the leader at time 0, the probability for Insurer 1 to be leader after t periods decreases quickly as t increases. After only 20
periods, Insurer 1 has losed its initial advantage.
Then, we look at the underwriting result by policy to see if some insurers underwrite adeliberate loss. As the first quartile is above zero, we observe that negative underwriting∗. When a2 < 0 and a21 + 4a2 < 0, the AR(2) is p-periodic with p = 2π arccos



√a1
2 −a2


.

121

tel-00703797, version 2 - 7 Jun 2012

Chapitre 2. Theorie des jeux et cycles de marché

Figure 2.2: A random path for PLN loss model and f˜jInsurer 1
Insurer 2
Insurer 3

Ruin beforet = 10Ruin beforet = 20

Leaderat t = 5

Leaderat t = 10

Leaderat t = 20

6.1e-0500.000244

6.1e-0500.000244

0.593
0.197
0.21

0.381
0.308
0.312

0.331
0.329
0.34

Table 2.10: Ruin and leadership probabilities

results are rather marginal. In fact, the probability of negative underwriting results are(0.0352, 0.0378, 0.0358) for Insurers 1, 2 and 3, respectively.

Insurer 1
Insurer 2
Insurer 3

Min.

1st Qu.

MedianMean

3rd Qu.

Max.

-0.7905-0.4340-0.47300.23090.22790.23080.36170.36000.36270.35630.35550.3563

0.48690.48690.4871

1.21401.14901.0950

Table 2.11: Summary statistics of underwriting result by policy at t = 20
On the left-hand plot of Figure 2.3, we analyze the (individual) market premium mt . We
plot quantiles at 5%, 50% and 95% as well as two random paths. The two plotted random
paths show a cyclic behavior, whereas the three quantiles are stable in time. On each random
path, we can fit an AR(2) model and estimate the cycle period. Only in 240 random paths, the2fitted AR(2) is not periodic,
 i.e. a2 ≥ 0 or a1 + 4a2 ≥ 0. Otherwise, the period is computed
as p = 2π arccos

122

√a1
2 −a2

. On the right-hand plot of Figure 2.3, we plot the histogram of

2.5. Conclusion
estimated periods: average period is around 10.

0

5

10

15

20

1000 2000 3000 4000 5000

Histogram of cycle periods

0Frequency

1.7
1.6
1.5
1.4

Q5%
Q50%
Q95%
path 1
path 2

1.3

market premium m_t

1.8

market premium

68

tel-00703797, version 2 - 7 Jun 2012

time t

10

12

14

16

18

20

Period (years)

Figure 2.3: Market premium

Finally, on a short time horizon, T = 5, we want to assess the impact of initial portfolio
size nj,0 and capital value Kj,0 on the probability to be leader at time T . We consider Insurer
1 as the leader and Insurers 2 and 3 as identical competitors. We take different values ofK1,0 and n1,0 for Insurer 1 and deduce capital values and portfolio sizes for Insurers 2 and 3√
as K2,0 = K3,0 = k0 σ(Y ) n2,0 and n2,0 = n3,0 = (n − n1,0 )/2, where k0 is a fixed solvency
coverage ratio. The sensitivity analysis consists in increasing both market shares and capitalvalues of Insurer 1 while other competitors have a decreasing market share and a constantcoverage ratio.
We look at the probability for insurer i to be a leader in terms of gross written premium
at period T = 5, i.e.
pi = P (∀k 6= i, GWPi,T > GWPk,T |Ni,0 = ni , Ki,0 = ki ).
We test two loss models NBLN and PGLN, for which the margingal claim distribution is acompound negative binomial distribution with lognormal distribution, but for PGLN, the loss
frequency among insurers is comonotonic, see Subsection 2.2.2.
On Figures 2.4, we observe the probability to a leader after five periods is an increasing
function of the initial market share. The initial capital does not seem to have any influence,
which can be explained by the high profit per policy. As one could expect, the comonotonic
loss model (Figure 2.4b) is favorable to Insurer 1 than the independent case (Figure 2.4a).

2.5

Conclusion

This paper assesses the suitability of noncooperative game theory for insurance market
modelling. The game-theoretic approach proposed in this paper gives first answers of theeffect of competition on the insurer solvency whose a significant part is linked to the abilityof insurers to sell contracts. The proposed game models a rational behavior of insurers insetting premium taken into account other insurers. The ability of an insurer to sell contracts
123Chapitre 2. Theorie des jeux et cycles de marché

Outsider1.0

Ca 1.6p.
So 1.8
l. R
atio

0.4
2.0

1.4

0.5

Ca 1.6p.
So 1.8
l. R
atio

0.4
2.0

(a) NBLN

0.5

0.7
0.6

re

re

0.7
0.6

Sha

re

Sha

0.5

Mar
ket

Ca 1.6p.
So 1.8
l. R
atio

0.0
1.4

0.1

0.4

0.7
0.6
Mar
ket

re

Sha

1.4

adershipadershipadershipadership

0.7
0.6

0.2

0.6

0.1

0.4

0.3

0.8

0.2

0.6

Prob. Le

Prob. Le

Prob. Le

Prob. Le

0.3

0.8

0.4

0.0
1.4

0.4
2.0

Ca 1.6p.
So 1.8
l. R
atio

Sha

0.4

Outsider

Mar
ket

1.0

Leader

0.5

Mar
ket

Leader0.4
2.0(b) PGLN

tel-00703797, version 2 - 7 Jun 2012

Figure 2.4: Leadership probabilities of Leader (Insurer 1) and Outsiders (Insurers 2 and 3)

is essential for its survival. Unlike the classic risk theory where the collection of premiums
is fixed per unit of time, the main source of risk for an insurance company in our game is apremium risk. We extends one-player model of Taylor (1986, 1987) and subsequent extensions
based optimal control theory. To our knowledge, the use of a repeated of noncooperative gameto model non-life insurance markets is new.
The game can be extended in various directions. A natural next step is to consider adverse selection among policyholders, since insurers do not propose the same premium to all
customers. In practice, insurers do not propose the same premium to all customers. Considering two risk classes of individuals would be an interesting extension of the game, but we would
also have to find a way to differentiate premium between individuals. A second extension is tomodel investment results as well as loss reserves. We could also consider reinsurance treaties
for players in addition a catastrophe generator. However, we must avoid not to complexify
too much the game as the game is already complex and deriving theoretical properties is an
issue.

2.6

Appendix

2.6.1

On the one-period game

Implicit function theorem
Below the implicit function theorem, see, e.g., (Zorich, 2000, Chap. 8).
Theorem. Let F be a bivariate C 1 function on some open disk with center in (a, b), such that
F (a, b) = 0. If ∂F∂y (a, b) 6= 0, then there exists an h > 0, and a unique function ϕ defined for]a − h, a + h[, such thatϕ(a) = b and ∀|x − a| < h, F (x, ϕ(x)) = 0.
Moreover on |x − a| < h, the function ϕ is C 1 and∂F∂x
ϕ0 (x) = − ∂F(x, y)∂y (x, y) y=ϕ(x)124.

2.6. Appendix
Computation details
Computation is based on a Karush-Kuhn-Tucker (KKT) reformulation of the generalized
Nash equilibrium problem (GNEP). We present briefly the problem reformulation and refer
the interested readers to e.g. Facchinei and Kanzow (2009), Dreves et al. (2011) or Dutang(2012a). In our setting we have I players and three constraints for each player. For each j of
the I subproblems, the KKT conditions areX
λjm ∇xj gjm (x) = 0,
∇xj Oj (x) −1≤m≤3

tel-00703797, version 2 - 7 Jun 20120 ≤ λj ⊥ gj (x) ≥ 0.
The inequality part is called the complementarity constraint. The reformulation proposed
uses a complementarity function φ(a, b) to reformulate the inequality constraints λj , gj (x) ≥ 0
and λjT gj (x) = 0.
A point satisfying the KKT conditions is also a generalized Nash equilibrium if the objective functions are pseudoconcave and a constraint qualification holds. We have seen that
objective functions are either strictly concave or pseudoconcave. Whereas constraint qualifications are always verified for linear constraints, or strictly monotone functions, see Theorem
2 of Arrow and Enthoven (1961), which is also verified.
By definition, a complementarity function is such that φ(a, b)√= 0 is equivalent to a, b ≥ 0
and ab = 0. A typical example is φ(a, b) = min(a, b) or φ(a, b) = a2 + b2 − (a + b) called the
Fischer-Burmeister function. With this tool, the KKT condition can be rewritten as∇xj Lj (x, λj ) = 0φ. (λj , gj (x)) = 0,

where Lj is the Lagrangian function for the subproblem j and φ. denotes the component wise
version of φ. So, subproblem j reduces to solving a so-called nonsmooth equation. In thispaper, we use the Fischer-Burmeister complementarity function. This method is implemented
in the R package GNE ∗ .
Graphs of lapse functions
Properties of multinomial logit function
We recall that the choice probability function is defined as
lgkj (x) = lgjj (x) δjk + (1 − δjk )efj (xj ,xk ) ,andlgjj (x) =

1+

P

1
,
efj (xj ,xl )

l6=jwhere the summation is over l ∈ {1, . . . , I} − {j} and fj is the price function. The pricefunction fj goes from (t, u) ∈ R2 7→ fj (t, u) ∈ R. Partial derivatives are denoted by∂fj (t, u)∂fj (t, u)
0
0
= fj1
(t, u) and= fj2(t, u).
∂t∂u∗. Dutang (2012c).

125

Chapitre 2. Theorie des jeux et cycles de marché
Total lapse rate func. (Pj) - price diff
1.0
0.8
0.6

0.8

0.4

r(x)

0.6

r(x)
1.5

2.0

2.5

0.2

3.0

1.5

x_j

2.0

2.5

P1
P2
P3
x_-j

0.0

price ratio
price diff
x_-1

0.0

0.0

0.2

P1
P2
P3
x_-j

0.2

0.4

0.4

r(x)

0.6

0.8

1.0

Total lapse rate func. (P1)

1.0

Total lapse rate func. (Pj) - price ratio

3.0

1.5

x_j

2.0

2.5

3.0

x_j

tel-00703797, version 2 - 7 Jun 2012

Figure 2.5: Total lapse rate functions

Derivatives of higher order use the same notation principle.
The lg function has the good property to be infinitely differentiable. We haveX
∂ lgjj (x)1∂ 
efj (xj ,xl ) 
=−
!2 .
∂xi
∂xi
P f (x ,x )l6=j
1+
ej j ll6=jSince we have
∂ X∂xi

efj (xj ,xl ) = δji

l6=j

X

0
0
fj1
(xj , xl )efj (xj ,xl ) + (1 − δji )fj2
(xj , xl )efj (xj ,xi ) ,l6=j

we deduce∂ lgjj (x)∂xi= −δji0 (x , x )0 (x , x )efj (xj ,xl )
X fj1
fj1
j
j
i
l
0
−
(1
−
δ
)f
(x
,
x
)
!2
!2 .
ji j2 j
l
P
P
0
l6=j
1+
efj (xj ,xl )1+
efj1 (xj ,xl )
l6=j

l6=j

This is equivalent tojX∂ lgj (x)
0
0
= −
fj1
(xj , xl ) lglj (x) lgjj (x)δij − fj2(xj , xl ) lgij (x) lgjj (x)(1 − δij ).
∂xi
l6=j

Furthermore,∂ lgjj (x) 
∂xi




δjk + (1 − δjk )efj (xj ,xk ) = − 


X

0
0
fj1
(xj , xl ) lglj (x) lgkj (x)δij −fj2
(xj , xi ) lgij (x) lgkj (x)(1−δij ).

l6=j

and also
lgjj (x)
126




∂ 
0
0
δjk + (1 − δjk )efj (xj ,xk ) = lgjj (x)(1−δjk ) δik fj2(xj , xk )efj (xj ,xk ) + δij fj1(xj , xk )efj (xj ,xk ) .
∂xi

2.6. Appendix
Hence, we get

tel-00703797, version 2 - 7 Jun 2012X∂ lgkj (x)00= −δij 
fj1(xj , xl ) lglj (x) lgkj (x) − (1 − δij )fj2
(xj , xi ) lgij (x) lgkj (x)
∂xi
l6=j
h
i
0
0
+ (1 − δjk ) δij fj1(xj , xk ) lgkj (x) + δik fj2(xj , xk ) lgkj (x) .
Similarly, the second order derivative is given by ∗


l
X
X
∂
lg
∂ 2 lgkj (x)
j k
00
00
0
= −δij δjm
fj11
(xj , xl ) lglj +(1 − δjm )fj12(xj , xm ) lgm
fj1(xj , xl )lgjj +∂xm ∂xi
∂xm
l6=j
l6=j


X
∂ lgkj0− δij fj1
(xj , xl ) lglj 
∂xm
l6=j
!
 i k
∂ lgij k∂ lgkj0000
00i−(1−δij ) δjm fj21 (xj , xi ) + δim fj22 (xj , xi ) lgj lgj +fj2 (xj , xi )lg +fj2 (xj , xi ) lgj
∂xm j
∂xm
!
 k
∂ lgkj0000
0+ (1 − δjk )δij
fj11 (xj , xk )δjm + fj12 (xj , xk )δkm lgj +fj1 (xj , xk )
∂xm
!
k

∂
lg
j
00
00
0
.
+ (1 − δjk )δik fj21
(xj , xk )δjm + fj22
(xj , xk )δim lgkj +fj2
(xj , xk )∂xm
Portfolio size function
We recall that the expected portfolio size of insurer j is defined as
XN̂j (x) = nj × lgjj (x) +nl × lgjl (x),l6=jwhere nj ’s denotes last year portfolio size of insurer j and lgkj is defined in equation (2.1).
The function φj : xj 7→ lgjj (x) has the following derivative


j
X
∂
lg
(x)
j
0
φ0j (xj ) == −
fj1(xj , xl ) lglj (x) lgjj (x).
∂xj
l6=j

For the two considered price function, we have0
fj1
(xj , xl ) = αj10
and f˜j1(xj , xl ) = α̃j ,xlwhich are positive. So, the function φj will be a decreasing function.
For l 6= j, the function φl : xj 7→ lgjl (x) has the following derivative
φ0l (xj ) =∂ lgjl (x)000= −fj2(xl , xj ) lgjl (x) lgjl (x)+fj2(xl , xj ) lgjl (x) = fj2(xl , xj ) lgjl (x)(1−lgjl (x)).
∂xj∗. We remove the variable x when possible.

127

Chapitre 2. Theorie des jeux et cycles de marché
For the two considered price function, we have0
fj2
(xj , xl ) = −αjxj0
and f˜j2(xj , xl ) = −α̃j ,x2lwhich are negative. So, the function φl will also be a decreasing function.
Therefore the portfolio size xj 7→ N̂j (x) function has the following derivativeXX∂ N̂j (x)00
= −nj fj1
(xj , xl ) lglj (x) lgjj (x) +nl fj2(xl , xj ) lgjl (x)(1 − lgjl (x)).
∂xj

tel-00703797, version 2 - 7 Jun 2012

l6=j

l6=j

P
Hence, it is decreasing from the total market size l nl to 0. So the function xj 7→ N̂j is botha quasiconcave and a quasiconvex function.
Therefore, using the C 2 characterization of quasiconcave and quasiconvex functions, wehave that∂ N̂j (x)∂ 2 N̂j (x)
=0⇒
= 0.
∂xj
∂x2j
Note that the function N̂j (x) is horizontal (i.e. has gradient of 0) when xj → 0 and xj → +∞for fixed x−j .
Finally, we also need


j
2
l
X
X
∂ lgj (x)∂ lgjj∂ lgj j00l=−
fj1 (xj , xl )lg −fj1 (xj , xl ) lgj
,
∂xj j
∂xj
∂x2j
l6=j

l6=j

00 is 0 for the two considered functions. Since,as fj11∂ lglj∂xj= − lglj
l6=j

X

0
0
fj1
(xj , xn ) lgnj + lglj fj1(xj , xl ) and∂ lgjj
∂xj

n6=j


= −


X

0
fj1
(xj , xl ) lglj  lgjj ,l6=jthen we get∂ 2 lgjj (x)
∂x2j

X
j


2
X

2
0
0
fj1
(xj , xl ) lglj +2 
fj1
(xj , xl ) lglj  lgjj .

l6=j

l6=j

= − lgj

Convexity concepts
Numerical applications for refined one-period game

2.6.2

On the dynamic game

Borel-Cantelli lemma and almost sure convergence
A random variable sequence (Xn )n is said to converge almost surely to X, if P (Xn →X) = 1. A simple characterization of almost sure convergence isp.s.

Xn −→ X ⇔ P (∩n0 ≥0 ∪n≥0 |Xn − X| ≥ ) = 0, ∀ > 0.
128

2.6. Appendix

Strictly
pseudoconvex

Strictly convex

Stronglyquasiconvex

Under
differentiability

Quasiconvex
Under lower
semicontinuity

ConvexStrictlyquasiconvex

Pseudoconvex
Under
differentiability

tel-00703797, version 2 - 7 Jun 2012

Figure 2.6: Convexity and generalized convexity, from Bazaraa et al. (2006)
x?1

x?2

x?3

||x? − a||2||x? − m||2

∆N̂1

∆N̂2

∆N̂3

Nb

1.40481.507
1.552712.41773
1.54951.66113
3
2.88772.85591
3
31.0537
1.17251.60311.48962.41072.82391.23593
32.93262.874
31.02141.79063

1.0642
1.59681.2504
1.52992.39942.80873
1.39932.90223
3
2.82793
1.5134
1.78390.14130.20590.22670.14494.0024
7.94413.0797
3.172
8.812
8.912
8.33068.07893.065
3.54796.436

0.08020.30610.35250.26754.6571
8.838
3.52773.67689.7701
9.877
9.273
9.00883.4201
3.8892
7.1316

-2583
-1609
-1359
3127
-83
-1191
-1182
-616
-305
-219
175
259
1277
-4498
-4487

1357
2695
-1217
-1790
-21
509
3479
-3197
-238
321
264
-759
1023
-357
-3167

1226
-1086
2576
-1338
104
683
-2297
3813
543
-102
-439
500
-2299
4856
7654

11
128
106
56
542
38
2
6
2
6
11
20
1
9
1

Table 2.12: Premium equilibria - PLN with price difference function
Lemma(Borel–Cantelli).
Let Bn be a sequence of events on a probability space. If the serie
P
P
P
(B
)
is
finiteP(Bnn ) < +∞, thennn
P (∩n0 ≥0 ∪n≥0 Bn ) = 0.
An application of the Borel-Cantelli lemma to almost sure convergence isXp.s.
∀ > 0,P (|Xn − X| ≥ ) < +∞ ⇒ Xn −→ X.
nNotation and definition of classic stochastic orders
Using the notation of Shaked and Shanthikumar (2007), we denote by ≤st the stochastic
order, which is characterised as X ≤st Y if ∀x ∈ R, P (X > x) ≤ P (Y > x). They are129

tel-00703797, version 2 - 7 Jun 2012

Chapitre 2. Theorie des jeux et cycles de marché
x?1

x?2

x?3

||x? − a||2||x? − m||2

∆N̂1

∆N̂2

∆N̂3

Nb

1.40481.507
1.552712.41773
1.54951.66113
3
2.88772.85591
3
31.0537
1.17251.60311.48962.41072.82391.23593
32.93262.874
31.02141.79063

1.0642
1.59681.2504
1.52992.39942.80873
1.39932.90223
3
2.82793
1.5134
1.78390.14130.20590.22670.14494.0024
7.94413.0797
3.172
8.812
8.912
8.33068.07893.065
3.54796.436

0.08020.30610.35250.26754.6571
8.838
3.52773.67689.7701
9.877
9.273
9.00883.4201
3.8892
7.1316

-2583
-1609
-1359
3127
-83
-1191
-1182
-616
-305
-219
175
259
1277
-4498
-4487

1357
2695
-1217
-1790
-21
509
3479
-3197
-238
321
264
-759
1023
-357
-3167

1226
-1086
2576
-1338
104
683
-2297
3813
543
-102
-439
500
-2299
4856
7654

11
128
106
56
542
38
2
6
2
6
11
20
1
9
1

Table 2.13: Premium equilibria - NBLN with price difference function

various other equivalent characterizations, including expectectation order for all increasing
function E(φ(X)) ≤ E(φ(Y )), quantile order FX−1 (u) ≤ FY−1 (u), distribution function order
FX (x) ≥ FY (x).
One important property of this stochastic order is the stability under convolutions, i.e. if
X ≤st Y and X̃ ≤st Ỹ , then X + X̃ ≤st Y + Ỹ , see theorem 1.A.3 of Shaked and Shanthikumar(2007). By this mean, we can show that an ordering of binomial distributions. If X ∼ B(n, p)
and Y ∼ B(n, q), such that p ≤ q, then X ≤st Y . Theorem 1.A.3 of Shaked and Shanthikumar(2007) also shows that the stochastic order is closed under mixtures.
The stochastic order is sometimes denoted by ≤1 since X ≤1 Y requires that for all
differentiable functions φ such that ∀x, φ(1) (x) ≥ 0, we have E(φ(X)) ≤ E(φ(Y )). With
this reformulation in mind, we define the convex order denoted by X ≤2 Y or X ≤cx Y asE(φ(X)) ≤ E(φ(Y )) for all convex functions φ. If we restrict to differentiable functions, itmeans ∀x, φ(2) (x) ≥ 0. This explains the relation between notations ≤1 and ≤2 .
Note that if expectations exist, then X ≤cx Y implies that E(X) = E(Y ), V ar(X) ≤V ar(Y ) and E((X − a)+ ) ≤ E((Y − a)+ ). By theorem 3.A.12 of Shaked and Shanthikumar(2007), the convex order is closed under mixtures and convolutions. We also have that X ≤cx Y
is equivalent to −X ≤cx −Y .
A third stochastic order is the increasing convex order: X ≤icx Y if for all increasing
convex functions φ, E(φ(X)) ≤ E(φ(Y )). For φ differentiable, it means that φ(1) (x) ≥ 0,
φ(2) (x) ≥ 0.

Notation and definition of majorization orders
Using the book of Marshall and Olkin (1979), the majorization order ≤m is defined asa ≤m ã if∀1 ≤ k ≤ n − 1,

k
X
i=1

130

ai ≤

k
X
i=1

ãi and

n
X
i=1

ai =

n
X
i=1

ãi .

BIBLIOGRAPHY
A direct consequence of property B.2.c of Marshalland OlkinPP (1979) is that if X1 , . . . , Xn arenexchangeable and a, ã ∈ R , a ≤m ã implies i ai Xi ≤cx i ãi Xi .

tel-00703797, version 2 - 7 Jun 2012

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Figure 2.7: A random path for NBLN loss model and f¯jBibliography
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135

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Chapitre 3

Synthèse des méthodes de calculd’équilibre de Nash généralisé—
A survey of GNE computation
methods: theory and algorithms

Theory attracts practice as the magnet attracts iron.
Carl Friedrich Gauss (1777–1855)

137

Chapitre 3. Calcul d’équilibre de Nash généralisé
We consider a generalized game of N players defined by their objective function θi : Rn 7→ R
and their constraint function g i : Rn 7→ Rmi . The generalized Nash equilibrium problem(GNEP for short) extends standard Nash equilibrium, since ones’ player strategy depends
on the rival players’ strategies. Thus, when each player’s strategy set does not depend on
the other players’ strategies, the GNEP reduces to standard Nash equilibrium problem. The
GNEP consists in finding x? such that for all i = 1, . . . , N , x?i solves the subproblem
min θi (xi , x?−i ) s.t. g i (xi , x?−i ) ≤ 0,

(3.1)

tel-00703797, version 2 - 7 Jun 2012

xi ∈RniP
n with n =where (xi , x−i ) denotes
thevector
(x
,
.
.
.
,
x
,
.
.
.
,
x
)
∈
R
1
i
N
i ni the total numberPof variables and m = i mi the total number of constraints.
GNEP arises from many practical problems, including telecommunications, engineeringand economics applications, see Facchinei and Kanzow (2009) and the references therein foran overview of GNEPs. This paper aims to make a survey of computational methods to solve
general GNEPs defined in Equation (3.1).
The paper is organized as follows: Section 3.1 present the different reformulations of
GNEPs. Section 3.2 describes the numerous optimization methods that can solved a nonlinear
reformulation of the GNEP. Finally, Section 3.3 carries out a numerical comparison of allalgorithms presented in the previous section, before Section 3.4 concludes.

3.1

Problem reformulations

As presented in Equation (3.1), the generalized Nash equilibrium problem is not directly
solvable. This section aims to present the different reformulations of the GNEP. On Figure
3.1, we present a basic flow-chart of the relationship among the different reformulations that
we present below.
GNEP

QVI

KKT

NIF

Compl. reform.

Constr. eq.

Figure 3.1: Map of GNE reformulations1383.1. Problem reformulations

3.1.1

The KKT conditions

The first reformulation uses the Karush-Kuhn-Tucker (KKT) conditions of the N optimization subproblems. We assume that both constraints and objective functions are twice
continuously differentiable C2 . Let x? be a solution of the GNEP. If a constraint qualification
holds for all players, then for all player i, there exists a Lagrange multiplier λi? ∈ Rmi such
that
X
i ?
∇xi θi (x? ) +λi?
(∈ Rni ).
j ∇xi gj (x ) = 0
1≤j≤mii?

i

0 ≤ λ , −g (x? ) ≥ 0, g i (x? )T λi? = 0(∈ Rmi ).

Concatening the N subproblems, we get the following “extended” KKT system

tel-00703797, version 2 - 7 Jun 2012

D(x, λ) = 0, λ ≥ 0, g(x) ≤ 0, λT g(x) = 0,

(3.2)

where the functions D, g are defined as


 1
 1

∇x1 L1 (x, λ1 )
λ
g (x)


 . 
 . 
..
n
m
m
D(x, λ) =  ∈ R , λ =  ..  ∈ R , g(x) =  ..  ∈ R ,.
∇xN LN (x, λN )λNg N (x)and Li is the Lagrangian function Li (x, λi ) = θi (x)+g i (x)T λi . The following theorem precises
the necessary and sufficient condition between the original GNEP in Equation (3.1) and the
KKT system in Equation (3.2).
Theorem. Let a GNEP with twice continuity and differentiability of objective and constraintfunctions.
(i) If x? solves the GNEP at which all the player’s subproblems satisfy a constraint qualification, then there exists λ? ∈ Rm such that x? , λ? solve equation 3.2.
(ii) If x? , λ? solve equation 3.2 and that the functions θi ’s are player convex and {yi , g i (yi , x−i ) ≤0} are closed convex sets, then x? solves the original GNEP.
Facchinei and Kanzow (2009) and Facchinei et al. (2009) report the previous theorem,
respectively in Theorem 4.6 and Proposition 1. Using Fritz John conditions, see, e.g., Simon(2011) or Bazaraa et al. (2006), the player convexity of θi in item (ii) can be relaxed to player
pseudoconvexity, i.e. xi 7→ θi (x) is pseudoconvexe.
The complementarity reformulation
A complementarity function φ : R2 → R is a function verifying the following propertyφ(a, b) = 0 ⇔ a ≥ 0, b ≥ 0, ab = 0.
√
Examples are φ∧ (a, b) = min(a, b), φF B (a, b) = a2 + b2 − (a + b), see, e.g., Facchinei andPang (2003).
The complementarity reformulation of the KKT conditions isD(x, λ)Φ(x, λ) = 0 where Φ(x, λ) =,(3.3)φ◦ (−g(x), λ)where φ◦ is the component-wise version of the complementarity function φ and D definedfrom the extended system. This reformulation of the GNEP is given in Facchinei et al. (2009),
Facchinei and Kanzow (2009) and Dreves et al. (2011). For a general discussion of semismoothreformulations of optimization problems, Fukushima and Qi (1999).
139

Chapitre 3. Calcul d’équilibre de Nash généralisé
The constrained equation reformulation
Dreves et al. (2011) also propose a constrained equation of the KKT system. Let ◦ denotethe component-wise product, i.e. x ◦ y = (x1 y1 , . . . , xN yN ). Let w ∈ Rm be slack variables(to transform inequality into equality). The KKT conditions are equivalent toD(x, λ)H(x, λ, w) = 0, (x, λ, w) ∈ Z, and H(x, λ, w) = g(x) + w ,
λ◦w

(3.4)

where Z is the set Z = {(x, λ, w) ∈ Rn+2m , λ ≥ 0, w ≥ 0}.

3.1.2

The QVI reformulation

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Let us first define the Variational Inequality (VI) and Quasi-Variational Inequality (QVI)
problems. Variational inequality problems VI(K, F (x)) consist in finding x ∈ K such that∀y ∈ K, (y − x)T F (x) ≥ 0,
where F : K 7→ Rn . VI problems typically arise in minimization problems. Quasi-variational
inequality problems are an extension of VI problems where the constraint set K depends on
x. The QVI problem is defined as∀y ∈ K(x), (y − x)T F (x) ≥ 0,
which is denoted by QVI(K(x), F (x)). Note that a QVI has very complex structure since ymust satisfy y ∈ K(x) for a vector x we are looking for.
The GNEP in Equation (3.1) can be reformulated as a QVI problem∇x1 θ1 (x)..
∀y ∈ X(x), (y − x)T F (x) ≥ 0, with F (x) = 
,
.

(3.5)

∇xN θN (x)and a constrained set X(x) = {y ∈ Rn , ∀i, g i (yi , x−i ) ≤ 0}. We have the following theorem
precising the equivalence between the GNEP and the QVI.
Theorem. If objective functions are C1 and player-convex, the action sets Xν (x−ν ) are closed
convex, then we have x? solves the GNEP if and only if x? solves the QVI(X(x), F (x)) definedin Equation (3.5).
This is Theorem 3.3 of Facchinei and Kanzow (2009) or Equation (3) Kubota and Fukushima(2010).
Penalized sequence of VI
We can express the KKT conditions for the QVI problem, but we naturally get back to
Equation (3.2). According to Facchinei and Kanzow (2009), methods to solve general QVI
140

3.1. Problem reformulations
problem arising from GNEP are still missing. However, Fukushima and Pang (2005) propose to solve the QVI(X(x), F (x)) as a sequence of penalized variational inequality problems
VI(X̃, F̃k ), where F̃k is defined as P1 (x)∇x1 θ1 (x)  .. ..
F̃k (x) = 
 +  . ,.


∇xN θN (x)(3.6)

PN (x)withPν (x) =mν Xuki + ρk giν (x)tel-00703797, version 2 - 7 Jun 2012

i=1


+

∇xν giν (x).

The set X̃ is either Rn or a box constraint set [l, u] ⊂ Rn , (ρk ) an increasing sequence ofpenalty parameters and (uk ) a bounded sequence. Theorem 3 of Fukushima and Pang (2005)
shows the convergence of the VIP solutions x?k to a solution of the QVI under some smoothness
conditions. We will see later that the QVI reformulation for a certain class of generalized games
reduces to a standard VI problem. In that case, it makes sense to use that reformulation.

3.1.3

Nikaido-Isoda reformulation

We present a last reformulation of the GNEP, which was originally introduced in thecontext of standard Nash equilibrium problem. We define the Nikaido-Isoda function as thefunction ψ from R2n to R byψ(x, y) =NX[θ(xν , x−ν ) − θ(yν , x−ν )].

(3.7)

ν=1

This function represents the unilateral player improvement of the objective function between
actions x and y. Let V̂ be the gap functionV̂ (x) = sup ψ(x, y).
y∈X(x)

Theorem 3.2 of Facchinei and Kanzow (2009) shows the relation between GNEPs and the
Nikaido-Isoda function.
Theorem. If objective functions θi are continuous, then x? solves the GNEP if and only if
x? solves the equationV̂ (x) = 0 and x ∈ X(x),(3.8)

where the set X(x) = {y ∈ Rn , ∀i, g i (yi , x−i ) ≤ 0} and V̂ defined in (3.7). Furthermore, thefunction V̂ is such that ∀x ∈ X(x), V̂ (x) ≥ 0.
As for the QVI reformulation, Equation (3.8) has a very complex structure. There is no
particular algorithm able to solve this problem for a general constrained set X(x). But a
simplification will occur in a special case.
141

Chapitre 3. Calcul d’équilibre de Nash généralisé

3.1.4

Jointly convex caseIn this subsection, we present reformulations for a subclass of GNEP called jointly convex
case. Firstly, the jointly convex setting requires that the constraint function is common to allplayers g 1 = · · · = g N = g. Then, we assume, there exists a closed convex subset X ⊂ Rn
such that for all player i,{yi ∈ Rni , g(yi , x−i ) ≤ 0} = {yi ∈ Rni , (yi , x−i ) ∈ X}.
The convexity of X implies that the constraint function g is quasiconvex with respect to all
variables. However, we generally assume that g is convex with respect to all variables.
KKT conditions for the jointly convex caseIn the jointly convex case, the KKT conditions (3.2) become

tel-00703797, version 2 - 7 Jun 2012∇xi θi (x? ) + ∇xi g(x? )λi? = 0, 0 ≤ λi? , −g(x? ) ≥ 0, g(x? )T λi? = 0,

(3.9)

since the constraint function is common to all players. But, there are still N Lagrange multipliers λi . Under the same condition as Subsection 3.1.1, a solution of this KKT system is alsoa solution of the original GNEP.
VI formulation for the jointly convex caseIn the jointly convex case, the QVI reformulation (3.5) simplifies to a variational inequality
problem VI(X, F )∇x1 θ1 (x)..
∀y ∈ X, (y − x)T F (x) ≥ 0, with F (x) = (3.10)
,.
∇xN θN (x)under certain conditions with X = {y ∈ Rn , ∀i, g(yi , x−i ) ≤ 0}. To understand that VI
problem solutions are a subclass of GNEs, we just compare the KKT conditions of the VIP(3.10) and Equation 3.9. This is given in the following theorem, see, e.g., Theorem 3.9 ofFacchinei and Kanzow (2009), Theorem 3.1 of Facchinei et al. (2007).
Theorem. Assuming θi and g are C1 functions and g is convex and θi player-convex. The
subset of variational equilibrium verifying Equation (3.10) are the solution of the KKT system(3.9) with a common multiplier λ1 = · · · = λN = λ? .
GNEs verifying the VI problem in Equation (3.10) are called variational or normalized
equilibrium, see also Kulkarni and Shanbhag (2010) for a detailed discussion of the VI representation of the QVI reformulation of GNEPs.
NIF formulation for the jointly convex case
Recalling that for the Nikaido-Isoda function (3.7), the gap function isV̂ (x) = sup ψ(x, y).
y∈X(x)

142

3.2. Methods to solve nonlinear equations
In the jointly convex case, we getV̂ (x) = 0 and x ∈ X(x),(3.11)

where the set X(x) = {y ∈ Rn , g(yi , x−i ) ≤ 0}. Still the computation of V̂ is a complex optimization over a constrained set X(x). As in the previous subsection, the class of GNE called
variational equilibrium can be characterized by the NI formulation. We have the folllowing
theorem.
Theorem. Assuming θi and g are C1 functions and g is convex and θi player-convex. x? isa variational equilibrium if and only if x? ∈ X and V (x? ) = 0 with V defined as
V (x) = sup ψ(x, y).

tel-00703797, version 2 - 7 Jun 2012y∈X

In the rest of the paper, we do not study all algorithms but rather focus on the mostpromising ones. We restrict our attention to general GNEPs and algorithms to solve the KKT
system presented in Subsection 3.1.1. So, we do not study jointly convex GNEPs for which
special methods have been proposed in the literature. These two situations differs widely,since in the general GNEP, we have to solve a nonlinear equation, while for the jointly convex
case, we solve a fixed point equation or a minimization problem.

3.2

Methods to solve nonlinear equations

As introduced in many optimization books, see, e.g., Dennis and Schnabel (1996); Nocedal and Wright (2006); Bonnans et al. (2006), an optimization method to solve a nonlinear
equation or more generally to find the minimum of a function is made of two components:
a local method and a globalization scheme. Assuming the initial point is not “far” from theroot or the optimal point, local methods use a local approximation of the function, generally
linear or quadratic approximation based on the Taylor expansion, that is easier to solve. The
globalization studies adjustments to be carried out, so that the iterate sequence still converges
when algorithms are badly initialized.
To emphasize the prominent role of the globalization, we first look at a simple example ofa nonlinear equation. Let F : R2 7→ R2 be defined as
 2

x + x2 − 2
F (x) = x11−1 2 3.
e+ x2 − 2This function only has two roots x? = (1, 1) and x̄ = (−0.7137474, 1.2208868). We notice thatthe second component of F explodes as x1 tends to infinity.
On Figure 3.2, we plot the contour level of the norm ||F (x)||2 , as well as two iteratesequences (xn ), (yn ) (see numbers 0, 1, 2,. . . ) starting from the point (x0 , y0 ) = (−1, −3/2).
The first sequence (xn ) corresponds to a “pure” Newton method, which we will present after,
whereas the second sequence (yn ) combine the Newton method with a line search (LS). We
can observe the sequence (yn ) converges less abruptly to the solution x? than the sequence(xn ).
On Figure 3.3, we plot the contour level of the norm ||F (x)||2 with two iterate sequences(xn ), (yn ), for pure and line-search Newton, respectively. But this time, sequences are initiatedat (x0 , y0 ) = (2, 1/2). Despite being close the solution x̄, the pure sequence (xn ) wanders in
143

Chapitre 3. Calcul d’équilibre de Nash généralisé

2

Contour of ||F(x)|| - Newton GLS (-1,-1.5)

2

Contour of ||F(x)|| - Newton Pure (-1,-1.5)

4 5
6
7
1
x_2

2

3

0

47
5
6

0

x_2

1

3

2

<1e-04
<0.01
<0.1
>0.1

-1

0

1

<1e-04
<0.01
<0.1
>0.1

0
-2

-2

0

-2

-1

1

-1

1

2

-2

-1

x_1

0

1

2

x_1

Figure 3.2: Pure and LS Newton methods, convergence to x?

52<1e-04
<0.01
<0.1
>0.1

Contour of ||F(x)|| - Newton GLS (2,0.5)

1
2

4
5
63

1

10

tel-00703797, version 2 - 7 Jun 2012

Contour of ||F(x)|| - Newton Pure (2,0.5)

2

0

x_2

0

14
13

0

x_2

10
4

12
11
-1

-5

10

<1e-04
<0.01
<0.1
>0.1

-10

9

-10

-5

-2

8
0

5

10

3
15

-2

-1

x_1

0

1

2

x_1

Figure 3.3: Pure and LS Newton methods, convergence to x̄

the whole space R2 whereas the sequence (yn ) converges very quickly to x̄. The sequence of(xn ) diverges because we use a pure local method.
This small example illustrates how essential a globalization scheme is. The globalization
step stabilizes the sequence and improves its convergence. In the sequel, we first consider asmooth equation F (z) = 0. Subsection 3.2.1 presents local methods whereas Subsection 3.2.2
describes globalization schemes in a smooth framework. Then in Subsection 3.2.3, we present
necessary extensions to deal with semismooth equations as encountered in Equation (3.3).
Finally, Subsection 3.2.4 details contrained equation methods.

3.2.1

Local methods

Local methods are sequences zk+1 = zk + dk where dk is a root of a certain equation
based on a local approximation of the root function F : Rn 7→ Rn . The first and most simple
method, the Newton method, uses a first-order Taylor expansion
F (z + h) = F (z) + J(z)h + o(h) ,144

3.2. Methods to solve nonlinear equations
where J denotes the Jacobian ∗ JacF . Let MkN (h) = F (zk ) + J(zk )h be the model function.
At the kth iterate, the Newton step consists in solving the system MkN (dk ) = 0. We get thefollowing equation
J(zk )dk = −F (zk ).
(3.12)
Note that there is no guarantee that J(zk ) is nonsingular.
Another method consists in replacing the Jacobian by an approximate matrix, which willbe always invertible. The direction dk solves
Hk dk = −F (zk ),(3.13)

where Hk is updated by a so-called quasi-Newton scheme. This is equivalent to a model
function MkQN (h) = F (zk ) + Hk h. A quasi-Newton scheme needs an iterative process to
update the approximate Jacobian from Hk to Hk+1 . The choice of the matrix is large, since
there are n2 terms. We can set first that

tel-00703797, version 2 - 7 Jun 2012

QN
F (zk ) = Mk+1
(zk+1 − zk ) ⇔ Hk+1 sk = yk ,where sk = zk+1 − zk and yk = F (zk+1 ) − F (zk ). The latter equation is called the secant
equation. We could have considered a scheme to approximate directly the inverse of theJacobian by Wk with the secant equation sk = Wk+1 yk . But still, we have an underdetermined
system n equations for n × n matrix.
Having no other property on the Jacobian of F (e.g. symmetry or positiveness), we generally consider the matrix that makes the smallest possible change to the preceding Jacobian
according to the Frobenius norm † , see (Nocedal and Wright, 2006, Chap. 11). For an implementation point of view, the smallest
 possible change feature reduces the linear algebra
computation work from O n3 to O n2 cost.
If we consider minimizing minH ||H − Hk ||F for all matrix H verifying the secant equation,
we get the (good) Broyden scheme
Hk+1 = Hk +(sk − Wk yk )ykT Wkyk − Hk sk T
s
⇔
W
=
W
+
,
k+1
k
k
sTk sksTk Wk yk

using the Sherman-Morrison formula. Otherwise, if we minimize minW ||W − Wk ||F for all
matrix W verifying the secant equation, then we will obtain the (bad) Broyden scheme
Hk+1 = Hk +(sk − Hk yk )ykT Hksk − Wk yk T
⇔ Wk+1 = Wk +yk .
Tyk Hk skykT ykAccording to Broyden (1965), this method appears often unsatisfactory in practice, so it will
be discused no further.
∗. The Jacobian of a function f is defined as usual by
 ∂f
1
...
 ∂x1
 ..
∂fi
Jac f (x) =  .
∂xj

∂fb
...
∂x1

∂f1
∂xa




.. 
(x).
. 


∂fb
∂xa

And the ∇f denotes the transpose of the Jacobian.
†. The Frobenius norm (also called the Euclidean norm) for matrix A is defined as ||A||F =qP

2i,j |aij | .

145

Chapitre 3. Calcul d’équilibre de Nash généralisé
For a general discussion of quasi-Newton methods, we refer to Dennis and Morée (1977)
and for details on quasi-Newton methods specialized to nonlinear equations, see Broyden(1965) or (Nocedal and Wright, 2006, Chap. 11).
Another way to solve the equation F (z) = 0 requires minimizing the norm f (z) =1
2
2 ||F (z)||2 . But not all minima of the function f are roots of F because we must have f (z) = 0.
A widely known method for this least square problem is the Gauss-Newton method. We minimize the model function 21 MkN (dk )T MkN (dk ). We get
J(zk )T J(zk )dk = −J(zk )T F (zk ).

(3.14)

tel-00703797, version 2 - 7 Jun 2012

To prevent the right-hand side matrix to be nonsingular, the Levenberg-Marquardt method
modifies Equation (3.14) toJ(zk )T J(zk ) + λk I dk = −J(zk )T F (zk ),(3.15)
where I denotes the n × n identity matrix and λk ≥ 0. Various choices of Levenberg-Marquartparameter are possible: Fan and Yuan (2005) consider terms λk = ||F (zk )||δ2 with δ ∈ [1, 2].
In practice, δ = 1, i.e. λ = ||F (zk )||2 , works much better than other δ’s. We may also use||Jk || or min(||F (zk )||2 , ||JkT F (zk )||2 ). The Levenberg-Marquart is sometimes referred to as
the modified Gauss-Newton method for its relation to Gauss-Newton method.
Summarizing this first subsection, local methods consider sequences of the form zk+1 =zk +dk where the direction dk is a solution of one of the above equations. Newton direction uses
Equation (3.12), quasi-Newton Equation (3.13) and Levenberg-Marquardt Equation (3.15).

3.2.2

Globalization schemes

Now, we focus on globalization schemes, as Figures 3.2 and 3.3 show their relevancy. There
are mainly two frameworks: line search and trust-region methods. To our knowledge, only
two other methods exist continuation/homotopy, e.g., Allgower and Georg (2003) or Chapter
8 of Facchinei and Pang (2003)), or successive approximations, see, e.g., Qi and Jiang (1997)
or Qi and Chen (1995).
Line-search techniques
Line-search techniques are a refinement of the local sequence by considering the sequencezk+1 = zk + tk dk where tk ∈]0, 1] is the stepsize in direction dk at the current iterate zk . Notethat the direction dk is not unitary, i.e. ||dk || >> 1 generally. Line-search techniques propose
criteria to choose tk . As the stepsize may reduces the (full) step from zk to zk+1 , line-searchversion of an algorithm is sometimes called the damped version of that algorithm.
Let f be a merit function. We define the function t 7→ φk (t) = f (zk + tdk ). We want tofind a good minimizer of φk . However, it is useless to find the global minimizer arg min φk (t),because we want to solve the outer problem F (z) = 0, and not the inner problem min φk (t).
In the following, we assume we have a descent direction dk for the merit function f , as aminimal condtion to choose tk is f (zk+1 ) < f (zk ). This descent direction condition translates
to φ0k (0) < 0. We are focused on two things, tk should be big enougth to ensure a sufficientdecrease of φ, and also tk should not be too small to guarantee a sufficient big step.
One could think that requiring f (zk+1 ) < f (zk ) is enough to show convergence, but unfortunately not. In literature, see, e.g., Dennis and Schnabel (1996); Nocedal and Wright (2006);
1463.2. Methods to solve nonlinear equations
Bonnans et al. (2006), two typical conditions are used to determine an appropriate stepsize.
Let 0 < c1 < 1/2 < c2 < 1. The Armijo condition ensures a decrease of fφk (t) ≤ φk (0) + tc1 φ0k (0) ⇔ f (xk + tdk ) ≤ f (xk ) + tc1 ∇f (xk )T dk .
The curvature condition ensures an increase of ∇φ, implying a decrease of f ,tel-00703797, version 2 - 7 Jun 2012

φ0k (t) ≥ c2 φ0k (0) ⇔ ∇f (xk + tdk )T dk ≥ c2 ∇f (xk )T dk .
These two conditions are referred to the Wolfe conditions. In this paper, we use a backtracking
algorithm, for which the curvature condition is always satisfied. Let tk,0 = 1 be the initial
guess of the stepsize. The backtracking algorithm is defined as follows
Repeat until f (xk + tdk ) ≤ f (xk ) + tk,i c1 ∇f (xk )T dk satisfied,– propose a new tk,i+1 using tk,i , . . . tk,0 .
end Repeat
This algorithm always tests a full step with tk,0 = 1, otherwise the above algorithm tries a
new stepsize. For the backtracking line search, a classic result shows that the full step will beeventually satisfied as zk tends to a solution. We test two stepsize proposal algorithms. The
geometric line search uses
tk,i+1 = ρ × tk,i ,with 0 < ρ < 1, whereas the quadratic line search uses a quadratic approximation of φ using
the information φk (tk,i ), φk (tk,i−1 ), φ0k (tk,i−1 ). We get
tk,i+1 = −φ0k (tk,i−1 )t2k,i1.
2 φk (tk,i ) − φk (tk,i−1 ) − φ0k (tk,i−1 )tk,i

Other proposal, such as cubic approximation, are given in Chapter 3 of Bonnans et al. (2006)
or Chapter 6 of Dennis and Schnabel (1996).
Until now, we do not specify the merit function f . For nonlinear equation, we generallychoose f (z) = 21 ||F (z)||22 , sometimes referred to the residual function. This merit function has
some defficiencies, since a local minimum is not necessarily a root of the function F . We will
see later in the GNEP context, that f has still some interesting properties.
Line-search methods require to a tractable formula for the gradient ∇f (z) = JacF (z)T F (z),when testing the Armijo condition. However, in a quasi-Newton framework, we do not necessarily have a tractable Jacobian. One way to deal with this is to use a numerical Jacobian,e.g., based on the forward difference. We use Dennis and Schnabel (1996)’s algorithm A5.4.1
defined by
F (z + hj ej ) − F (z)∈ Rn ,
hj√
where ej is the jth unit vector and hj a small step, typically, hj = zj where  is the epsilon
machine ( = 1e−16 ).
D(F )(z) = (D1 , . . . , Dn ), with Dj =Trust-region approach
Trust-region strategies relaxe the constraint that dk is a descent direction. Line searchassumes the “best” point from zk lies on the half-line zk + R+ dn . Quoting Bonnans et al.
(2006), “what is magic in this half line? answer: nothing”. Trust-region approach will look for147

Chapitre 3. Calcul d’équilibre de Nash généraliséappropriate steps hk in a “small” region around zk . Such regions are not the half-line as inline search.
To find the root of F (x) = 0, trust-region methods minimizes a local quadratic approximation mk of a merit function f : Rn 7→ R on a bounded subset: the trust region{z, ||z − zk || ≤ ∆k }. The name comes from viewing ∆k as providing a region in which we cantrust mk to adequately model f . Classic trust region methods consider the following model
function1
mk (h) = f (zk ) + g(xk )T h + hT H̃k h,
2
with g(xk ) the approximate gradient of f and H̃k a (ideally) positive definite matrix approximating the Hessian of f .
To adapt the trust region radius ∆k , we define the following ratio

tel-00703797, version 2 - 7 Jun 2012ρk (h) =f (xk ) − f (xk + h).
mk (0) − mk (h)ρk (h) is the ratio of the actual reduction and the predicted reduction of the merit function fora step h. The higher is ρk (h), the higher is the reduction of the merit function f for a givenh. Now, we can define a generic algorithm for a model function mk , see, e.g., Nocedal andWright (2006).
Init ∆0 = 1, η0 > 0, 0 < η1 < η2 < 1 and 0 < γ1 < 1 < γ2 .
Iterate until a termination criterion is satisfied,– get hk = arg min mk (h) (approximately),||h||<∆k– compute ρk (hk ),– if ρk (hk ) < η1 then ∆k+1 = γ1 ∆k (unsuccessful),– else if ρk (hk ) > η2 and ||hk || = ∆k then ∆k+1 = min(γ2 ∆k , ∆max ) (very successful),– else ∆k+1 = ∆k .
– next iterate– if ρk (hk ) > η0 then xk+1 = xk + hk ,– else xk+1 = xk .
end Iterate
Typical values of parameters are ∆0 = 1 or ||g100 || , ∆max = 1010 for radius bounds, η0 = 10−4 ,η1 = 41 , η2 = 34 for ratio threshold, γ1 = 12 and γ2 = 2 for radius expansion coefficients.
If readers have been attentive, then they should have noticed that the algorithm cannot be
used directly. In fact, we have to determine how to compute the solution hk of the following
minimization problem
min mk (h).
||h||<∆k

As for line search techniques, this problem has to be solved approximately as this problem
is not our primary concern. There are two main methods to achieve this: Powell’s doglegand double dogleg methods. These two methods are heuristics to compute in one step anapproximate solution hk .
Let pSk be the scaled steepest descent direction and pN
k be the Newton point defined asg T gk−1
pSk = − T k
gk and pN
k = −H̃k gk .
gk H̃k gk
The Powell dogleg method uses a linear approximation of the model function mk (h). The
Powell dogleg method is as follows, see, e.g., Chapter 6 of Powell (1970).
148

3.2. Methods to solve nonlinear equations

tel-00703797, version 2 - 7 Jun 2012

– If ||pN || ≤ ∆k , then h? = pN .
– Else if ||pS || ≥ ∆k , then h? = ∆k /||pS || × pS .
– Otherwise, we choose a convex combination between the two points pS and pN . That is
we find a λ ∈ [0, 1] such that ||pS + λ(pN − pS )|| = ∆k . We get h? = λ? pN + (1 − λ? )pSwithp− < pS , pN − pS > + < pS , pN − pS >2 −||pN − pS ||2 (||pS ||2 − ∆k )?
λ =,||pN − pS ||2
where < ., . > denotes the scalar product and ||.|| denotes the Euclidean norm.
Nocedal and Wright (2006) also propose a “simple” dogleg method which remove the step||pS || ≥ ∆k , see Algorithm 11.6.
The double dogleg method finds an approximate solution of the model function mk (h)assuming the Hessian matrix is H̃k = LT L. The double dogleg method is a variant Powelldogleg method. Dennis and Schnabel (1996) propose the following procedure.
– If ||pN || ≤ ∆k , then h? = pN .
– Else if ηk ||pN || ≤ ∆k , then h? = pN × ηk /∆k .
– Else if ||pS || ≥ ∆k then h? = pS × ∆k /||pS ||.
– Otherwise, we choose λ such that ||pS + λ(ηk pN − pS )|| = ∆k . We get back to the Powelldogleg case with ηk pN instead of pN .
where the parameter ηk ≤ 1 is defined asηk = 0.2 + 0.8

α2,
β|gkT dN |with α = ||gk ||2 , β = ||Lgk ||2 .
As for line-search techniques, we use the merit function f (z) = 12 ||F (z)||22 . We recall thatthe gradient is given byg(z) = JacF (z)T F (z).
Therefore, the approximated Hessian H̃k is JacF (zk )T JacF (zk ) as in the Gauss-Newton
model. Hence, the steepest descent point and the Newton point have the following expressiong T gkT
pSk = − T kT
gk and pN
k = −Jk gk .
gk Jk Jk gk
As in the previous subsection, when working a quasi-Newton method, the Jacobian is numerically approximated by a forward difference.
The special case of the Levenberg-Marquardt method
Until now, all globalization methods are adapted for the Newton or the Broyden direction
defined in Equations (3.12) and (3.13). We need to precise how to globalize the LevenbergMarquardt direction. This method was introduced in the context of the least-square problem min 12 ||F (z)||22 . In fact, there is a relation between the trust-region approach and theLevenberg-Marquardt method. The solution to the quadratic problem
1
min f (zk ) + g(xk )T h + hT H̃k h
2||h||<∆k149

Chapitre 3. Calcul d’équilibre de Nash généralisé
is equivalent to the problem of finding λ? , h? such that

tel-00703797, version 2 - 7 Jun 2012(H̃k + λ? I)h? = −g(xk ) and λ? (∆k − ||h? ||) = 0,

(3.16)

with the condition that H̃k + λ? I is a positive semidefinite matrix. We note in Equation (3.16)
that variable λ? has the same role as in the Levenberg-Marquardt method.
We can easily interpret Equation (3.16). If h? lies strictly inside the trust-region thenparameter λ? is zero. Otherwise, h? hits the radius ∆k , and then parameter λ? is set to a
positive value. With this interpretation, the use a trust-region approach with the LevenbergMarquardt method is redundant.
However, we still consider a globalization strategy for the Levenberg-Marquardt method.
Firstly, we test the geometric line-search strategy, defined in Subsection 3.2.2, which is proposed in Fan and Yuan (2005), Yamashita and Fukushima (2000). Secondly, we use theadaptive Levenberg-Marquardt method discussed in Fan (2003). The method consists in adjusting the parameter λk based on λk = µk ||F (zk )||2 , where µk is udpated at each iteration
depending on the value of the ratio ρk . Their Algorithm 2.2 updates µk as in the genericalgorithm of the previous Subsection 3.2.2. So, we do not restate here the updating scheme.

3.2.3

Extension to a semismooth setting

In the two previous subsections, a blanket assumption of differentiability of the root function F is assumed. In the GNEP context, the root function is defined asΦ(x, λ) =D(x, λ),φ◦ (−g(x), λ)where the bottom part has some specificities: the function Φ is not everywhere differentiable.
Hence, some adjustments have to be made to use local methods and globalization strategies,
previously presented.
To emphasize the pitfalls between the assumed conditions of the previous methods andour current setting, we present local/global convergence results in the differentiable setting.
Then, we describe the necessary adjustment for the GNEP context.
Differentiable setting
By definition, the Newton and Levenberg-Marquardt methods require the root function
F to be differentiable, when the Broyden method does not. Let us analyze the convergenceconditions for these methods. We concatenate all theorems in one, see, e.g., Theorems 11.2
and 11.5 of Nocedal and Wright (2006) for Newton and Broyden, respectively, Theorem 2.1 ofYamashita and Fukushima (2000) or Theorem 10 of Fischer (2002) for Levenberg-Marquardt.
Theorem. Suppose that F is continuously differentiable in a open convex set O ⊂∈ Rn . Letz ? be a solution of F (z) = 0 and let (zkN )k be the sequence generated by the Newton method.
If the Jacobian at the solution J(z ? ) is nonsingular, then (zkN )k converges superlinearly to z ? .
If in addition, F is Lipschitz continuously differentiable ∗ near z ? , then the convergence rate
is quadratic.
∗. F is continuously differentiable C1 and the Jacobian is Lipschitzian.

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3.2. Methods to solve nonlinear equations
Let (zkB )k be the sequence generated by the Broyden method. If for the starting points, there
exist δ,  > 0, such that||z0B − z ? || ≤ δ, ||B0 − J(z ? )|| ≤ ,

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then (zkB )k converges superlinearly to z ? .
Let (zkLM )k be the sequence generated by the Levenberg-Marquardt method. If F is Lipschitz continuously differentiable on O and ||F (x)|| provides a local error bound, then (zkLM )kconverges superlinearly to z ? .
In summary, convergence theorems require (i) F is Lipschitz continuously differentiable(LC) 1 in an open convex set around a solution z ? and (ii) the Jacobian at the solution J(z ? )is nonsingular for Newton and Broyden method or the function F verifies a local error bound.
Let f be the merit function. The global convergence of line-search techniques is guaranteed
when we have the following conditions: (i) the set L = {z, f (z) ≤ f (z0 )} is bounded, (ii) the
Jacobian is bounded in an open convex set around a solution z ? and (iii) line-search always
satisfies the Wolfe conditions with a descent direction, see, e.g. Theorem 11.6 of Nocedal andWright (2006). We have seen that the backtracking algorithm satisfies the Wolfe conditions
at each step.
The convergence of trust-region strategies is proved with similar conditions and requires
also that the set L and the Jacobian are bounded. Furthermore, the approximated solutionof the quadratic local problem min mk (h) such that ||h|| < ∆k verifies two conditions: (i)mk (0) − mk (h? ) ≥ c1 ||JkT Fk || min(∆k , ||JkT Fk ||/||JkT Jk ||) for some c1 > 0 and (ii) ||h? || ≤ γ∆kfor some γ ≥ 1.
Overall, local methods and globalization strategies need differentiability by definition andLipschitz continuity and some additional conditions for convergence. In the next subsection,
we will see how these conditions can be weakened.
Toward the non-differentiable setting
Getting back to our original GNEP, we want to solve the KKT conditions (3.3) using thecomplementarity reformulation of Subsection 3.1.1. Thus, the root function Φ : Rn × Rm 7→
Rn × Rm is defined asD(x, λ)Φ(x, λ) =,φ◦ (−g(x), λ)where the first component D(x, λ) is composed of N derivatives of the Lagrangian function
Li (x, λi ) and the second component φ◦ (−g(x), λ) is the component-wise application of thecomplementarity function φ on the overall constraint function g : Rn 7→ Rm . Firstly, only thebottom component has some non-differentiability problem, because most complementarity
functions φ(., .) are non-differentiable at (0, 0). In this paper, we use √the minimum functionφ∧ (a, b) = min(a, b) and the Fischer-Burmeister function φF B (a, b) = a2 + b2 − (a + b).
To deal with non-differentiability, Clarke (1975) introduced the generalized gradient. Few
laters, its famous book, Clarke (1990), provides a comprehensive presentation of the mathematical fundamentals of nonsmooth analysis. We briefly present here some elements necessary
to our paper, refer to Appendix 3.5.1 for details.
Let G : Rn 7→ Rm a function with component Gj . By the Rademacher theorem, a locally
Lipschitzian function is almost everywhere differentiable. We define first the limiting Jacobian,also called B(ouligand) subdifferential, denoted by ∂B G(x).
151Chapitre 3. Calcul d’équilibre de Nash généraliséDefinition (limiting Jacobian). The limiting Jacobian of G at x is defined as∂B G(x) = {Vx , ∃(xk )k ∈ DG , xk → x, Jac G(xk ) → Vx },where DG is the differentiability set of G.
For his desirable properties, the Clarke’s generalized Jacobian, based on the limiting Jacobian, is commonly used.

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Definition (generalized Jacobian). The generalized Jacobian ∂G(x) of G at x is the convex
hull of the limiting Jacobian ∂B G(x) ∗ .
Let us start with an example. Using the Example 7.1.2 of Facchinei and Pang (2003), we
define a function G : R2 7→ R2 with components G1 (x) = min(x1 , x2 ) and G2 (x) = |x1 |3 − x2 .
G is not differentiable at (0,0). By splitting R2 into 4 parts, we can compute the limiting
Jacobian

 

1 0
0 1
∂B G((0, 0)) =
,
.
0 −1
0 −1
Then, the generalized Jacobian is ∂G((0, 0)) = co ∂B G((0, 0)). From a computational pointof view, using a component-wise version of the generalized Jacobian ∂G(x) ⊂ ∂G1 (x) × · · · ×∂Gm (x) is sometimes useful, see Appendix 3.5.1.
Now, we focus on the properties of the generalized Jacobian and more flexible criterion thancontinuous differentiability. Before introducing the semismoothness, we present directional
derivative of a function.
Definition (directional derivative). For a locally Lipschitzian function G, the (classical) directional derivative at x along h ∈ Rn is defined as
G(x + th) − G(x).
t↓0tG0 (x; h) = lim
The Hadamard directional derivative is
G0H (x; h) =G(x + th0 ) − G(x).
h →h,t↓0tlim
0

The difference between classical and Hadamard directional derivatives lies the fact that
we look at all directions h0 → h and not only at h. Now, we can define the semismoothness.
Definition (semismooth). A locally Lipschitzian function G is semismooth at x, if for alld ∈ Rn , the following limit exists
˜
lim
V d.
˜
V ∈∂G(x+td)

˜
d→d,t↓0

In the case, this limit exists and equals to G0H (x; h).
∗. Note that if m = 1, the generalized Jacobian reduces to the generalized gradient, see Theorem 2.5.1 ofClarke (1990) for this characterization of the generalized gradient.

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3.2. Methods to solve nonlinear equations
Therefore, the semismoothness of function H at x requires the Hadamard directional
derivative at x to exists along any direction converging to h and not only along h.
Examples of semismooth functions are smooth, convex and piecewise linear functions. An
important property is that composite, scalar products, sums, minimum, maximum preserve
semismoothness. Indeed for the minimum function, the breakpoint is at (0, 0). By standard calculations, the directional derivate of φ∧ at this point is φ∧ ((0, 0); (a, b)) = min(a, b).
Futhermore, for all nonzero vector (a, b), φ∧ is differentiable and is uniquely given by


11a≤b
∇φ∧ (a, b) =
.

11a>b

We deduce

tel-00703797, version 2 - 7 Jun 2012   



1
0
λ
∂B φ∧ (0, 0) =,and ∂φ∧ (0, 0) =, λ ∈ [0, 1] .
0
1
1−λFurthermore, for all V ∈ ∂φ∧ (c, d), such that (c, d) → (a, b), we have a− φ∧ ((0, 0); (a, b)) = a11c≤d + b11c>d − min(a, b) = o((a, b)) .
Vb(3.17)(3.18)

By Appendix 3.5.1, we conclude that φ∧ is semismooth at (0,0).
Finally, we introduce the strong semismoothness, also called 1-order semismoothness, thatwill be used in most convergence theorems.
Definition (strongly semismooth). A locally Lipschitzian function G is strongly semismoothat x if for all d → 0, ∀V ∈ ∂G(x + d), we haveV d − G0 (x, d) = O ||d||2 .
Based on Equation (3.18), we conclude that the minimum function φ∧ is not strongly
semismooth at (0, 0). But, the Fischer-Burmeister function is strongly semismooth at (0, 0).
In fact, by standard calculations, for all nonzero vector (a, b), we have
!
√ a
−
1
2
2
a +b
∇φF B (a, b) =.
√ b
−1
a2 +b2
We deduce∂B φF B (0, 0) = {∇φF B (a, b), (a, b) 6= (0, 0)} and ∂φF B (0, 0) = B̄((−1, −1), 1),(3.19)

where B̄ denotes the closed ball.
Furthermore, for all V ∈ ∂φF B (c, d), such that (c, d) → (a, b), we have aV
= φF B (a, b) and φF B ((0, 0); (a, b)) = φF B (a, b).
b
Hence, we have aV
− φF B ((0, 0); (a, b)) = 0.
b
153Chapitre 3. Calcul d’équilibre de Nash généralisé
We conclude that φF B is strongly semismooth at (0, 0), as proved in Kanzow and Kleinmichel(1998).
Now, we can express appropriately the generalized Jacobian of the GNEP. We denote byJΦ (z) elements of the generalized Jacobian ∂Φ(z). Using chain rules and previous definitions,
we have 

Jac x D(x, λ)diag ∇xi g i (x) iJΦ (z) =,(3.20)−Da (z)Jac x g(x)Db (z)
where Jac x denotes the Jacobian with respect to x and diag[...] represents a block diagonal
matrix, see Appendix 3.5.1 for a extended representation of the generalized Jacobian JΦ .
The diagonal matrices Da and Db are given by

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Da (z) = diag[a1 (x, λ1 ), . . . , aN (x, λN )] and Db (z) = diag[b1 (x, λ1 ), . . . , bN (x, λN )],with ai (x, λi ), bi (x, λi ) ∈ Rmi defined as( 
φ0a (−gji (x), λij ), φ0b (−gji (x), λij )iiii(aj (x, λj ), bj (x, λj )) =(ξij , ζij )if (−gji (x), λij ) 6= (0, 0),
if (−gji (x), λij ) = (0, 0),

where φ0a (resp. φ0b ) denotes the derivative of φ with respect to the first (second) argument a(b) and (ξij , ζij ) ∈ B̄(pφ , cφ ), the closed ball at pφ of radius cφ .
Let us precise the top-left part.
Jac x1 L1 (x, λ1 ) . . . Jac xN L1 (x, λ1 )


..
..
Jac x D(x, λ) = ,.
.
Jac x1 LN (x, λN ) . . .

Jac xN LN (x, λN )P iiiwhere Li (x, λi ) = ∇xi θi (x) + m
j=1 ∇xi gj (x)λj . The top-right part is block diagonal and givenbyJac x1 g 1 (x)T
0


..
.

.
0

Jac xN g N (x)TLet us specify the parameters pφ and cφ for the two considered complementarity functions.
For the minimum function, using Equation (3.17), we have pφ = (1/2, 1/2) and cφ = 1/2.
For the Fischer-Burmeister function, using Equation (3.19), we have pφ = (−1, −1) and cφ =1. We refer to Kanzow and Kleinmichel (1998) and Facchinei and Pang (2003) for othercomplementarity functions.
If functions θi and gi are Ck+1 , by the chain rule, the root function Φ defined in Equation (3.20) is Ck except at points z such that gji (x) = λij = 0. At these points, when thecomplementarity function is φ∧ , Φ is semismooth, while for φF B , Φ is strongly semismooth.
Extension and local convergence in the semismooth framework
As the Jacobian of the root function is not available, the direction computation of local
methods presented in Subsection 3.2.1 must be adapted. The solution consists in replacing
the Jacobian by an element of the generalized Jacobian.
Considering the Newton method (3.12), the direction solves
Jk dk = −F (zk ),154(3.21)

3.2. Methods to solve nonlinear equations
whereas for the Levenberg-Marquardt method (3.15), the direction solves
 T
Jk Jk + λk I dk = −JkT F (zk ),(3.22)

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with Jk ∈ ∂F (zk ). Corresponding sequences are called generalized Newton and generalized
Levenberg-Marquardt methods, respectively. For the quasi-Newton direction does not require
any modification. Some authors also use the B-subdifferential ∂B F (zk ) or the component wise
B-subdifferential ∂B F1 (zk ) × · · · × ∂B Fn (zk ) instead of the generalized Jacobian in Equations(3.21) and (3.22).
Now, we present theorems for local convergence. Local convergence theorems for smoothfunctions have been extended to nonsmooth functions by Qi and Sun (1993), cf. Theorem 3.2.
We give below a slightly more general version than the original version.
Theorem. Let z ? a solution of F (z ? ) = 0. If F is locally Lipschitzian and semismooth at z ?
and all elements ∗ J ? ∈ ∂B F (z ? ) are nonsingular, then the generalized Newton method is welldefined and converges superlinearly to z ? . If in addition, F is strongly semismooth at z ? , thenthe convergence rate is quadratic.
A version of the previous theorem exists when the generalized Newton method use thelimiting Jacobian Jk ∈ ∂B F (zk ) (instead of the generalized Jacobian) in Equation (3.21), see,e.g., Sun and Han (1997), Jiang and Ralph (1998) or Facchinei et al. (2009). On a similaridea, Jeyakumar (1998) presents a convergence theorem for a generalized Newton method
when working with approximate Jacobian, which reduces to a single valued function undercertain assumptions.
For the quasi-Newton approach, extensions have been proposed in the literature, e.g. Ip
and Kyparisis (1992) and Qi (1997), where the differentiability is needed at a solution rather
than in an open convex. Lopes and Martinez (1999) give a minimal condition (lesser thansemismoothness) for a general quasi-Newton method to converge linearly.
As in the differentiable setting, e.g., Dennis and Schnabel (1996), the convergence of quasiNewton methods for semismooth functions is done in two steps: (i) a theorem gives conditionsof local linear convergence based on the limited difference between approximate Jacobian andelements in the generalized Jacobian, and (ii) another theorem gives an additional condition
for a general quasi-Newton method to converge superlinearly. We report here Theorems 4.1
and 4.2 of Sun and Han (1997).
Theorem. Let z ? a solution of F (z ? ) = 0. If F is locally Lipschitzian in the open convex
D ⊂ Rn such as z ? ∈ D. Consider the sequence z0 ∈ D and zk+1 = zk − Vk−1 F (zk ) with Vk an × n matrix updated by a quasi-Newton scheme.
Suppose F is semismooth at z ? and for all J ? ∈ ∂b F (x? ) are nonsingular. There exist constant, ∆ > 0 such that if ||z0 − z ? || ≤  and there exists Wk ∈ ∂b F (zk ) such that ||Vk − Wk || ≤ ∆,then the quasi-Newton sequence is well defined and converges linearly to z ? .
Theorem. Let F be locally Lipschitzian in a open convex D ⊂ Rn . Assume F is semismoothand ∀J ? ∈ ∂b F (z ? ) are nonsingular. Consider a sequence of nonsingular matrices Vk and∗. Originally, Qi and Sun (1993) use the generalized Jacobian and not the limiting Jacobian. But as
mentioned in Qi and Jiang (1997) and Qi (1993), there is a weaker condition for superlinear convergence tohold, that is all elements in the limiting Jacobian are nonsingular.

155

Chapitre 3. Calcul d’équilibre de Nash généralisépoints zk+1 = zk − Vk−1 F (zk ). If (zk )k converges to z ? then (zk )k converges superlinearly toz ? and F (z ? ) = 0 is equivalent to ∃Wk ∈ ∂b F (zk ),||(Vk − Wk )sk ||
= 0,
k→+∞
||sk ||
lim

with sk = zk+1 − zk .
Local convergence of the generalized Levenberg-Marquardt method is studied by Facchinei
and Kanzow (1997), cf. Theorem 6.

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Theorem. Let z ? a solution of F (z ? ) = 0, with F a locally Lipschitzian, semismooth at z ?
and ∀J ? ∈ ∂B F (z ? ) are nonsingular. If the Levenberg-Marquardt parameters (λk ) converge to0, then the (generalized) Levenberg-Marquardt converges superlinearly to z ? . If in addition, F
is strongly semismooth and locally directionnally differentiable at z ? , and λk = O(||Jk gk ||) or
λk = O(||gk ||), then the sequence converges quadratically.
We turn our attention to the assumption analysis of preceding theorems. All theorems
require the root function to be semismooth at a solution z ? . This is verified for our function
Φ defined in Equation (3.3) as long as the complementarity function φ is semismooth. Furthermore, strong semismoothness improves the convergence rate. In the previous subsection,
we have seen this requires for φ to be strongly semismooth, as e.g., the Fischer-Burmeisterfunction.
The nonsingularity condition is required only for elements J ? of the limiting Jacobian ata solution z ? . As analyzed in Facchinei et al. (2009), the limiting Jacobian (3.20) might havesome identical rows at the bottom part. Let us investigate first this issue. We recall first theexpression of generalized Jacobian 
Jac x D(x, λ)diag ∇xi g i (x) iJΦ (z) =.
−Da (z)Jac x g(x)Db (z)As only terms (ξij , ζij ) in diagonal matrices Da and Db change between the generalized andthe limiting Jacobian of Φ, we study the nonsingularity condition directly on the generalized
Jacobian. In a detailed form, the bottom part has the following structure−D1a (x, λ1 )Jac xN g 1 (x) D1b (x, λ1 )−D1a (x, λ1 )Jac x1 g 1 (x) . . .
0


..
..
..

,
.
.
.
a (x, λN )Jac g N (x) . . .
−DNx1

a (x, λN )JacN
−DN
xN g (x)b (x, λN )DN
(3.23)
where Dia and Dib are mi × mi diagonal matrices. In the following, we denote by Da -part andDb -part the left and right parts of Equation (3.23).
Assume the generalized Jacobian has two identical rows, say for players i and ĩ and components ji and jĩ . The, the Db -part requires that the ji th row of Dib and the jĩ th row of Dĩbequals zero
(3.24)bij (x, λij ) = bĩjĩ (x, λĩjĩ ) = 0,0with bij (x, λij ) = φ0b (−gji (x), λij ). Identical rows in the Da -part is equivalent to the n dimensional equationaij (x, λij )Jac x gji (x) = aĩjĩ (x, λĩjĩ )Jac x gjĩĩ (x).
(3.25)
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3.2. Methods to solve nonlinear equations
If φ = φ∧ for which φ0∧b (a, b) = 11b<a , then Equation (3.24) leads to gji (x) < λij andgjĩ (x) < λĩj . Since at a solution z ? both gji (x) and λij are nonnegative and (at least) one is zero.
ĩ

ĩ

Hence, gji (x) = gjĩ (x) = 0, i.e. active constraints. For the Da -part using φ0∧a (a, b) = 11a<b ,ĩEquation (3.25) leads to aij (x, λij ) = aĩj (x, λĩj ) = 1. Furthermore, Jac x gji and Jac x gjĩ are
ĩ

ĩ

ĩ

identical. So if the constraints gji , gjĩ are active and shared, then both Equations (3.24) andĩ
(3.25) are satisfied, i.e. the generalized Jacobian has two identical (null) rows.
If we use the Fischer-Burmeister function, Equation (3.24) is equivalent togji (x) = 0 and λij > 0,
and the same for j̃. This is equivalent to gji , gjĩ are strongly active constraints. Theseĩconditions are such that aij (x, λij ) and aĩj (x, λĩj ) are non-zero and expressed as aij (x, λij ) =tel-00703797, version 2 - 7 Jun 2012ĩĩaĩj (x, λĩj ) = −1. Equation (3.25) again requires that the constraint functions gji and gjĩ haveĩĩĩidentical derivatives.
We conclude that for these two complementarity functions, the bottom part of generalized
Jacobian Φ has two identical rows if and only if there are two shared and active constraints.
For the top part, there are less problem in general. As shown in Appendix 3.5.1, matrices areless sparse: identical rows implies that the constraint functions are linear and two objective
derivatives are identical.
If we exclude the pathological case described above, the nonsingularity of limiting Jacobian ∗ (3.20) requires that J ? has a non-zero determinant. Using the determinant formula forpartitionned matrix and the Schur complement, the nonsingularity condition can be expressedin two different formulations(i) that Jac x D(x, λ) and
Db (z) + Da (z)Jac x g(x)Jac x D(x, λ)−1 diag ∇xi g i (x) i(3.26)
are nonsingular,(ii) Db (z) andJac x D(x, λ) + diag
∇xi g i (x) i Db (z)−1 Da (z)Jac x g(x)(3.27)

are nonsingular.
Note, since Db (z) is a diagonal matrix, its nonsingularity is equivalent to have nonnull terms.
We will see in subsection 3.2.3 that (3.26) is typical. Following the above discussion, we
conclude that both the generalized Newton and the generalized Levenberg-Marquardt method
converges superlinearly to a solution z ? if one of the two conditions (3.26) and (3.27) aresatisfied and no constraint functions are shared.
Finally, we focus on the convergence of the Broyden method. Theorems of Sun and Han(1997) give minimal conditions for a quasi-Newton method to converge (superlinearly). In
Sun and Han (1997) or Qi and Jiang (1997), the Broyden method is used on smooth partsof the root function F . As emphasized in Jiang et al. (1996), when applying the Broyden
method directly on F , it is hard to show the convergence for a general semismooth function.
The weakest condition seems to be the Lipschitz continuity of B-derivative of F proved in Ip
and Kyparisis (1992). Let us note that the function F (x, λ) is not differentiable only when∗. We do not have to require all elements of ∂F but only elements of ∂B F .

157

Chapitre 3. Calcul d’équilibre de Nash généralisé
λij = gji (x) = 0. Otherwise, F is differentiable for which the convergence is established. In
this paper, we test the Broyden method without proving the convergence. To our knowledge,
the Broyden method is not used in the GNEP context.
Global convergence
Let us study the global convergence starting by line-search techniques. The merit functionused is the same function as for the differentiable setting the residual norm.

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f (z) = || F (z)||2 .
2The gradient is given by ∇f (z) = V T F (z), where V ∈ ∂F (z). As mentioned in Qi and Sun(1998), the merit function may be C1 , even if F is only semismooth.
Jiang and Ralph (1998) and Qi and Sun (1998) show the convergence of the Newton
method globalized with a backtracking (geometric) line search. When using the generalized
Jacobian, Jiang (1999) shows the convergence of the corresponding algorithm. All proofs relyon the fact that after a large number of iteration, the full Newton step is accepted, i.e. we getback to local convergence.
Theorem. Suppose that the root function F is a semismooth function and the merit function
f is C1 . Then any accumulation points z ? of the line-search generalized Newton method is astationary point of f , i.e. ∇f (z ? ) = 0. If z ? is a solution of F (z) = 0 and all matrices in∂B F (z ? ) are nonsingular, then the whole sequence converges superlinearly (resp. quadratically)
to z ? (if F is strongly semismooth at z ? ).
To our knowledge, the global convergence of general quasi-Newton methods for nonsmoothequation is not established. However, for the Levenberg-Marquardt with a backtracking linesearch technique, Jiang and Ralph (1998) and Jiang (1999) show the convergence. We present
below a light version of their theorem.
Theorem. Let (zk )k be a sequence of the generalized LM method globalized with a backtracking
line search to solve F (x) = 0 for a semismooth function F and a C1 merit function f . Assuming the direction step is solvable at each iteration, we denote by z ? an accumulation point.
If λk = min(f (zk ), ||∇f (zk )||) and elements in ∂B F (z ? ) are nonsingular, then (zk )k converges
superlinearly. If in addition F is strongly semismooth at z ? , then it converges quadratically.
The attentive reader may have noticed that in global convergence not all stationary points
z ? of the merit function f are not necessarily a solution of the equation F (z) = 0. Thisis already the case in the smooth setting, e.g. F (x) = sin(5x) − x has only three roots (0and ±0.519148..) but (sin(5x) − x)2 /2 has many local minima, see Figures 11.1 and 11.2 of
Nocedal and Wright (2006).
Now, we focus on the second globalization strategy: the trust-region approach. We present
first a convergence result of Jiang and Ralph (1998), based on a result of Jiang et al. (1998)
in the context of complementarity problems.
Theorem. Consider F (z) = 0 with F semismooth and a C1 merit function f ? Let (zk )be generated by the trust-region algorithm with the submodel given mk (h) = JkT F (xk )h +
1 T T
?
?
2 h Jk Jk h. If z is an accumulation point of the sequence, then z is a stationary point of?
f . If for all element of ∂B F (z ) are nonsingular, then the entire sequence converges to x?
superlinearly. If in addition, F is strongly semismooth at z ? , then the convergence rate isquadratic.
158

3.2. Methods to solve nonlinear equations
In our GNEP context, the root function Φ is given in Equation (3.3) and the merit function
is defined as 21
D(x, λ).
f (z) =φ◦ (−g(x), λ) 2
2
The gradient of f can be expressed as T Jac x D(x, λ)diag ∇xi g i (x) iD(x, λ).
φ◦ (−g(x), λ)−Da (z)Jac x g(x)Db (z)



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∇f (z) =using Equation (3.20). As mentioned in Dreves et al. (2011), the gradient ∇f is single-valued
since only the bottom part of the generalized Jacobian ∂Φ(z) contains multi-valued expressions(Da and Db when gji (x) = λij = 0) and the bottom part of Φ(x, λ) has zero entries in thatcase. Hence, f is C1 as long as the objective and constraint functions are C2 . Therefore, theline-search is well defined for our GNEP reformulation.
The nonsingularity assumption of elements in the limiting Jacobian ∂B Φ was alreadystudied in the previous subsection for local convergence. Furthermore, Dreves et al. (2011)’s
theorem 3 analyzes the additional conditions for a stationary point of f to be a solution of
the nonsmooth equation, hence of the original GNEP. The condition is that Jac x D(x, λ) is
nonsingular and the matrix
M = Jac x g(x)Jac x D(x, λ)−1 diag ∇xi g i (x) i(3.28)
is a P0 -matrix ∗ . The nonsingularity condition (3.28) for local convergence requires thatDb (z) + M Da (z) is nonsingular. If diagonal matrices Da and Db have non zero terms, then M
is P0 -matrix implies that Db (z) + M Da (z) is nonsingular. Thus, Conditions (3.28) and (3.26)
are closely related.

3.2.4

Specific methods for constrained equations

This subsection aims to present methods specific to solve constrained (nonlinear) equations, first proposed by Dreves et al. (2011) in the GNEP context. The KKT system can bereformulated as a constrained equation, see Equation (3.4) of Subsection 3.1.1. Techniques tosolve such equation may provide good alternatives to standard optimization procedures. In a
VI problem context, Facchinei and Pang (2003) devotes a chapter to interior-point methods
for solving such constrained equations (CE). Here, we focus on the method of Monteiro andPang (1999) providing a general framework for CE problems.
A constrained equation is defined as
H(z) = 0,z ∈ Ω,(3.29)

where Ω is a closed subset of Rn . Generally, the constraint set Ω has a simple structure, e.g.
the nonnegative orthant Ω = Rn+ or a hypercube Ω = [l, u]. As mentioned in Wang et al.
(1996), in practice, the root function H has also a structured form


F (z)
H(z) =.
G(z)∗. A m × m square matrix M is a P0 -matrix if for all subscript set α ⊂ {1, . . . , m} the determinantdet(Mαα ) ≥ 0.

159

Chapitre 3. Calcul d’équilibre de Nash généralisé
The KKT reformulation of the GNEP (3.2) falls within this framework
Let Ω̊ denote the interior of Ω. In the contrained equation literature, we assume that (i) Ωis a closed subset with nonempty interior Ω̊, (ii) there exists a closed convex set S ⊂ Rn , suchthat 0 ∈ S, H −1 (S̊) ∩ Ω̊ is nonempty but H −1 (S̊) ∩ bd Ω is empty and (iii) H is C1 function.
The set S contains zero, so that the set H −1 (S̊) contains a solution to Equation (3.29) andlocal points around. The second assumption requires that such points should not be on theboundary of Ω. The third assumption is just differentiability. In the following, these threeassumptions will be referenced as the constrained equation blanket assumptions.
Potential reduction algorithms with line-searchMonteiro and Pang (1999) build a framework for potential functions, where these functions
play a major role. A potential function S̊ 7→ R satisfies the following properties:

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1. For all sequences (uk )k in S̊ such that either ||uk || tends to infinity or (uk ) tends to apoint on the boundary bd S \ {0}, then p(uk ) tends to infinity.
2. p is C1 function on its domain and the curvature condition uT ∇p(u) > 0 for all nonzero
vectors.
3. There exists a pair (a, σ̄) in Rn ×]0, 1], such that for all u ∈ S̊, we have ||a||2 uT ∇p(u) ≥σ̄(aT u)(aT ∇p(u)).
The potential function has the dual objective to keep the sequences (H(xk ))k away from theset bd S \ {0} and to help the convergence to the zero vector. The parameter a, known as thecentral vector, will play a crucial role to generate iterates in the constrained set Ω.
For example, if the subset S is the nonnegative orthant Rn+ , then a typical potentialfunction isnX2p(u) = ζ log ||u||2 −log ui for u > 0.
i=1Monteiro and Pang (1999) prove that this function verifies the above conditions when ζ > n/2
and with the pair (a, σ̄) = (11n , 1), 11n being the n-dimensional one P
vector.
2m.
n.
In the GNEP context, thePsubset S is R × R+ where n. = i ni is the total numberof player variables and m. = i mi is the total number of constraints, i.e. n = n. + m. . The
function H has components F and G given by
g(x) + w
F (z) = D(x, λ) and G(z) =,λ◦wwhere z = (x, λ, w), see, e.g., Wang et al. (1996). Dreves et al. (2011) propose the following
potential function2m.
 X
p (u) = ζ log ||u1 ||22 + ||u2 ||22 −log(u2i ),i=1.
where u = (u1 , u2 ) ∈ Rn. × R2mand ζ > m. in order to enter the potential framework. The+
pair of constants is (a, σ̄) = ((0n. , 11m. ), 1).
The difficulty, compared to a classical nonlinear equation, is to ensure that all the iterates
remains in the constrained set Ω. In order to solve Equation (3.29), Monteiro and Pang
(1999) use a modified Newton globalized with a backtracking line-search. We report below
their potential reduction Newton algorithm. The algorithm is divided into two parts: (i)160

3.2. Methods to solve nonlinear equations
compute the direction using the central vector a and (ii) find an appropriate stepsize witha geometric line-search for which the merit function is ψ(u) = p(H(u)). Note that H(u) isvalued in Rn. × R2m. .
Init z0 ∈ Ω̊, 0 < ρ, α < 1 and choose σ0 ∈ [0, σ̄[Iterate until a termination criterion is satisfied,– Solve the system ∗ to get dk
H(zk ) + JacH(zk )d = σkaT H(zk )a.
||a||22(3.30)– Find the smallest integer mk such that

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ψ(zk + ρmk dk ) ≤ ψ(zk ) + αρmk ∇ψ(zk )T dk , and zk + ρmk dk ∈ Ω̊.
– zk+1 = zk + ρmk dk .
end Iterate
Due to the special structure H and a might have, the computation of dk in Equation (3.30), a
modified Newton direction because of the right-hand side term, may be further simplified bydecomposing into its components F and G. In this form, the algorithm is defined when the
Jacobian JacH is nonsingular at zk ∈ Ω̊. Lemma 2 of Monteiro and Pang (1999) shows thatthe direction computed in the first step is a descent direction for the merit function ψ. So, thealgorithm is well-defined. Their Theorem 3 shows the convergence of the potential reduction
algorithm.
Theorem. Assume p is a potential function, the constrained Equation (3.29) satisfies theconstrained equation blanket assumptions, the Jacobian Jac H(z) is nonsingular for all z ∈ Ω̊and we have lim supk σk < σ̄. Let (zk ) be a sequence generated by the potential reduction
Newton algorithm. We have (i) the sequence (H(zk )) is bounded and (ii) any accumulation
point, if there exists, solves the constrained Equation (3.29). In particular, if (zk ) is bounded,
the constrained equation has a solution.
Application to GNEP
As already mentioned, Equation (3.4) of the GNEP can be reformulated as a constrained
equation. The root function H : Rn × R2m 7→ Rn × R2m is defined asD(x, λ)H(x, λ, w) = g(x) + w ,λ◦wwhere the dimensions n, m correspond to the GNEP notation and (a, σ̄) is given by ((0n , 11m ), 1).
The potential function is given by
p (u) = ζ log||x||22 + ||λ||22 + ||w||22



−

m
X
k=1

log(λk ) −mX

log(wk ),k=1∗. In Monteiro and Pang (1999), they use the directional derivative along d in the left-hand side of Equation(3.30), which is equivalent to this formulation since H is C1 under the blanket assumptions.

161

Chapitre 3. Calcul d’équilibre de Nash généralisémwhere u = (x, λ, w) ∈ Rn × Rm?+ × R?+ and ζ > m. This reformulation of the potentialfunction emphasizes the three components u = (x, λ, w). For the line-search, the gradient ∇p
is given by


2ζ
x
2
2
2
||x||2 +||λ||2 +||w||2


2ζ
−1 
∇p(x, λ, w) = 
 ||x||22 +||λ||22 +||w||22 λ − λ  ,2ζ
w − w−1||x||2 +||λ||2 +||w||22

22

where λ and w have positive components and terms λ−1 and w−1 correspond to the component-

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wise inverse vector. Compared to the semismooth reformulation, the root function H is now
C1 . The Jacobian is given byJac x D(x, λ) diag ∇xi g i (x) i0JacH(x, λ, w) =  Jac x g(x)0I .
0diag[w]
diag[λ]As reported in Dreves et al. (2011), the computation of the direction dk = (dx,k , dλ,k , dw,k ) in
Equation (3.30) can be simplified due to the special structure of the above Jacobian matrix.
The system reduces to a linear system of n equations to find dx,k and the 2m components
dλ,k , dw,k are simple linear algebra. Using the classic chain rule, the gradient of the merit
function is given by∇ψ(x, λ, w) = JacH(x, λ, w)T ∇p(H(x, λ, w)).
Again the computation of this gradient can be simplified due to the sparse structure of JacH.
Theorem 4.3 of Dreves et al. (2011) is the direct application of the previous theorem in the
GNEP context. We do not restate here their theorem, but present their nonsingularity result
given in Theorem 4.6. The Jacobian matrix is nonsingular, if the matrix Jac x D(x, λ) is
nonsingular and
M = Jac x g(x)Jac x D(x, λ)−1 diag ∇xi g i (x) i(3.31)
is a P0 -matrix. This is exactly Equation (3.28) given in the semismooth setting.

3.3

Numerical results

In this section, we perform a numerical illustration to compare the different methods
presented in this paper. The implementation is done in the R statistical software and the
package GNE, freely available on internet.
Our test problem is a simple two-player polynomial-objective game for which there are
four generalized Nash equilibria. The objective functions (to be minimized) are given byθ1 (x) = (x1 − 2)2 (x2 − 4)4 and θ2 (x) = (x2 − 3)2 (x1 )4 ,for x ∈ R2 , while the constraint functions are
g1 (x) = x1 + x2 − 1 ≤ 0 and g2 (x) = 2x1 + x2 − 2 ≤ 0.
Objective functions are player strictly convave. This problem is simple but not simplistic,since second-order partial derivatives of objective functions are not constant, as for other162

3.3. Numerical results

z ?1
z ?2
z ?3
z ?4

x?1

x?2

λ?1

λ?1

2
-2
0
1

-2
3
1
0

0
8
4 × 3429

5 × 2500
6

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Table 3.1: Four GNEs

test problem such as the river basin pollution game of Krawczyk and Uryasev (2000) or theexample of Rosen (1965).
Due to the simple form of the objective function, we can solve the KKT system for this
GNEP, manually. The solutions are listed in Table 3.1.
Before discussing the results, we detail the stopping criteria used in our optimization
procedure. They are based on Chapter 7 of Dennis and Schnabel (1996). Algorithms always
stop after a finite number of iterations with an exit code specifying whether the sequence
converges or not: (1) convergence is achieved ||F (z)||∞ < f tol with f tol = 10−8 , (2) algorithm
is stuck because two consecutive iterates are too close ||(zk − zk−1 )/zk ||∞ < xtol with xtol =10−8 , (3) stepsize is too small tk < xtol or radius is too small ∆k < xtol, (4) the iteration
limit is exceeded k > kmax with kmax = 300 or generalized Jacobian is too ill-conditionned(5) or singular (6).
On this example, we compare the following methods: the Newton and Broyden methods with a globalization scheme (line search or trust-region), the Levenberg-Marquardt (LM)method with line search, the Levenberg-Marquardt with adaptive parameter, the constrainedequation modified Newton. In the following tables, the results are labelled as follows: GLS
stands for geometric line search, QLS quadratic line search, PTR Powell trust-region, DTR
double dogleg trust-region. We report the number of calls to the root function and the generalized Jacobian, the time (sec), the final iterate z ? (when successful convergence), the valueof Euclidean norm at z ? and the exit code. Meanings of exit code are 1 for successful convergence, 2 or 3 consecutive iterates too small, 4 iteration limit exceeded, 5 or 6 ill-conditionned
Jacobian.
In Table 3.2, we report the result with the complementarity function φF B and the starting point z0 = (5, 5, 0, 0), while for the constrained equation method, the starting point is
z0 = (5, 5, 2, 2, 2, 2). Most methods converge to different equilibria. Surprisingly, the Newton
method with geometric line search converges to z ?1 , whereas all Broyden methods converge toz ?2 and Newton trust-region methods converge to z ?3 , despite using the same initial points.
There is only one method diverging to a local minimum of the merit function 1/2||F (z ? )||22
which is not a root of F : the Newton method with quadratic line search. The LM methodwith adaptive parameter and modified Newton of constrained equation are stuck on singularmatrices. In overall, there is a clear advantage for classic semismooth methods solving theextended KKT system on this example.
With the minimum complementarity function φ∧ , we get similar results, see Table 3.4
in Appendix 3.5.2. Newton methods converges to a different GNE than convergent Broyden
methods. This time, Levenberg-Marquardt method with adaptive parameter is convergent
in relative few iterations. But again, the constrained equation Newton method is divergent
because of a singular Jacobian.
163

Chapitre 3. Calcul d’équilibre de Nash généraliséFct. callJac. call

Time

x?1

x?2

λ?1

λ?1

||F (z ? )||Code

Newton - GLS
Newton - QLS
Newton - PTR
Newton - DTR

96
67
322
317

24
20
217
217

0.008
0.007
0.102
0.056

2

-2

-2.8e-12

160

7e-04
0.00066

1
1

324
324

-2.3e-17-4.6e-17

2.8e-1211
8.6e-09
6.9e-09

1
3
1
1

Broyden - GLS
Broyden - QLS
Broyden - PTR
Broyden - DTR

78
52
91
127

4
3
3
3

0.005
0.005
0.006
0.008

-2
-2
-2
-2

3
3
3
3

8
8
8
8

2.4e-094.8e-138.5e-09
6.6e-13

3e-09
1.2e-09
1.2e-08
1.1e-091
1
1
1

LM min - GLS
LM adaptive

29
368

29
184

0.02
0.111

-2

3

8

-1.4e-09

3.7e-09
0.000191
6Mod. CE Newton

1782

158

0.295

2500

6

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Table 3.2: Results with starting point (5, 5, 0, 0) and φF B

min Newton GLSFB Newton GLS
min Broyden PTRFB Broyden PTR

z ?1

z ?2

z ?3

z ?4

∞

58
183
106
104

213
198
362
381

280
211
45
35

394
238
385
248

55
170
102
232

Table 3.3: Number of GNEs found for 1000 random initial points

To further compare these methods and the complementarity function, we draw uniformly
1000 random initial points such that z0 ∈ [−10, 10] × [−10, 10] × {1} × {1} and run algorithms
on each of them. For simplicity, we restrict out comparison to Newton GLS and Broyden
PTR methods test the Newton GLS method both with the minimum and Fischer-Burmeistercomplementarity functions. Results are summarized in Table 3.3, the first four columns store
the number of sequences converging to a particular GNE, while the last column contains
the number of diverging sequences (termination criteria remain the same as in the previous
example.). With this comparison, the best method seems to be the Newton GLS method
combined with the minimum function. We observe that using the minimum function tends
to get only two GNEs, namely z ?2 and z ?4 . The method finding almost equally all GNEs is
the Newton GLS method with the Fischer-Burmeister function. Finally, the Broyden PTR
method with the Fischer-Burmeister function seems very poor on this example.

3.4

Conclusion

The generalized Nash equilibrium problem (GNEP) is a useful tool for modelling many
concrete applications in economics, computer science and biology, just to name a few. The
demand for computational methods of the GNEP in general form is increasing. This survey
paper aims to present and to compare the current optimization methods available for the
GNEP. Our numerical experiments show an advantage for the KKT reformulation of the
GNEP compared to the constrained equation reformulation. But, in Dreves et al. (2011), the164

3.5. Appendix
constrained equation reformulation was better. A method working for any general GNEP has
yet to be found and its convergence to be proved.

3.5

Appendix

3.5.1

AnalysisNonsmooth analysis
Definition (locally Lipschitzian). G is locally Lipschitzian (on Rn ) if ∀x ∈ Rn , ∃U ∈ N (x), ∀y, z ∈U, ∃kx > 0, ||G(y) − G(z)|| ≤ kx ||y − z||.
From Clarke and Bessis (1999), the Rademacher theorem is

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Theorem. Let f : Rn 7→ R be a locally Lipschitz function. Then f is almost everywhere
differentiable.
From (Clarke, 1990, Cor 2.2.4, Chap. 2), for a function f : Rn 7→ R locally Lipschitzian atx, we have that the generalized gradient ∂f (y) is a singleton for all y ∈ B(x, ) is equivalent
to f is C1 on B(x, ).
From (Clarke, 1990, Prop 2.6.2, Chap. 2), we have the following properties of the generalized Jacobian.
Proposition.
– ∂G(x) is a nonempty, convex, compact subset of Rm×n , while ∂B G(x) is
nonempty and compact.
– ∂G is upper semicontinuous and closed at x and ∂B G is upper semicontinuous.
– ∂G(x) ⊂ ∂G1 (x) × · · · × ∂Gm (x), where the right-hand side is a matrix set where the ith
row is the generalized gradient.
The term ∂G1 (x) × · · · × ∂Gm (x) is sometimes denoted by ∂C G(x). But it is not Clarke’s
subdifferential, which seems to refer only to real-valued function, i.e. ∂G(x) = ∂C G(x).
From Theorem 2.3 of Qi and Sun (1993), we have the following equivalences
Proposition.
– G is semismooth at x.
– ∀V ∈ ∂G(x + h), h → 0, V h − G0 (x; h) = o(||h||).
– ∀x ∈ DG , G0 (x + h; h) − G0 (x; h) = o(||h||).
From Lemma 2.2 of Qi and Sun (1993) and Lemma 2.1 of Sun and Han (1997), we havethe following properties
Proposition. If G is semismooth at x, then d 7→ G0 (x; d) is a Lipschitz function; ∀h, ∃V ∈G(x), V h = G0 (x; h) and ∀h → 0, G(x + h) − G(x) − G0 (x; h) = o(||h||) .
The KKT system
The generalized Jacobian of the complementarity formulation has the following formJac x1 L1 (x, λ1 ) . . . Jac xN L1 (x, λ1 )Jac x1 g 1 (x)T
0
..
..


..


.
.
.


N
N
N
T

Jac x1 LN (x, λ ) . . . Jac xN LN (x, λ )0Jac xN g (x) J(z) = 
.
a
1
1
b
1
 −Da (x, λ1 )Jac x g 1 (x) . . .

−D
(x,
λ
)Jac
g
(x)
D
(x,
λ
)
0
x
1
1
1
N
1




..
..
..


.
.
.
a
N
N
a
N
N
b
N
−DN (x, λ )Jac x1 g (x) . . . −DN (x, λ )Jac xN g (x)0
DN (x, λ )165

Chapitre 3. Calcul d’équilibre de Nash généralisé

3.5.2

Numerical results
Fct. callJac. callTime

Newton - GLS
Newton - QLS
Newton - PTR
Newton - DTR

14
9
38
34

6
6
17
16

0.003
0.003
0.005
0.052

Broyden - GLS
Broyden - QLS
Broyden - PTR
Broyden - DTR

1866
93
21
21

4
4
2
2

0.079
0.005
0.002
0.003

LM min - GLS
LM adaptive

33
18

33
9

0.023
0.006

Mod. CE Newton

1782

158

0.295

x?1

x?2

λ?1

λ?1

||F (z ? )||

Code

71
71
1.4e-11
4.7e-29

5
6
1
1
1
5
1
1

2
2

-2
-2

1e-25
-3.4e-29

160
160

1

4.1e-18

512

6

1
1

-7.6e-15-4.6e-15

512
512

6
6

7e-13
71
1.3e-112.3e-12-2

38-3.9e-14

4.9e-073.3e-11

3
1

2500

6

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Table 3.4: Results with starting point (5, 5, 0, 0) and φ∧

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Jiang, H. and Ralph, D. (1998), Global and local superlinear convergence analysis of Newtontype methods for semismooth equations with smooth least squares, in M. Fukushima andL. Qi, eds, ‘Reformulation - nonsmooth, piecewise smooth, semismooth and smoothing
methods’, Boston MA: Kluwer Academic Publishers. 155, 158
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Chapitre 3. Calcul d’équilibre de Nash généraliséKanzow, C. and Kleinmichel, H. (1998), ‘A new class of semismooth Newton-type methods for nonlinear complementarity problems’, Computational Optimization and Applications
11(3), 227–251. 154
Krawczyk, J. and Uryasev, S. (2000), ‘Relaxation algorithms to find Nash equilibria witheconomic applications’, Environmental Modeling and Assessment 5(1), 63–73. 163
Kubota, K. and Fukushima, M. (2010), ‘Gap function approach to the generalized Nashequilibrium problem’, Journal of optimization theory and applications 144(3), 511–531. 140
Kulkarni, A. A. and Shanbhag, U. V. (2010), Revisiting generalized Nash games and variational inequalities. preprint. 142

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Lopes, V. L. R. and Martinez, J. M. (1999), On the convergence of quasi-Newton methods fornonsmooth problems. preprint. 155
Monteiro, R. and Pang, J.-S. (1999), ‘A Potential Reduction Newton Method for Constrained
equations’, SIAM Journal on Optimization 9(3), 729–754. 159, 160, 161
Nocedal, J. and Wright, S. J. (2006), Numerical Optimization, Springer Science+BusinessMedia. 143, 145, 146, 148, 149, 150, 151, 158
Powell, M. (1970), A hybrid method for nonlinear algebraic equations, in P. Rabinowitz, ed.,‘Numerical Methods for Nonlinear Algebraic Equations’, Gordon & Breach, chapter 6. 148
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Methods, Technical report, Applied Mathematics Report AMR 98/10, School of Mathematics, the University of New South Wales. 158
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Yamashita, N. and Fukushima, M. (2000), On the rate of convergence of the LevenbergMarquardt method, Technical report, Kyoto University. 150

169

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Chapitre 3. Calcul d’équilibre de Nash généralisé170

tel-00703797, version 2 - 7 Jun 2012

Théorie de la ruine

171tel-00703797, version 2 - 7 Jun 2012

tel-00703797, version 2 - 7 Jun 2012

Chapitre 4

Asymptotiques de la probabilité deruine dans un modèle de risque avec
dépendance—
The A + B/u rule for discrete andcontinuous time ruin models withdependence

A teacher can never truly teach unless he is still learning himself.
A lamp can never light another lamp unless it continues to burn its own
flame. The teacher who has come to the end of his subject, who has noliving traffic with his knowledge but merely repeats his lessons to his
students, can only load their minds ; he cannot quicken them.
Rabindranath Tagore (1861-1941)

Ce chapitre se base sur l’article Dutang et al. (2012) soumis à l’Insurance: Mathematics
and Economics.

173

Chapitre 4. Asymptotiques de la probabilité de ruine

4.1

Introduction

Traditionally, the free surplus (Ut )t of an insurance company at time t is represented by
Ut = u + ct −

Nt
X

Xi ,

tel-00703797, version 2 - 7 Jun 2012i=1

where u is the initial surplus, c is the premium rate, (Xi )i are the successive claim amounts
and (Nt )t is the claim arrival process (the claim waiting times are denoted by (Ti )i ). In the
Cramér-Lundberg model, (Nt )t is modelled by a Poisson process, (Xi )i are independent andidentically distributed (i.i.d.) random variables and claim severity (Xi )i are independent of
the claim waiting times (Ti )i . Andersen (1957) generalized the Cramér-Lundberg model by
proposing a general renewal process for the claim arrival process (Nt )t .
Since then, extensions have been proposed in many directions. Asmussen and Rolski(1991) studied ruin models with phase-type distributions for both claim severities Xi andclaim waiting times Ti . Gerber and Shiu (1998) unified the analysis of ruin measures in the
Cramér-Lundberg model, including the deficit at ruin, the claim causing the ruin or the ruinprobability, by introducing a so-called discounted penalty function. Gerber and Shiu (2005),
Song et al. (2010) and many others extended the Gerber-Shiu approach to a wider class ofrisk models. Various generalizations of the Sparre Andersen model have been proposed, suchas for non-homogeneous claim arrivals (e.g. Lu and Garrido (2005), Albrecher and Asmussen(2006)), reinsurance treaties (e.g. Centeno (2002a,b)), multivariate risks (e.g. Cai and Li(2005),Collamore (1996)) and dependent risks (e.g. Albrecher and Boxma (2004),Boudreault
et al. (2006),Albrecher and Teugels (2006)).
The ultimate ruin probability, i.e. ψ(u) = P (∃t > 0 : Ut < 0|U0 = u), is a major ruinmeasure and has received a considerable attention in the literature. For the Sparre Andersen
model, with light-tailed claim amounts, ψ(u) ∼ Ce−γu as u → ∞, where γ is the positive rootof a simple equation involving the moment generating function of Xi (see, e.g., Asmussen and
Albrecher (2010)). For heavy-tailed claim amounts, the ruin probability decreases at a slower
polynomial rate since ψ(u) ∼ C/uα as u → ∞ (e.g., Embrechts and Veraverbeke (1982);
Klueppelberg and Stadtmueller (1998)). Concerning models with dependence, Albrecher andTeugels (2006), e.g., studied the ruin probability when claim size and claim waiting times,(Xi , Ti )i , are correlated; they obtained again an exponential decrease for ψ(u) in the case oflight-tailed claim sizes. In a recent paper, Albrecher et al. (2011) investigated study the ruinprobability when there is dependence by mixing among the claim sizes (Xi )i or the claim
waiting times (Ti )i , see also Constantinescu et al. (2011). They derived here an asymptotic
formula ψ(u) − A ∼ B/u for Pareto correlated claims or inter-occurence times.
The main purpose of the present work is to show that the asymptotic rule A+B/u applies toa wide class of dependent risk models in discrete and continuous time. That dependence will beincorporated through a mixing approach among claim amounts (Xi )i or claim interarrival times(Ti )i . This translates a systemic risk behavior; by comparison, a dependence between claim
sizes and waiting times would correspond to risks of catastrophes. Sufficient conditions arealso given under which the ruin probability can be expanded as a series of terms 1/uk , k ≥ 0.
Much care is paid on risk models that are formulated in discrete time. In fact, suchmodels are often more appropriate in insurance because the surplus of the company is usually
examined after regular time periods. Li et al. (2009) provided a review of standard risk models
in discrete time. Our starting point is when claim amounts have a geometric distribution,174

4.2. Model formulation
which implies an exponential decrease for ψ(u). Adopting a mixing approach, we will focus
on three particular cases of special interest. We also obtain asymptotics for the tail of theresulting claim distributions and then discuss the dependence structure involved.
The paper is organized as follows. Section 4.2 describes the mixing approach for both
continuous and discrete time models. Section 4.3 establishes the asymptotic rule A + B/u
and some variants. Section 4.4 focuses on special features of the discrete time model. Except
mentioned otherwise, all numerical illustrations are done with the R statistical software (RCore Team (2012)).

4.2

Model formulation

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This section is devoted to the presentation of dependent risk models, first in the continuous
time framework and then in the discrete time framework. In addition to a general formulaof the ruin probability under the mixing approach, we present two and three special cases of
mixing distributions for both time scales.

4.2.1

Continuous time framework

In this subsection, we present the continuous time framework based on the classic CramérLundberg model.
Surplus process
The free surplus of an insurance company at time t is modeled by
Ut = u + ct −

Nt
X

Xi ,

i=1

where u is the initial surplus, c is the premium rate, (Xi )i are the claim amounts and (Nt )t≥0
is the Poisson claim arrival process with intensity λ. We assume that the (Xi )i are i.i.d.
conditionally on a latent variable Θ (distributed as X|Θ = θ, say); they are independent of
the claim arrival process. Θ can be interpreted as the heterogeneity in the claim process. In
such setting, the claim sizes (X1 , . . . , Xn ) are dependent random variables.
Ruin probabilities
Ruin occurs as soon as the surplus process becomes negative. Conditionally on Θ = θ, theruin probability is thus defined asψ(u, θ) = P (∃t > 0 : Ut < 0|U0 = u, Θ = θ).
To determine such a probability, a standard method consists in looking at the state of thesurplus after the first claim arrival. This leads to an integro-differential equation that canbe solved by using Laplace-Stieltjes transforms, see, e.g., Asmussen and Albrecher (2010). In
the case of exponentially distributed claims (Xi )i ∼ E(θ), we have the well-known following
formula
λ −u(θ− λ )cψ(u, θ) = min
e
,1 ,
θc
175

Chapitre 4. Asymptotiques de la probabilité de ruine
where the min is equivalent to the net profit condition θ > λ/c. Integrating over the parameterθ yields the ruin probability,
ψ(u) = FΘ (θ0 ) + I(u, θ0 ),(4.1)
where θ0 = λ/c andZ +∞I(u, θ0 ) =θ0θ0 −u(θ−θ0 )edFΘ (θ).
θ(4.2)(4.1) is nothing else than Equation (5) of Albrecher et al. (2011).
Two special cases

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Now, we briefly present the results for two particular distributions of the latent variable Θ,
reported in Albrecher et al. (2011). Firstly, we consider for Θ a gamma distribution Ga(α, λ)with density
λα α−1 −λθγ(α, λθ)fΘ (θ) =θe , thus FΘ (θ) =, θ > 0,Γ(α)Γ(α)
where γ(., .) (resp. Γ(.)) denotes the incomplete lower gamma function (the gamma function),see, Olver et al. (2010). The resulting claim generic variable X has a Pareto distribution withparameter Pa(α, λ), whose survival function is
P (X > x) =1
α , x ≥ 0.
1 + λxUsing the change of variable y = θ(λ + u), the integral I(u, θ0 ) can be expressed in terms of
the incomplete upper gamma function Γ(., .), see Appendix 4.6.1 for the definition of Γ(., .).
We getγ(α, θ0 λ) λα θ0 θ0 u Γ(α − 1, θ0 (λ + u))
ψ(u) =
+
e ×
.
Γ(α)
Γ(α)
(λ + u)α−1
Note that the formula is only valid when the shape parameter verifies α > 1, i.e. the densityof X/Θ = θ is log-concave.
Secondly, consider for Θ a Lévy distribution with densityαα−α2 /4θ√fΘ (θ) = √e, thus FΘ (θ) = erfc
, θ > 0,2 θ
2 πθ3where erfc(.) denotes the complementary error function, see, Olver et al. (2010). The resulting
claim distribution is a Weibull distribution We(1/2, 1/α2 ) for which the distribution tail is√

P (X > x) = e−α x , x ≥ 0.
Unlike the previous case, the computation of I(u, θ) in the Lévy case is more complicated.
Using this time the change of variable x = uθ, we get√Zθ0 α u3 uθ0 +∞ 1 −x−α2 u/(4x)√ e√ eI(u, θ0 ) =
dx.
(4.3)
2 π
x5
uθ0
The latter integral is related to the generalized error function, a particular case of the generalized incomplete upper gamma function, which is defined asZ +∞Γ(a, x; b) =ta−1 e−t−b/t dt,
x176

4.2. Model formulationsee, e.g., Chaudry and Zubair (1994, 2002). In Equation (4.3), we use Γ(−3/2, θ0 u; α2 u/4). As
for the classic incomplete gamma function, the function Γ(., ., .) satisfies a recurrence equationon the parameter a,Γ(a + 1, x; b) = aΓ(a, x; b) + bΓ(a − 1, x; b) + xa e−x−b/x ,tel-00703797, version 2 - 7 Jun 2012

see Theorem 2.2 of Chaudry and Zubair (2002). Using this equation, we are able to computeΓ(−3/2, x; b) in terms of Γ(−1/2, x; b) and Γ(1/2, x; b), which can be both expressed in termsof the (classic) error function, see Appendix 4.6.2 for details. We get

 √
√
1
θ0 u uθ0
I(u, θ0 ) =1− √eeα u erfc (d+ )
α
α u



√
1
2
2
e−α u erfc (d− ) − √e−uθ0 −α /(4θ0 ) ,+ 1+ √
α u
πuθ0
√
√
√
√
where d+ = uθ0 + α/(2 θ0 ) and d− = uθ0 − α/(2 θ0 ). √The constant term for the ruinprobability appearing in Equation (4.3) is FΘ (θ0 ) = erfc(λ/2 θ0 ).

4.2.2

Discrete time framework

The compound binomial risk model, introduced by Gerber (1988), is the discrete timeanalog of the Cramér-Lundberg model. Here too, we construct an extended version of thismodel by using a mixing approach. We are going to derive the ruin probability, for this risk
process, as well as explicit formulas for three special cases.
Surplus process
The insurance portfolio is now examined at times t ∈ N. Here too, the successive claim
amounts form a sequence of i.i.d. random variables conditionally on Θ = θ (distributed as
X|Θ = θ).
The units of time and money are chosen such that the premium for each time unit is equal
to one. The surplus of the insurance company at time t is then given by
Ut = u + t −

t
X

Xi ,

i=1

where u is the initial surplus. When the claims are independent, this model is named compound binomial, because the number of strictly positive claims has a binomial distribution
B(t, q) where q = P (X > 0). The net profit condition is E(X) < 1 in order to avoid thecertain ruin.
Ruin probability in infinite time
The definition of ruin probability has to be made precise since there is a non-zero probability for the surplus to be zero. In other words, we must specify if the ruin of the insurancecompany occurs when Ut < 0 or Ut ≤ 0. Gerber (1988) considers the ruin as the first time
the process U reaches 0, i.e.
ψG (u) = P (∃t ∈ N+ : Ut ≤ 0|U0 = u).
177Chapitre 4. Asymptotiques de la probabilité de ruine
Shiu (1989) considers the ruin as the first time the process U becomes strictly negative:

tel-00703797, version 2 - 7 Jun 2012ψS (u) = P (∃t ∈ N+ : Ut < 0|U0 = u).
Graphically, ψG is the probability that the surplus process crosses the level 0 while ψS is
the probability that the surplus crosses the level -1. We can switch from one formula to theother using the relation ψG (u) = ψS (u − 1). For the rest of the paper, we consider the ruinprobability ψS .
Closed formulas for the ruin probability ψS are available (see, e.g., Willmot (1993), Sundt
and dos Reis (2007)). Sundt and dos Reis (2007) derived the ruin probability when X isgeometrically distributed. More precisely, assuming a geometric decreasing tail for the ruinprobability, they proved that the claim amount distribution is of geometric type (see proof ∗of Theorem 1 of Sundt and dos Reis (2007)).
In the Sundt and dos Reis (2007) framework, when the claim distribution is geometricGe(q, ρ, 1 − α), see Appendix 4.6.3 for details, then the ultimate ruin probability is given byu (1 − q)(1 − ρ) 1 − q
ψS (u) = min(1 − ρ) + α , 1 ,q(1 − α)q
where the minimum is equivalent to the net profit condition 1−q
q (1 − ρ) + α < 1. The net profit
condition ensures the term in power of u does not explode. From this result, we can easily
deduce the 0-modified geometric case, when ρ = 1 − α. When X is geometrically distributedGe(q, ρ), we have!


1 − q 1 − ρ u+1(4.4)
ψS (u) = min,1 .
ρqAgain the net profit condition (i.e. ρ > 1 − q) ensures that the term ((1 − ρ)/q)u+1 does not
explode.
At our disposal, we have two closed formulas for the infinite time ruin probability. Now,
let us extend the formula (4.4) by using again a mixing approach. We choose this formula
rather than the previous one because of its tractability. Specifically, we suppose that Xi /Θ =
θ ∼ Ge(q, e−θ ), then the overall ruin probability isψ(u) = F̄Θ (θ0 ) + I(u, θ0 ),(4.5)

where θ0 = − log(1 − q) andZ θ0I(u, θ0 ) =
0

1−q
e−θ



1 − e−θqu+1dFΘ (θ).

(4.6)

Compared to the continuous setting, (4.1) and (4.2), the integral in (4.6) is done over theinterval [0, θ0 ] for I(u, θ) rather than the interval [θ0 , +∞[. This is due to the fact thatψS (u, θ) is decreasing function of θ in the considered parametrization.
We do not choose the classic geometric distribution Ge(ρ), because the net profit condition(ρ > 1/2) is restrictive on the type of parametrization for the parameter ρ. However, in that∗. Sundt and dos Reis (2007)’s
P reasoning works because substracting the recurrence equation at u and u + 1
cancels the terms in the sum u+1
x=1 . If we assume another type for the ruin probability, such as α/(u + β), it
is far more difficult to get back to the claim distribution.

178

4.2. Model formulationcase, one could consider, for example, a geometric distribution X/Θ = θ ∼ Ge(1/(1 + θ)).
This leads to a ruin probability
Z 1ψ(u) =
θu+2 dFΘ (1) + F̄Θ (1).
0Choosing a uniform distribution Θ ∼ U(0, p) with p ≤ 1 yields the surprisingly simple formulaψ(u) = pu+2 /(u + 3). This simple ruin probability is particularly interesting, because whether
p < 1 or p = 1, the decrease of the ruin probability switches a geometric speed to a polynomial
speed. In this special setting, the ruin probability is also explicit when Θ is beta distributed.

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Three special cases
We present here results for three different distributions of Θ. Firstly, we consider an
exponential distribution Θ ∼ E(λ). We use the following definite integral
Z 1
Z x  
b
a
dp
−θ
−θ
pa (1 − p)bdθ =I1 (a, b, x) =e
1−e= β̄(a, b + 1, e−x ),pe−x0
for x > 0. I1 (a, b, x) reduces to the beta function β(a, b + 1) when x tends to infinity. Using
I1 (λ + 1, k, +∞), the mass probability function of the claim distribution is given
P (X = k) = qδk0 + (1 − δk0 )λ(1 − q)β(λ + 1, k),where δij denotes the Kronecker product. With the presence of a beta function in this massprobability function, one can recognize the zero-modified Yule-Simon distribution, see, e.g.,
Simon (1955). This distribution appears in the study of word frequency. The survival function
is given by
P (X > k) = λ(1 − q)β(λ, k + 1).
Using I(u, θ0 ) = I1 (λ − 1, u + 1, θ0 ), the ruin probability can be derived.
Proposition 4.2.1. Let us consider the discrete time framework of Subsection 4.2.2 with alatent variable Θ exponentially distributed E(λ).
ψ(u) = (1 − q)λ +λ(1 − q)β̄(λ, u + 1, 1 − q), ∀u ≥ 0.
q u+1

Secondly, consider Θ follows a gamma distribution Ga(α, λ). We use the following integral
I2 (a, n, b, x) =

Z x

e

−θ

a 

1−e

−θ

0

yielding

n

Z x
n  
X
n
n−j
θ dθ =(−1)
e−θ(a+n−j) θb dθ,
j
0
b

j=0

Z x̃
n  
X
n
yb
n−j
e−y dy.
I2 (a, n, b, x) =
(−1)
b+1
j
0 (a + n − j)j=0with x̃ = x(a + j). Substituting n − j to j gives
I2 (a, n, b, x) =

n  
X
n
j=0

j

(−1)j

γ(b + 1, x̃),(a + j)b+1
179

Chapitre 4. Asymptotiques de la probabilité de ruine
where γ(., .) denotes the incomplete lower gamma function. When x tends to infinity, only
the term γ(b + 1, x̃) changes and tends to Γ(b + 1). With the integral I2 (λ, k − 1, α − 1, +∞),
the resulting claim distribution has mass probability function
P (X = k) = qδk0 + (1 − δk0 )(1 − q)

k−1
X

j
Ck−1
(−1)j

j=0

λα
.
(λ + j)αSimilarly with I2 (λ, k, α − 1, +∞), the survival function is given by
k  
X
k
λα
P (X > k) = (1 − q)(−1)j.
(λ + j)αj
j=0

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Using I2 (λ − 1, u + 1, α − 1, θ0 ), the ruin probability can be deduced.
Proposition 4.2.2. Let us consider the discrete time framework of Subsection 4.2.2 with alatent variable Θ gamma distributed Ga(α, λ).


α
u+1 
Γ(α, λθ0 ) 1 − q X u + 1
λj γ(α, θ0 (λ + j − 1))+ u+1,
ψ(u) =
(−1)
Γ(α)
q
j
Γ(α)
λ+j−1
j=0

with λ > 1, θ0 = − log(1 − q) and for u ≥ 0.
Finally, consider Θ is Lévy distributed Le(α). We use the integral
Z x  nab
1 − e−θ θ−3/2 e− θ dθ.
I3 (a, n, b, x) =e−θ0Using the change of variable, we haveZ ∞ a+n−j
n  
X
n
2
−
n−j
I3 (a, n, b, x) =(−1) 2e y2 e−by dy.
j
x̃
j=0

with x̃ = x−1/2 . This integral is linked to the generalized incomplete upper gamma function.
Using Appendix 4.6.4, we get
!
√
√ " √
n  
X
√
n
π
b p
j
2 b(a+j)
I3 (a, n, b, x) =(−1) √ eerfc √ + a + j x
j
x
2
b
j=0
!#
√
√
p
√
b
.
+e−2 b(a+j) erfc √ − a + j xx
When x tends to infinity, we haveI3 (a, n, b) =√√ √π(−1) √ e−2 b a+j .
j
b

n  
X
n
j=0

j

Using I3 (0, k − 1, α2 /4) and I3 (0, k, α2 /4), the mass probability and survival functions are
given by

k−1 
k  
√
√
X
X
k−1
k
(−1)j e−α j .
P (X = k) = (1 − q)(−1)j e−α j and P (X > k) = (1 − q)
j
j
j=0

180

j=0

4.3. Asymptotics – the A + B/u rule
The expressions derived when Θ is Lévy distributed, are much more complex than in the continuous time framework. In Subsection 4.3.3, we study asymptotics for the survival function.
The ruin probability can be computed using I3 (−1, u + 1, α2 /4, θ0 ).
Proposition 4.2.3. Let us consider the discrete time framework of Subsection 4.2.2 with alatent variable Θ Lévy distributed Le(α).
ψ(u) = erfc

α
√
2 θ0

with the convention ∗tel-00703797, version 2 - 7 Jun 2012

4.3



√


 √

u+1 
p 
p
1−q X u+1
αj
α j−1
√ + j − 1 θ0
+ u+1(−1) e
erfc
4q
j
2
θ0
j=0

√
p 
p
α
−α j−1√ − j − 1 θ0 ,
+e
erfc
2 θ0
−1 = i, θ0 = − log(1 − q) and for u ≥ 0.

Asymptotics – the A + B/u ruleThis section is the core of the paper, where we establish the A + B/u asymptotic rule forthe ultimate ruin probability for both continuous and discrete time models. We also obtainan expansion of the ruin probability as a power series of 1/u. Finally, we investigate theasymptotic behavior of the resulting claim distribution, which requires a special treatment
with complex analysis.

4.3.1

Notation

We recall basics and notation of the asymptotic analysis; see e.g. Jones (1997), Olveret al. (2010). We introduce the standard Landau notation O(), o() and ∼. One says that f isasymptotically bounded by g as x → x0 , denoted by
f (x) = O(g(x)) ,x0

if there exists K, δ > 0, such that for all 0 < |x − x0 | < δ, we have |f (x)| ≤ K|g(x)|. In otherwords, in a neighborhood of x0 excluding x0 , |f (x)/g(x)| is bounded.
Then, f is said to be asymptotically smaller than g as x → x0 , denoted by
f (x) = o(g(x)) ,x0

if for all  > 0, there exists δ > 0, such that for all 0 < |x − x0 | < δ, we have |f (x)| ≤ |g(x)|.
That is to say, in a neighborhood of x0 excluding x0 , |f (x)/g(x)| tends to 0.
And finally, f is asymptotically equivalent to g around x0 , if the ratio of f over g tends to
1, i.e.,
f (x)
−→ 1.
f (x) ∼ g(x) ⇔x0g(x) x→x0
This is equivalent to f (x) = g(x) + o(g(x)).
The asymptotic idea aims to approximate a complicated function f at x0 by a sum ofknown and tractable terms g(x), controlling the error by o(g(x)). Note that x0 can be +∞.
∗. One can check that the term j = 0 is still a real number.

181

Chapitre 4. Asymptotiques de la probabilité de ruine
The usual way to achieve this is to take a series expansion of f around x0 as g. f is said totake a series expansion at x0 if for all N ∈ N, we havef (x) =x0NX

an φn (x) + o(φN (x)) ,n=0

where (φn )n is a sequence of so-called gauge functions such that ∀n ∈ N, φn+1 (x) = o(φn (x))
around x0 . This condition is equivalent tof (x) −N
−1
X

an φn (x) = O(φN (x)) ,x0

n=0

for all N ∈ N. In this paper, we also use the following notation for a series expansion of f at
x0+∞
X
f (x) ∼
an φn (x).

tel-00703797, version 2 - 7 Jun 2012

x0

n=0

When x0 ∈ R, we generally choose φn (x) = (x − x0 )n , whereas for x0 = +∞ we use
φn (x) = x−n . Integration by part is a standard tool to study integral asymptotics and derive
asymptotics, as pointed in Olver et al. (2010). Integration by part will be extensively used in
the next two subsections. In Appendix 4.6.5, we recall two integration by part theorems.

4.3.2

Continuous time framework

In this subsection, we present and show the A + B/u rule for the continuous time model.
Theorem 4.3.1. Let us consider the continuous time framework of Subsection 4.2.1 with apositive latent variable Θ and θ0 = λ/c.
(i) For all u > 0, the ruin probability is boundedψ(u) ≤ FΘ (θ0 ) +1 FΘ (θ0 )
×
.
u
θ0

(ii) If Θ has a continuous distribution with density fΘ such that fΘ is almost everywhere
differentiable on [θ0 , +∞[ and fΘ0 being a Lebesgue-integrable, then we have fΘ (θ0 )1
ψ(u) = FΘ (θ0 ) +
+o
.
u
u
(k)

(iii) If in addition fΘ is Ck-1 almost everywhere on [θ0 , +∞[ and fΘ is Lebesgue integrable
and bounded on [θ0 , +∞[, then we haveψ(u) = FΘ (θ0 ) +k−1 (i)
X
h (0)
i=0

ui+1

 
1
+o k ,uwhere h(x) = θ0 fΘ (x + θ0 )/(x + θ0 ), so thath(i) (0) =

i
X
j=0

182

(−1)j

i!
(i − j)!θ0j

(i−j)

fΘ

(θ0 ).

4.3. Asymptotics – the A + B/u rule(iv) If fΘ is C∞ on [θ0 , +∞[, then we have
ψ(u)

∼

u→+∞

FΘ (θ0 ) ++∞ (i)
X
h (0)
i=0

ui+1

.

tel-00703797, version 2 - 7 Jun 2012

Proof. (i) From (4.1) and (4.2), the ruin probability is given byZ +∞θ0ψu (θ)dFΘ (θ), with ψu (θ) = e−u(θ−θ0 ) ,ψ(u) = FΘ (θ0 ) +θθ0where θ0 = λ/c.
Both ψu and FΘ are bounded functions on [θ0 , +∞[. They also have bounded variations
since they are monotone. In addition, ψu is continuous. So by Corollary 7.1.23 of Silvia (1999)
or Theorem 12.1 of (Hildebrandt, 1971, Chap. 2), FΘ is Stieltjes integrable with respect tothe function ψu .
R
Then, we apply the integration by part theorem on ψu dFΘ reported in Appendix 4.6.5.
We getZ +∞ψ(u) = FΘ (θ0 ) + lim ψu (b)FΘ (b) − ψu (θ0 )FΘ (θ0 ) −FΘ (t)dψu (t).
b→+∞

θ0

Since ψu is continuously differentiable, the Stieltjes integral
integral. We haveRFΘ dψu reduces to a Riemann−1θ0ψu0 (θ) = 2 θ0 e−u(θ−θ0 ) + (−u)e−u(θ−θ0 ) = −θ0



u
1
+
2
θ
θ

θ

θ



e−u(θ−θ0 ) .

Furthermore, ψu (θ0 ) = 1 andlim ψu (b)FΘ (b) = 0.

b→+∞Therefore, we obtain Z +∞Z +∞1u −u(t−θ0 )1u
ψ(u) = θ0edt ≤ θ0 max FΘ (t) 2 +×e−u(t−θ0 ) dt.
FΘ (t) 2 +
t
t
t
t
t∈[θ
,+∞[
0
θ0
θ0
We getψ(u) ≤ FΘ (θ0 ) +1 FΘ (θ0 )
×
.
u
θ0

R +∞
(ii) Let I(u, θ0 ) = θ0 ψu (θ)dFΘ (θ). We assume a continuous distribution for the mixing
variable Θ and make the change of variable t = θ − θ0 , we getZ +∞θ0I(u, θ0 ) =fΘ (t + θ0 )e−ut dt.
θ
+
t
0
0
We easily recognize a Laplace transform of the function h defined ash(t) =

θ0fΘ (t + θ0 ).
θ0 + t

The minimum condition to apply an integration by part theorem is to require h to be absolutely
continuous, see Appendix 4.6.5. Heil (2007) reports a version of the Fundamental Theorem of183

Chapitre 4. Asymptotiques de la probabilité de ruine
Calculus for absolutely continuous functions. So, absolute continuity of h on [a, b] is equivalent
to h is almost everywhere differentiable [a, b] with h0 being Lebesgue integrable on [a, b].
Since t 7→ θ0 /(θ0 + t) is C∞ on [0, b] for b > 0, h is absolutely continuous on [0, b] if andonly if fΘ is. By assumption, fΘ is almost everywhere differentiable on R+ with fΘ0 being
Lebesgue integrable, hence h is absolutely continuous. Thus we have
Z b

−ut

h(t)e
0


b
Z
Z
e−ut
1 b 0
h(0) h(b)e−bu
1 b 0
−ut
+
h (t)e dt =−+h (t)e−ut dt.
dt = h(t)−u 0 u 0uu
u 0

As b tends to infinity, we get
h(0) 1I(u, θ0 ) =
+
u
u

Z +∞

h0 (t)e−ut dt.

0

Using a property of the Laplace transform, see, e.g., Chapter 19 of Jeffrey and Dai (2008), we
haveZ

tel-00703797, version 2 - 7 Jun 2012+∞h0 (t)e−ut dt −→ 0.
u→+∞0Finally, we conclude fΘ (0)
1
ψ(u) = FΘ (θ0 ) +
.
+o
u
u
(k)

(iii) As fΘ is Ck − 1 almost everywhere on [θ0 , +∞[ and fΘ is Lebesgue integrable, then(i)h is absolute continous for all i ≤ k. Applying k times the integration by part theorem, wegetZ +∞k−1 (i)Xh (0)1I(u, θ0 ) =h(k) (t)e−ut dt.
+ k
ui+1
u 0
i=0

Similarly if h(k) (t) is bounded on [θ0 , +∞[, then the latter term is controlled by o 1/uk . Let0g be the function t 7→ θ0θ+t. The ith-order derivative of h, if it exists, can be derived by the
Leibniz formula(i)h (t) =

i  
X
i
j=0

j

(i−j)

g (j) (t)fΘ

(t + θ0 ) with g (j) (t) =(−1)j j!θ0.
(θ0 + t)j+1

Thus, we haveψ(u) = FΘ (θ0 ) +k−1 (i)
X
h (0)
i=0

ui+1

 
i
X
1
i!
(i−j)
+o k
with h(i) (0) =(−1)jf(θ0 ).
j Θ
u(i − j)!θ0
j=0(iv) if fΘ is C∞ , we have
ψ(u)

∼

u→+∞

FΘ (θ0 ) ++∞ (i)
X
h (0)
i=0

ui+1

.

Unsurprisingly, we get back to asymptotic result (2.3.2) of (Olver et al., 2010, Chapter 2),
since I(u, θ) is a Laplace transform.
184

4.3. Asymptotics – the A + B/u rule
Remark 4.3.2. A sufficient condition for fΘ to be almost everywhere differentiable is local
Lipchitzness. This is a consequence of the Rademacher theorem, see Appendix 4.6.5.
Remark 4.3.3. A similar approach can be done when mixing the waiting times (T1 , T2 , . . . ).
Using Albrecher et al. (2011)’s Section 3, we haveZ λ0
λ −u/θ(1−λ/λ0 )ψ(u) = F̄Λ (λ0 ) +ψu (λ)dFΛ (λ), with ψu (λ) =e, λ0 = θc.
λ0
0

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We give here only the first terms of the series expansion assuming Λ has a continuous distribution 11ψ(u) = F̄Λ (λ0 ) + fΛ (λ0 ) + o.
cuuBelow, we present asymptotics for the two special cases analyzed in Subsection 4.2.1, basedon known asymptotics listed in Appendix 4.6.1. When Θ is gamma distributed, we have γ(α, θ0 λ) λα θ0α−1 −λθ011
α−1ψ(u) =
+o 2
.
+
e
+
2
Γ(α)
Γ(α)
λ + u (λ + u) θ0u
If we use Theorem 4.3.1, we getγ(α, λθ0 ) λα θ0α−1 −λθ0
ψ(u) =
+
e
Γ(α)
Γ(α)



1
1
+ 2
u u




 
α−1
1
.
−λ +o 2θ0uThese two expressions are similar with only different denominators 1/u against 1/(λ + u), but
it does not matter for large values of u.
When Θ is Lévy distributed, the term I(u, θ0 ) contains two terms linked with the error
complementarity function. There exists expansion formula for the error function, cf. Appendix√
4.6.1, but unfortunately the asymptotic of I(u, θ0 ) leads to an explosive term eα u . We
conclude that a term-by-term asymptotic is not appropriate, a uniform expansion of theoriginal function Γ(3/2, x, b) is needed, when both x and b are large. But, we can still use
Theorem 4.3.1 to get

 2


 

α
α
α
1
1
3
1
−α2 /4θ0√√ −ψ(u) = erfc
+ p 3e
+ 2
+o 2
.
u u
u
4 θ0 2θ0
2 θ0
2 πθ0

4.3.3

Discrete time framework

Now, let us turn our attention to the discrete-time framework, where the approach of thissubsection shares strong similarities with the previous subsection.
Theorem 4.3.4. Let us consider the discrete time framework of Subsection 4.2.2 with a positive latent variable Θ and θ0 = − log(1 − q).
(i) For all u ≥ 0, the ruin probabiliy is lower boundedqF̄Θ (θ0 ) + FΘ (θ0 )≤ ψ(u).
u+2(ii) If Θ has a continuous distribution with density fΘ such that fΘ is almost everywhere
differentiable on [0, θ0 ] with fΘ , fΘ0 being bounded, then we have1qfΘ (θ0 )1
ψ(u) = F̄Θ (θ0 ) +
×
+o
.
u+2
1−q
u+2
185

Chapitre 4. Asymptotiques de la probabilité de ruine(iii) If in addition fΘ is Ck-1 almost everywhere on [0, θ0 ] and successive derivatives of fΘare bounded on [0, θ0 ], then we haveψ(u) = F̄Θ (θ0 ) +

k−1
X
i=0



h̃(i) (0)1,+o(u + 2) . . . (u + 2 + i)(u + 2) . . . (u + 2 + k − 1)with h̃(x) = fΘ (− log(1 − xq))/(1 − xq)2 .
(iv) If fΘ is C∞ on [0, θ0 ], then we have
ψ(u)

∼

u→+∞

F̄Θ (θ0 ) +

+∞
X
i=0

h̃(i) (0).
(u + 2) . . . (u + 2 + i)

Proof. (i) From (4.5) and (4.6), the ruin probability is given by
Z θ0

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ψ(u) = F̄Θ (θ0 ) +01−qψu (θ)dFΘ (θ), with ψu (θ) = −θe



1 − e−θqu+1,

where θ0 = − log(1 − q).
First, we change the right-hand side Stieltjes integral by using the survival function F̄Θrather than the cumulative distribution function. We get
Z θ0ψ(u) = F̄Θ (θ0 ) −ψu (θ)dF̄Θ (θ).
0Then, it is easy to see that both ψu and F̄Θ are also of bounded variation on [0, θ0 ]. They
also have bounded variations since they are monotone. In addition, ψu is continuous. So by
Corollary 7.1.23 of Silvia (1999), F̄Θ is Stieltjes integrable
R with respect to the function ψu .
Then we apply the integration by part theorem on ψu dF̄Θ reported in Appendix 4.6.5.
We getZ θ0
Z θ0
ψ(u) = F̄Θ (θ0 ) − ψu (θ0 )F̄Θ (θ0 ) + ψu (0)F̄Θ (0) +F̄Θ (t)dψu (t) =F̄Θ (t)dψu (t),00using ψu (θ0 ) = 1 and ψu (0) = 0.
R
Since ψu is continuously differentiable, the Stieltjes integral F̄Θ dψu reduces to a Riemannintegral. We haveψu0 (θ) = (1 − q)eθ

1 − e−θqu+1

1 − e−θqu

1−q(u + 1)F̄Θ (t)
q



1−q
+
(u + 1)q.

Therefore, we obtainZ θ0t(1 − q)e F̄Θ (t)ψ(u) =
0

Let J(u) =R θ00

1 − e−t
q

u+1

Z θ0
dt +
0

1 − e−tqudt.

((1 − e−t )/q)u dθ. Making the change of variable qx = 1 − e−t , we have
Z 1

J(u) = q
0

186



1
1
q
1
xu dx and q ×≤ J(u) ≤×.
1 − xqu+1
1−q u+1

4.3. Asymptotics – the A + B/u rule
Furthermore, we havemax F̄Θ (θ)eθ =θ∈[0,θ0 ]1, min F̄Θ (θ)eθ = F̄Θ (θ0 ).
1 − q θ∈[0,θ0 ]Therefore, the ruin probability is bounded as(1 − q)F̄Θ (θ0 )J(u + 1) + (u + 1)

1−q
1−qF̄Θ (θ0 )J(u) ≤ ψ(u) ≤ J(u + 1) + (u + 1)
J(u).
q
q

This yields

tel-00703797, version 2 - 7 Jun 2012FΘ (θ0 )q
1q+ FΘ (θ0 ) ≤ ψ(u) ≤
×
+ 1.
u+2
1−q u+2Rθ(ii) Let I(u, θ0 ) = 0 0 ψu (θ)dFΘ (θ). We assume a continuous distribution for the mixing
variable Θ and make the change of variable x = (1 − e−θ )/q, for which qdx = e−θ dθ, we get
Z 1fΘ (− log(1 − xq)) u+1
I(u, θ0 ) = q(1 − q)x dx.
(1 − xq)20
Let h be fΘ ◦ g with g(x) = − log(1 − xq). The minimum condition to apply an integration bypart theorem is to require the integrand function (h(x)/(1 − xq)2 ) to be absolutely continuous,
see Appendix 4.6.5. As x 7→ 1/(1 − xq)2 are C∞ , we must show h is absolutely continuous.
But h = fΘ ◦ g is not necessarily continuous if both fΘ and h are absolutely continuous.
According to Merentes (1991), if g is absolutely continous, then fΘ ◦ g is absolutely continuous
if and only if fΘ is locally Lipschitzian. Using the Rademacher theorem, see Appendix 4.6.5,
we deduce that fΘ is locally Lipschitizan, so h is absolutely continuous.
We obtain

1
 u+2
Z 1
h(x) xu+2
h0 (x)
2qh(x)xI(u, θ0 ) = q(1 − q)
− q(1 − q)
+
dx .
2
3
(1 − xq)2 u + 2 0
(1
−
xq)
(1
−
xq)
u
+2
|0
{z
}
J(u)

The first term equals toqfΘ (θ0 )(1 − q)(u + 2)
while the integral term is controlled as
qfΘ0 (g(x)) 2qfΘ (g(x))+
3(1 − xq)3x∈[0,1] (1 − xq)

Z 1|J(u)| ≤ sup

0



xu+2
1
1
dx = C=o,u+2(u + 2)(u + 3)
u+2

since fΘ and fΘ0 are bounded on [0, θ0 ]. Combining the two preceding results, we get to1qfΘ (θ0 )ψ(u) = F̄Θ (θ0 ) ++o.
(1 − q)(u + 2)
u+2(k)(iii) As fΘ is Ck − 1 almost everywhere on [0, θ0 ] and fΘ is Lebesgue integrable, then h(i)is absolute continous for all i ≤ k. Applying k times the integration by part theorem, we getI(u, θ0 ) =

k−1
X
i=0

h̃(i) (0)+(u + 2) . . . (u + 2 + i)Z θ00h̃(k) (t)xu+2+k−1dx,(u + 2) . . . (u + 2 + k − 1)187Chapitre 4. Asymptotiques de la probabilité de ruine(i)where h̃(x) = h(x)/(1 − xq)2 . Since
 successive derivatives fΘ are bounded on [0, θ0 ], the intek
gral term is controlled by o 1/u . The expression of the ith order derivative for a compositionf ◦ g is complex, see Huang et al. (2006).
(iv) If fΘ is C∞ on [0, θ0 ], we have
ψ(u)

∼

u→+∞

F̄Θ (θ0 ) +

+∞
X
i=0

h̃(i) (0).
(u + 2) . . . (u + 2 + i)

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We examine below the three special cases studied in Subsection 4.2.2. Only one asymptotic
is available via known asymptotics of the incomplete beta function asymptotic, see Appendix
4.6.1. Indeed, when Θ is exponentially distributed, we have


1
λ
λ 1
ψ(u) = (1 − q) + λ(1 − q)
.
+o
u+2
u+2
Using Theorem 4.3.4, with h̃(x) = λ(1 − xq)λ /(1 − xq)2 , leads to the same expansion.
For the two other distributions, gamma and Lévy, we have to use Theorem 4.3.4, as no
asymptotic is available. When Θ is gamma distribution, the function h̃ is α−1 α11
λλh̃(x) =
(1 − xq) log.
2(1 − xq)
1 − xq
Γ(α)
Thus,



Γ(α, λθ0 )λα
1
λ−1 α−1 1
ψ(u) ∼+(1 − q) θ0
+o
.
u→+∞
Γ(α)
Γ(α)
u+2
u+2

with λ > 1 and θ0 = − log(1 − q).
When Θ is Lévy distibuted, the function h̃ isαh̃(x) = √ log
2 π



1
1 − xq−3/2 −
α2
1
4 log( 1−xq
).
e

Thus,

ψ(u)

∼

u→+∞

erfc

α
√
2 θ0





α2
α −3/2 − 4θ11
+ √ θ0 e 0
+o
,
u+2
u+2
2 π

with θ0 = − log(1 − q).

4.3.4

Tail claim distributions

In this subsection, we analyze the tail of the claim distribution, i.e. P (X > x) for large
values of x. For the present model by mixing, the survival function is the following Stieltjes
integralZ+∞

P (X > x) =P (X > x|Θ = θ)dFΘ (θ).
0

In the continuous time framework, it leads toZ +∞P (X > x) =e−θx dFΘ (θ),
0188

4.3. Asymptotics – the A + B/u rule
which is the Laplace transform of the random variable Θ. Here too, one can see that asimilar argument works only when Θ has a light-tailed distribution. In fact, we cannot obtain
interesting results by applying the integration by part directly on this Stieltjes integral (as forthe ruin probability). So, we assume that Θ has a continuous distribution and, similarly tothe first subsection, we are going to derive the asymptotic survival function.
Proposition 4.3.5. Let us consider the continuous time framework of Subsection 4.2.1 andassume Θ has a continuous distribution with density fΘ .
(i) If fΘ is almost everywhere differentiable on R+ with fΘ0 being a Lebesgue-integrable,
then for x > 0, fΘ (0)
1
P (X > x) =.
+oxx(ii) If fΘ is C∞ in the neighborhood of the origin, then for x > 0,

tel-00703797, version 2 - 7 Jun 2012

P (X > x)∼+∞ (k)
X
f (0)
Θ

x→+∞

k=0

xk

.

(iii) If fΘ can be expanded in the neighborhood of the origin asfΘ (t) ∼

t→0

+∞
X

k+η

fk t µ−1,

k=0

for η, µ > 0, then for x > 0,

+∞ 
X
fk
k+η
P (X > x) ∼
Γ
k+η .
x→+∞
µ
x µ
k=0

Proof. (i) fΘ satisfies the minimum requirement for an application of the integration by parts.
We get
+∞
 
Z
e−θx
1 +∞ −θx 0fΘ (0)
1
P (X > x) = fΘ (t)+e fΘ (θ)dθ =
+o
.
−x 0
x 0
x
x


(ii) It is a direct application of Property 2.3(i) of Olver et al. (2010).
(iii) It is a direct application of the Watson lemma, e.g. 2.3(ii) of Olver et al. (2010).
Remark 4.3.6. Parts (i) and (ii) of this proposition may be not applicable when the density
is not defined or zero at the origin. This justifies the part (iii).
Remark 4.3.7. The reason, why the behavior of the integrand function fΘ at the origin
matters, is explained by the Laplace’s method. The Laplace method studies the asymptotic of
the following integral
Z b
I(x) =
exp(t) q(t)dt,
awhere p and q are continuous functions around the point a, assumed to be the minimum of pin [a, b[. In our case, p(t) = t, hence the minimum of the exponent on R+ is attained at theorigin. See, e.g., 2.3(iii) of Olver et al. (2010).
189

Chapitre 4. Asymptotiques de la probabilité de ruine
Let us see if the two special cases studied in the previous section fall within the frameworkof the previous proposition. Firstly, assume that Θ follows a gamma distribution Ga(α, λ).
Using the integral representation of the exponential function, the density function can be
expanded as+∞
λα X (−λ)k α+k−1fΘ (t) =
t
.
Γ(α)
k!
k=0

Thus, we get
P (X > x)

+∞
X

∼

x→+∞

(−1)

k=0

 k+αλ.
Γ(α)k!
xk Γ (k + α)with η = α and µ = 1. This (asymptotic) polynomial decrease of the survival function isconsistent with the fact that X is Pareto distributed

tel-00703797, version 2 - 7 Jun 2012

P (X > x) =1
α .
1 + λxWhen Θ follows a Lévy distribution Le(α),α
2fΘ (θ) = √e−α /4θ .
32 πθ
Although this function is not defined at zero, the density converges to zero, since we havefΘ (1/t) =αt3/2 −α2 t/4
√ e
−→ 0.
t→+∞
2 π

However, we cannot find a valid series expansion of fΘ (1/t) at +∞, or equivalently of fΘ (θ)
at 0, based on the series expansion of the exponential function. Therefore, the preceding
proposition is of limited use in the Lévy case, where we already know that√

P (X > x) = e−α x , x ≥ 0.
More generally, Proposition 4.3.5 is difficult to apply when the density function of Θ is notdefined.
Now, we look at the tail of the claim distribution in the discrete time framework. We have
Z +∞

u
P (X > u) =(1 − q) 1 − e−θ dFΘ (θ).
0

One way to deal with such an integral is to use an integration by part directly on the integral.
But, it does not lead to satifying results as for the ruin probability. Even if we assume Θ hasa continuous distribution, we do not get a Laplace transform of a certain function as in thecontinuous time:
Z +∞

u
P (X > u) =(1 − q)fΘ (θ) 1 − e−θ dθ.
0

To get a term easily integrable, one can try to make a change of variable, e.g. x = 1 − e−θ or
x = − log(1 − e−θ ). The latter is not possible on R+ because the derivative of the function
θ 7→ − log(1 − e−θ ) is unbounded near 0. Let us try x = 1 − e−θ . We get
Z 1fΘ (− log(1 − x)) u
x dx.
P (X > u) =(1 − q)1−x0190

4.3. Asymptotics – the A + B/u rule
Using an integration by part theorem requires that the following limit to existfΘ (− log(1 − x))= lim fΘ (t)et .
t→+∞
x→1
1−x
lim

This requirement is strong and will be satisfied only for light-tailed distributions and a certain
range of parameter values. For example, when Θ is exponentially distributed E(λ), the previous
constraint imposes λ > 1.
Another way to deal with such integral asymptotic is to apply the Pascal formula, assuming
u is an integer. We get
P (X > u) =

u  
X
u

tel-00703797, version 2 - 7 Jun 2012k=0

k(1 − q)(−1)kZ +∞e−kθ dFΘ (θ).

0

The integral is (once again) the Laplace transform LΘ of the random variable Θ at k. Thisbinomial alternating sum requires a special treatment because finding an asymptotic for LΘ (k)will not help us to derive an asymptotic of the sum. This issue is studied in the next subsection.

4.3.5

Binomial alternating sum for claim tails

In the discrete time framework, the survival function can be expressed as
u  
X
u
(1 − q)(−1)k LΘ (k),P (X > u) =k
k=0

where LΘ denotes the Laplace transform of the random variable Θ.
This integral falls within the framework of the alternating binomial sum defined as
n  
X
n
Sn (φ) =(−1)k φ(k),
k

(4.7)

k=n0

where 0 ≤ n0 ≤ n, φ is a real function and n ∈ N is large. n0 can be used to exclude first few
points of the sum that would not be defined.
Lettingλαφ(k) = (1 − q),
(λ + j)αin Equation (4.7), we get the distribution function of X when Θ is gamma distributedofR +∞Subsection 4.2.2. Note that, having started with the integral representation 0 P (X =k|Θ = θ)dFΘ (θ) of P (X = k), we know that the alternating sum is valued on [0, 1]. This is
not immediate without that integral representation.
Let us point out that the probability P (X = k) is a decreasing function of k. This is
not easy to see by using the alternating binomial sum representation (4.7). Here is a simle
proof. To indicate the dependence on the parameter λ, denote(X
λ .From algebraic
 by P = k)nn−1
n−1
manipulation and using the binomial recurrence equation k = k + k−1 , we get
P (X = k + 1)λ = P (X = k)λ − P (X = k)λ+1

λα.
(λ + 1)α191

Chapitre 4. Asymptotiques de la probabilité de ruine
So, as announced, the probability mass function of X is decreasing. The binomial alternating
sum representation of P (X > k) is
k  
X
k
λα
P (X > k) = (1 − q)
(−1)j
.
j
(λ + j)αj=0
There should be an exponential cancelling in the sum, since kj tends quickly to infinity forlarge values of k (we recall nk ∼ e−k nk /k!) and P (X > k) is a decreasing function. A studyof the alternating sum seems to be rather complex.
Going back to the alternating sum with n0 = 0, the first few terms can be expressed as
S0 (φ) = φ(0), S1 (φ) = φ(0) − φ(1), S2 (φ) = φ(0) − 2φ(1) + φ(2).

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Let ∆ be the forward difference operator. Then we have S0 = ∆0 φ(0), S1 = −∆φ(0) andS2 = ∆2 φ(0). More generally, the binomial alternating sum can be rewritten as
n  
X
n
Sn (φ) =(−1)k φ(k) = (−1)n ∆n (φ)(0).
kk=n0Some complex analysis
To deal with such sums, a standard method consists in using complex analysis and contour
integrals. Flajolet and Sedgewick (1995) provide a complete overview of this topic. In thissubsection, we consider that the complex extension of φ of the sum Sn (φ). Their Lemma 1
gives the so-called Rice integral representation of Sn (φ)
I
n  
X
n!
n
(−1)k φ(k) = (−1)n φ(z)
dz,
k
z(z
−
1)
. . . (z − n)
γ

(4.8)

k=n0

where φ is assumed to be analytic on a domain Ω containing [n0 , n[ and γ is a closed curve inΩ encircling [n0 , n[ but not [0, n0 − 1]. Let f be the integrand of the right-hand side of (4.8).
By the residue theorem (e.g. Chapter 10 of Bak and Newman (2010)), if the integrand isanalytic except at a countable number of isolated singularities inside the domain γ, then thecontour integral equals to the sum of the residues of the integrand taken at the singularities
inside γ. The right-hand side of Equation (4.8) is still cumbersome to compute, since theintegrand has at least n + 1 singularities at 0, 1, . . . , n. So, in the present situation, thecomplex contour integration does not really simplify the problem.
As we want to derive some asymptotics when n tends to infinity, the domain γ of thecontour integration has to be extended to the entire positive real half-plane {z, Re(z) > 0}.
However, we do not want to compute the sum of the residuals at integers {n0 , n0 +1, . . . , +∞}.
Furthermore, we do not know if f (z) does not explode as Re(z) → +∞. Nevertheless, thesolution does come by extending the contour of integration. Let γ be the circle C(0,R) of radius
R centered at 0 excluding poles in N. Assuming f is of polynomial growth towards infinity,
the integralIf (z)dz
C(0,R)

192

4.3. Asymptotics – the A + B/u ruleH
tends to 0 as R → +∞. By the residue theorem, the contour integral C
f (z)dz also equals(0,∞)
to the sum of residuals of f at integers {0, . . . , n0 − 1} and {n0 , n0 + 1, . . . , +∞}. The firstresidual contribution is a finite sum, while the second contribution is the binomial alternatingsum Sn (φ).
Thus, in the particular case of polynomial growth, the binomial sum Sn (φ) reduces to thecomputation of a limited number of residuals at {0, . . . , n0 − 1}, see the proof of Theorem 1of Flajolet and Sedgewick (1995).
Theorem. Let φ be a rational analytic function on [n0 , ∞[. Then we have
n  
X
n
k=n0

k

k

n

(−1) φ(k) = −(−1)

XsRes φ(s)n!
s(s − 1) . . . (s − n),

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where the summation of residues is done over poles not on [n0 , ∞[.
Theorem 2(i) of Flajolet and Sedgewick (1995) applies the same approach when f is meromorphic (i.e. complex differentiable everywhere except at a countable number of points) andnot necessarily of polynomial growth.
The same argument applies when we replace the circle C(0,R) by a semicircle S(0,R,d) ={z ∈ C, Re(z) > d, |z| < R}, see part (ii) of Theorem 2 of Flajolet and Sedgewick (1995).
But this time, we only get an asymptotic for the original problem. Furthermore, things are
intrinsically more complicated when the function φ is not a rational function, because we
consider the complex extension of φ. For instance, function z 7→ 1/z 2 has a second-order poleat z = 0 but function z 7→ 1/z 1.414 has algebraic singularity at z = 0.
To deal with algebraic singularities, the integration contour γ must exclude the singularities. Flajolet and Sedgewick (1995) examplify a keyhole structure approach (i.e. Hankel
contour) when φ(z) has a (non polar) algebraic singularity 1/xλ , see proof of their Theorem
3. The keyhole structure captures the effect of the singularity at 0 by decreasing the radius of
the hole in 1/ log(n) as n tends to infinity. Dealing with non-isolated singularities (i.e. branch
points) need even more care than just skipping it as with a semicircle.
√
Let us consider, for example, the complex square root z, the branch point is the negative
real axis ] − ∞, 0]. The branch point is of finite order, compared to the complex logarithm for
instance. Indeed we have( √q√
− ρei(θ/2)
if k = 1i(θ/2+kπ)ρei(θ+2kπ) = ρe=√ i(θ/2)ρe
if k = 2
Flajolet and Sedgewick (1995) consider a local approximation of z 1/2 around the origin onlyat the contour part of the keyhole structure surrounding the origin. Otherwise, we keep thepolynomial growth of the square root for the rest of the contour, see Example 7. Handlingbranch points of infinity order, say with the complex logarithm function log(z), is similar
except that the resulting asymptotic as n tends to infinity is different, see Example 8.
We report below a table of correspondences between singularity and asymptotics.
Two simple illustrations
Let us consider for phi two particular functions of interest below. Firstly, we choose f1 (z)be 1/(z + β)α where z ∈ C. It has a singularity at −β, which is a multiple pole if α ∈ N.
193

Chapitre 4. Asymptotiques de la probabilité de ruine
Singular part s0 ∈/N

Asymptotics

simple pole (z − s0 )−1−Γ(−s0 )ns0

multiple pole (z − s0 )−mn)−Γ(−s0 )ns0 (log(m−1)!

algebraic singularity (z − s0 )λn)−Γ(−s0 )ns0 (logΓ(−λ)+ logarithmic singularity (z − s0 )λ (log(z − s0 ))rn)−Γ(−s0 )ns0 (logΓ(−λ)

m−1
−λ−1

−λ−1

(log log n)r

Table 4.1: Correspondences between singularity and asymptotic
Using a keyhole structure centered at −β and Table 4.1, we have an asymptotic of the form
Sn (f1 )Γ(β) (log n)α−1.
n→+∞ Γ(α)
nβ∼tel-00703797, version 2 - 7 Jun 2012√

Secondly, we choose f2 (z) be e−α z . The function f2 has a branch point at z = 0, becauseof the complex square root. First, we use an inifinitesimal asymptotic of the exponential√
around 0. That is f2 (z) = 1 − α z + o(z). Since the contour integral of a sum is the sumof contour integrals and that the contour integral of an analytic function is zero, we can drop
the constant 1.
We use a right-oriented half-plane keyhole structure centered at 0 for Re(z) > d (with−∞ < d < 0), similar to Theorem 3 of Flajolet and Sedgewick (1995), since the function f2
has a exponential growth on the half-plane Re(z) < d and cannot be integrated as the radius
tends to infinity.
We cannot use the singularity correspondence table for the square root, because the singularity is zero. But, the square root can be approximated by Theorem 3 of Flajolet andSedgewick (1995). And the term o(z) is controlled by the small circle of the keyhole structureon which |z| < 1/ log(n). Thus we get the following asymptotic of the alternating sum
Sn (f2 )

∼

n→+∞

√

α
αγe
− q
,
π log(n) 2 π log3 (n)where γe = 0.5772156649 is the Euler-Mascheroni constant.
Claim tail asymptotics
Based on the previous subsections, we are able to derive tail asymptotics of a claim X
given a mixing distribution Θ in the discrete time framework presented in Subsection 4.2.2.
When Θ follows an exponential distribution E(λ), we use the asymptotic of the beta
function β(a, b) for large values of b, see Appendix 4.6.1. We get that the tail of the distribution
is asymptoticallyΓ(λ + 1)
P (X > k) ∼ (1 − q),k→+∞(k + 1)λwhich decreases like a discrete Pareto distribution (i.e. a Zipf distribution). This tail behaviorof a Yule-Simon distribution was already reported in Simon (1955).
When Θ follows a gamma distribution Ga(α, λ), we use asymptotics of the alternatingbinomial sum with the function f1 (z) = 1/(z + λ + 1)α . Therefore, the tail distribution is194

4.4. Focus on the dependence structure
asymptoticallyλα Γ(λ + 1) (log k)α−1
,
k→+∞
Γ(α)
k λ+1
which decreases slightly slower than a Zipf distribution due to the logarithm in the numerator.
When Θ follows a Lévy distribution Le(α),
again we use an asymptotic of alternating√−αzbinomial sums with the function f2 (z) = e. Thus, the tail distribution is asymptotically


1
γe
,
P (X > k) ∼ (1 − q)α  √− qk→+∞
π log(k) 2 π log3 (k)P (X > k)∼(1 − q)which decreases extremely slowly. Such a tail behaviour is heavier than for a Pareto distribution. With continuous distributions, a similar behaviour is obtained for the log-Cauchy
distribution, for example.

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Numerical illustrations
On Figure 4.1, we plot the tails of the distributions derived above. The exponentialgeometric distribution has a very tractable survival function, since incomplete beta function isavailable most softwares, e.g., in R via the pbeta function. Therefore, we can benchmark theasymptotic with the true value. However, for the two other distributions, we have to computetwo binomial alternating sums. These sums are particularly unstable because the central termn/2
Cn reaches very quickly infinity, which drives the binomial alternating sum between +∞ or−∞.
In modern computers, a real number is stored in eight bytes (i.e. 64 bits), but only
53 bits are reserved for the precision (see, e.g., http://en.wikipedia.org/wiki/Double-precision_
floating-point_format). In our numerical experiment, the alternating binomial sum Sn (φ)
becomes unstable for n ≥ 48 with the standard double precision. To compute the alternatingsum Sn (φ) for large n, we have no other option than to use high precision floating-point
arithmetic libraries such as the GMP library of Grandlund Torbjoern & the GMP Devel.
Team (2011) and the MPFR library of Fousse et al. (2011). Using GMP and MPFR libraries
allow us to a high number of bits, say 500 or 1000. Reliability of those libraries has been
established in Fousse et al. (2007). Those libraries are interfaced in R via the Rmpfr packageof Maechler (2012).
Figures 4.1a, 4.1b and 4.1c correspond to a mixing distribution when Θ is exponential,
gamma and Lévy-stable distributed, respectively. Note that all plots have log scale for x andy axes. On Figures 4.1a, 4.1b, the distribution tail show a Pareto-type behavior, as we observea straight line. The Lévy stable mixing on Figure 4.1c shows clearly a heavier tail.

4.4

Focus on the dependence structureThis section studies the dependence structure of the dependent risk models described in
Subsections 4.2.1 and 4.2.2, respectively for discrete-time and continuous-time settings. Let
us start by recalling Property 2.1 of Albrecher et al. (2011) for the continuous-time model.
Proposition. When claim sizes fullfill for each n ≥ 1,
P (X1 > x1 , . . . , Xn > xn |Θ = θ) =nYe−θxi ,

i=1

195

Chapitre 4. Asymptotiques de la probabilité de ruine

exact
asympto

1e-10

0.15

1e-08

0.20

P(X>x) - log scale

1e-04

0.25

1e-02

asympto
exact

1e-06

P(X>x) - log scale

1e-03
1e-05

Survival function - stable mixing
0.30

Survival function - gamma mixing

exact
asympto

1e-07

P(X>x) - log scale

1e-01

Survival function - exp mixing

1

10

100

1000

x - log scale(a) Exponential

10000

2

5

10 20

50

200 500

2000

x - log scale(b) Gamma

5

10

50

500

5000

x - log scale(c) Lévy stabletel-00703797, version 2 - 7 Jun 2012

Figure 4.1: Survival functions

then, they have a dependence structure due to an Archimedean survival copula with generator
φ = L−1
Θ , the inverse Laplace transform of Θ.
Therefore, in continuous time, the dependence structure is simply an Archimedean copula.
Regarding the discrete-time setting, things are more complicated: the dependence among
discrete random variables is a complex topic. Genest and Neslehova (2007) present issues
linked to discrete copula. To better understand the problem, we recall the Sklar theorem, see,e.g., Joe (1997); Nelsen (2006).
Theorem (Sklar). Let H be the bivariate distribution function of a random pair (X, Y ). Thereexists a copula C such that for all x, y ∈ R,
H(x, y) = C(FX (x), FY (y)).

(4.9)

Furthermore, C is unique on the Cartesian product of the ranges of the marginal distributions.
As a consequence, the copula C is not unique outside the support of the random variables
X and Y . When X, Y are discrete variables in N, C is only unique on N2 but not on R2 \ N2 .
The non-identifiability is a major source of issues. An example of discrete copulas is theempirical copula for observation sample (Xi , Yi )1≤i≤n .
Some problems in the discrete case arise from the discontinuity of the distribution functionof discrete random variables. Let B be a Bernoulli variable B(p). The distribution function
FB , given by FB (x) = (1 − p)11x≥0 + p11x≥1 , is discontinuous. Thus, the random variable
FB (B) is not a uniform variable U(0, 1).
Let us introduce Genest and Neslehova (2007)’s notation. Let A be the class of functions
verifying Equation (4.9) for all x, y ∈ R for a given FX and FY . Let us define the function
B as for all u, v ∈ [0, 1], B(u, v) = (FX−1 (u), FY−1 (v)). We also denote by D the distributionfunction of the pair (FX (X), FY (Y )).
With a simple bivariate Bernoulli vector, Example 1 of Genest and Neslehova (2007) showsthat (i) functions B and D are different, (ii) B is not a distribution function despite both B
and D belong to the class A. Even in that simple support {0, 1}2 , the identifiability issue ofthe copula C cannot be discarded. Proposition 1 of Genest and Neslehova (2007) extends to196

4.4. Focus on the dependence structure
any bivariate pair (X, Y ): B is not a distribution, whereas D is a distribution function but
not a copula.
Let us now consider two exponential random variables that are conditionally independent
given a factor Θ. We choose to focus here only on the bivariate case. Suppose that thesevariables are discretized by taking their integer parts. We will see that the two resulting
random variables have geometric distributions; these are, of course, correlated. Our purposein Subsections 4.4.1 - 4.4.4 below is precisely to study the dependence involved.

4.4.1

Dependence induced by the mixing approach

We start by comparing the joint distributions of two mixed exponential random variables
and of geometric approximations obtained by discretization.

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Continous caseNow, consider Yi , i = 1, 2 to be conditionnaly independent exponential random variables,
i.i.d.
i.e. Yi |Θ = θ ∼ E(θ). We haveZ ∞FY1 (x) = P (Y1 ≤ x) =P (Y1 ≤ x|Θ = θ)dFΘ (θ) = 1 − LΘ (x),0where LΘ stands for the Laplace transform of the random variable Θ, assuming LΘ exists.
Using this formulation, we can check that the random variable FYi (Yi ) is uniformly distributed.
Indeed we have
−1
P (FY1 (Y1 ) ≤ u) = P (Y1 ≤ L−1
Θ (1 − u)) = 1 − LΘ (LΘ (1 − u)) = u.

Furthermore, the (unique) copula of the couple (Y1 , Y2 ) can be derived by−1
CY1 ,Y2 (u, v) = P (FY1 (Y1 ) ≤ u, FY2 (Y2 ) ≤ v) = P (Y1 ≤ L−1
Θ (1 − u), Y2 ≤ LΘ (1 − v))
Z ∞
−1
−1
=
(1 − e−θLΘ (1−u) )(1 − e−θLΘ (1−v) )dFΘ (θ).
0Hence the copula function is given by−1
CY1 ,Y2 (u, v) = u + v − 1 + LΘ L−1
Θ (1 − u) + LΘ (1 − v) .

(4.10)We retrieve the fact that the couple (Y1 , Y2 ) has an Archimedean survival copula with generator
L−1Θ , see, e.g., Albrecher et al. (2011). In other words, the joint distribution of the tail is given
by


−1
P (F̄Y1 (Y1 ) > u, F̄Y2 (Y2 ) > v) = LΘ L−1
Θ (u) + LΘ (v) .
From Theorem 4.6.2 of Nelsen (2006), the above expression can be extended to any dimension. A n-dimension function C(u1 , . . . , un ) defined by a generator φ and C(u1 , . . . , un ) =φ−1 (φ(u1 ) + · · · + φ(un )) is a n-copula for all n ≥ 2 if and only if the generator φ−1 iscompletely monotone, i.e. for all k ∈ N,(−1)kdk φ−1(t) ≥ 0.
dtk

As the generator is φ−1 (t) = LΘ (t) in our model, then the Laplace transform is completely
monotone. In particular, LΘ is a decreasing convex function. As φ is the generator of an−1
Archimedean copula, φ is a convex decreasing function. Since φ(t) = L−1
Θ (t), LΘ is also a
convex decreasing function.
197

Chapitre 4. Asymptotiques de la probabilité de ruine
Discrete case
If Y follows an exponential distribution E(θ), then W = bY c follows a geometric distribution G(1 − e−θ ), where bxc denotes the floor function. Indeed, we have
P (W = k) = P (Y ∈ [k, k + 1[) = e−θk (1 − e−θ ).
Furthermore, P (W > x) = F Y (bxc + 1) = e−θ(bxc+1) and P (W ≤ x) = FY (bxc + 1) =1 − e−θ(bxc+1) for all x ∈ R. Hence, the distribution function FW and FY only coincide for
integer values with a shift, i.e. FW (n) = FY (n + 1) for all n ∈ N.
Using the same assumption for Y1 and Y2 as in Subsection 4.4.1, we look at the dependenceof W1 , W2 with Wi = bYi c, i = 1, 2. The distribution function is given by
FW1 (x) = P (W1 ≤ x) = P (bY1 c ≤ x) = P (Y1 < bxc + 1) = P (Y1 ≤ bxc + 1) = 1 − LΘ (bxc + 1),

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for x ∈ R+ . In this case, the random variable FWi (Wi ) is not uniform on [0,1] since−1
P (FW1 (W1 ) ≤ u) = P (W1 ≤ L−1
Θ (1 − u) − 1) = 1 − LΘ bLΘ (1 − u)c 6= u.
Their joint distribution function is given by−1
DW1 ,W2 (u, v) = P (FW1 (W1 ) ≤ u, FW2 (W2 ) ≤ v) = P (W1 ≤ L−1
Θ (1 − u) − 1, W2 ≤ LΘ (1 − v) − 1)
Z ∞
−1
−1
(1 − eθ(bLΘ (1−u)−1c+1) )(1 − eθ(bLΘ (1−v)−1c+1) )dFΘ (θ).
=
0

The distribution function of (FW1 (W1 ), FW2 (W2 )) can be rewritten as
DW1 ,W2 (u, v) = 1 − LΘ (blu c) − LΘ (blv c) + LΘ (blu c + blv c) ,(4.11)

where lp = L−1
Θ (1 − p). Equations (4.10) and (4.11) differ by the use of the floor function in
the arguments of LΘ .
Remark 4.4.1. If one uses another parametrization for Wi than the geometric distribution
G(1 − e−θ ), then we have to replace all the Laplace transform by an appropriate expectation.
For example, if we consider G(e−θ ), we use E((1 − e−Θ )x ) instead of LΘ (x).

4.4.2

Differences between CY1 ,Y2 and DW1 ,W2

Before looking at the differences between CY1 ,Y2 and DW1 ,W2 , we check that these distribution functions (defined in Equations (4.10) and (4.11)) are identical on the support of(FW1 , FW2 ). Let Im(FWi ) be the inverse image by FWi of the integer set N. Let u ∈ Im(FW1 )and v ∈ Im(FW2 ), i.e. there exist n, m ∈ N, such that u = FW1 (n) and v = FW2 (m). Firstly,
we haveu = FW1 (n) = FY1 (n + 1) and v = FW2 (m) = FY2 (m + 1).
Secondly, functions CY1 ,Y2 and DW1 ,W2 can be written as
CY1 ,Y2 (u, v) = P (FY1 (Y1 ) ≤ FY1 (n + 1), FY2 (Y2 ) ≤ FY2 (m + 1)) = P (Y1 ≤ n + 1, Y2 ≤ m + 1).
and
DW1 ,W2 (u, v) = P (FW1 (W1 ) ≤ FW1 (n), FW2 (W2 ) ≤ FW2 (m)) = P (W1 ≤ n, W2 ≤ m).
Hence, both functions CY1 ,Y2 (u, v) and DW1 ,W2 (u, v) are equal for u, v ∈ Im(FW1 ) × Im(FW2 ).
Now, we are going to determine the maximal distance between CY1 ,Y2 (u, v) and DW1 ,W2 (u, v).
For that, we begin by deriving two elementary lemmas.
198

4.4. Focus on the dependence structure
Lemma 4.4.2. Let f be a continuous concave (resp. convex) function on a set S. Then thesequence (f (xi ) − f (xi−1 ))i with xi = x0 + iδ is decreasing (resp. increasing).
Proof. Let F̃i : [xi−1 , xi+1 ] 7→ R be defined as
F̃i (x) =x − xi−1
xi+1 − xf (xi−1 ) +f (xi+1 ).
xi+1 − xi−1
xi+1 − xi−1

Since f is concave, we have for all x ∈ [xi−1 , xi+1 ], f (x) ≥ F̃i (x). In particular for x = xi , wegetf (xi ) ≥ (f (xi−1 ) + f (xi+1 ))/2 ⇔ f (xi ) − f (xi−1 ) ≥ f (xi+1 ) − f (xi ).

Lemma 4.4.3. Let f be a completely monotone function and c > 0. The function fc : x 7→
f (x) − f (x + c) is also completely monotone.

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Proof. As f being completely monotone, (−1)n f (n) is decreasing. Furthermore, we have(−1)n fc(n) (x) = (−1)n f (n) (x) − (−1)n f (n) (x + c) ≥ 0
as x < x + c. Thus fc is completely monotone.
In particular, if f is convex decreasing, then x 7→ f (x) − f (x + c) is also convex decreasing,and x 7→ f (x + c) − f (x) is concave increasing.
Let ∆i,j be the height of stairstep in the graphical representation of the distributionfunction DW1 ,W2 . We have ∆i,j = DW1 ,W2 (xi , yj ) − DW1 ,W2 (xi−1 , yj−1 ), where xi , yj ∈Im(FW1 ) × Im(FW2 ), i.e. xi = FW1 (i) and yj = FW2 (j).
Proposition 4.4.4. The maximal stairstep, representing the highest difference between distribution functions DW1 ,W2 and CY1 ,Y2 , is ∆∞,0 = ∆0,∞ .
Proof. We are looking for the maximum of ∆i,j . By algebraic manipulations, we have
DW1 ,W2 (xi , yj ) = P (W1 ≤ i, W2 ≤ j) = 1 − LΘ (i + 1) − LΘ (j + 1) + LΘ (i + j + 2).
Since LΘ is completely monotone, the function x 7→ LΘ (x + c) − LΘ (x) is concave increasing,cf. Lemma 4.4.3. Similarly, for a constant c > 0, the function gc : x 7→ 1 − LΘ (c + 1) − LΘ (x +1) + LΘ (x + 2 + c) is concave increasing.
Furthermore, we have∆i,j = DW1 ,W2 (xi , yj ) − DW1 ,W2 (xi , yj−1 ) + DW1 ,W2 (xi , yj−1 ) − DW1 ,W2 (xi−1 , yj−1 )Using the function f defined above, we have DW1 ,W2 (xi , yj )−DW1 ,W2 (xi , yj−1 ) = gi (j)−gi (j −1). Hence, (DW1 ,W2 (xi , yj ) − DW1 ,W2 (xi , yj−1 ))j is a decreasing sequence, using Lemma 4.4.2
and gi is concave. DW1 ,W2 (xi , yj−1 ) − DW1 ,W2 (xi−1 , yj−1 ) = gi+1 (j + 1) − gi+1 (j) + LΘ (i) −LΘ (i + 1). By Lemma 4.4.3, the function gi+1 is concave and by Lemma 4.4.2 the sequence(H(xi , yj−1 ) − H(xi−1 , yj−1 ))j is decreasing.
Therefore for a fixed i, (∆i,j )j is a sum of two decreasing sums of j. Since Archimedean
copulas are symmetric, we deducemax ∆i,j = max ∆i,j = max max ∆i,j = max ∆i,0 .
i,j≥0

i≥j

i≥0 i≥j≥0i

199

Chapitre 4. Asymptotiques de la probabilité de ruine
Getting back the original definition, we have∆i,0 = P (W1 ≤ i, W2 ≤ 0) = P (W1 ≤ i, W2 ≤ 0) = 1 − LΘ (i + 1) − LΘ (1) + LΘ (i + 2).
Since LΘ is convex, the sequence LΘ (i + 2) − LΘ (i + 1) is an increasing sequence. We conclude
that (∆i,1 )i is increasing, so the maximum is attained at +∞. Therefore,max ∆i,j = ∆+∞,0 = P (W1 ≤ +∞, W2 ≤ 0) = P (W2 = 0).
i,jwith α = 2, λ = 4. We plot the unique continuous copula CY1 ,Y2 and the distribution function
DW1 ,W2 , left-hand and right-hand graphs, respectively.
Continuous case - Gamma G(2,4)

1.0Discrete case - Gamma G(2,4)

1.0

0.8

0.8
0.6

H(u,v

0.6

0.4

)

0.4
1.0
0.8

0.2
0.0
0.0

0.6
0.2

v

0.4

0.4
u 0.6

0.2
0.8

1.0
0.8

0.2
0.0
0.0

0.6
0.2

0.4

v

)

C(u,v

tel-00703797, version 2 - 7 Jun 2012

On Figure 4.2, we investigate the numerical differences with a given distribution for thelatent variable. We consider Θ follows a gamma distribution Ga(2, 4). In other words, we
assume
α

λ
−1/α
⇔ L−1
LΘ (t) =− 1),
Θ (z) = λ(z
λ+t

0.4
u 0.6

0.2
0.8

1.0 0.0(a) Continuous case

1.0 0.0(b) Discrete case

Figure 4.2: Comparison of copula CY1 ,Y2 and distribution function DW1 ,W2CY1 ,Y2 passes through all the top-left corners of the stairs described by HW1 ,W2 . The grid ofpoints used for the graphs is {(FW1 (n), FW2 (m)), n, m ∈ N}. As n increases, the distributionfunction FW1 (n) tends 1, so there infinitely many stairs towards the point (1, 1). Graphically,
the maximal stairstep corresponds to (u, v) = (1, FW2 (0)) or (FW1 (0), 1) and ∆∞,0 = 0.36.
The maximal stairstep is also the maximum difference between copula CY1 ,Y2 and distributionfunction DW1 ,W2 .

4.4.3

Non-identifiability issues

The function DW1 ,W2 is not a copula but only a distribution function. The maximal
stairstep leads to the question of the maximal differences between two copulas satisfying
200

4.4. Focus on the dependence structure
Equation (4.9)
H(x, y) = C(FX (x), FY (y)),−+
when X, Y take discrete values. The answer is given by the Carley bounds CH, CH. For allcopulas CH verifying Equation (4.9), we havetel-00703797, version 2 - 7 Jun 2012−+
CH
≤ CH ≤ CH,

cf. Proposition 2 of Genest and Neslehova (2007).
The non-identifiability issue matters, since with discrete copulas the dependence measure,
such as the tau of Kendall or the rho of Spearman, are no longer unique. Furthermore, if X
and Y are independent, it does not imply that the copula of (X, Y ) is the independent copula.
In other words, the copula alone does not characterize the dependence: we need assumptions
on margins.
In the copula literature, efforts have been done to tackle the non-identifiability issue ofz , which is a bilinear
dependence measure. The current answer is the interpolated copula CX,Y
interpolation of the distribution function D of the pair (F (X), F (Y )) on the discrete set
Im(FX ) × Im(FY ).
Let (ui , vj ) ∈ Im(FX ) × Im(FY ), i.e. ui = FX (i) and uj = FX (j). For all (u, v) ∈[ui , ui+1 ] × [vj , vj+1 ], the interpolated copula is defined asz
CX,Y(u, v) = λ̄u λ̄v D(ui , vj )+λu λ̄v D(ui+1 , vj )+ λ̄u λv D(ui , vj+1 )+λu λv D(ui+1 , vj+1 ), (4.12)

where λu = (u − ui )/(ui+1 − ui ), λv = (v − vj )/(vj+1 − vj ). This copula was already mentioned
in Lemma 2.3.5 of Nelsen (2006) to prove the Sklar theorem. The interpolated copula canalso be interpreted as the copula of (X + U, Y + V ) where U, V are two independent uniform
random variables, see, e.g., Section 4 of Denuit and Lambert (2005). This formulation is usefulwhen doing random generation.
The properties of the interpolated copula of the pair (X, Y ) are: (i) Kendall’s tau τ (X, Y ) =z ) and Spearman’s rho ρ(X, Y ) = ρ(C z ), (ii) C z
τ (CX,YX,Y
X,Y is absolutely continuous and (iii)zzX ⊥ Y ⇔ CX,Y = Π, the independent copula. Unfortunately, CX,Y
depends on marginalsz
and when FX = FY does not imply CX,Y (u, v) = min(u, v), as well as FX = 1 − FY ;z (u, v) = (u + v − 1) , see, e.g., Genest and Neslehova (2007).
CX,Y+
As the interpolated copula is absolutely continuous, a density can be derived. Indeed, bydifferentiating Equation (4.12) with respect to u and v, we getcz
X,Y (u, v) =D(ui , vj ) − D(ui+1 , vj ) − D(ui , vj+1 ) + D(ui+1 , vj+1 ),(ui+1 − ui )(vj+1 − vj )(4.13)for (u, v) ∈ [ui , ui+1 ] × [vj , vj+1 ].
z
To better see the differences between the copula CY1 ,Y2 and the interpolated copula CW,
1 ,W2
we choose to plot the densities on Figure 4.3 and not the distribution functions. On Figure 4.3,
we plot densities of the continuous copula of the pair (Y1 , Y2 ) and the interpolated copula ofthe pair (W1 , W2 ), in the same setting as the previous subsection, i.e. Θ is gamma distributedGa(2, 4).

4.4.4

Zero-modified distributions

In the previous Subsections, we worked only with a simple geometric distribution. Following the model presented in Subsection 4.2.2, we use hereafter a zero-modified distribution
201

Chapitre 4. Asymptotiques de la probabilité de ruine

Continuous copula density - Gamma G(2,4)

Templar copula density - Gamma G(2,4)

3.0
2.5

3

c+(u,v

1.5

)

c(u,v

2.0
2

)

1.0
0.8

0.5

0.6

0.0
0.2

v

0.4

0.4
u 0.6

0.2
0.8

0.6

0.0
0.2

0.4

0.4
u 0.6

0.2
0.8

1.0 0.0(a) Continuous casetel-00703797, version 2 - 7 Jun 2012

1.0
0.8

1

v

1.0

1.0 0.0(b) Discrete case

Figure 4.3: Comparison of copula cY1 ,Y2 and interpolated cz
W1 ,W2Ge(q, ρ) where the mixing is done over the parameter ρ. We want again to study the associateddependence structure.
Continuous case
Let Yi , i = 1, 2 be conditionnaly independent exponential random variable Yi |Θ = θ ∼ E(θ)
and Ii be independent Bernoulli variable B(1 − q). We define Zi = Ii Yi . By conditioning on
the variable Ii , we get
P (Zi ≤ x) = 1 − (1 − q)LΘ (x),andP (Z1 ≤ x, Z2 ≤ y) = 1 − (1 − q)(LΘ (x) + LΘ (y)) + (1 − q)2 LΘ (x + y),
for x, y ≥ 0. Let DZ1 ,Z2 be the distribution function of the pair (FZ1 (Z1 ), FZ2 (Z2 )). We have
DZ1 ,Z2 (u, v) =0
P (Z1 ≤ lu , Z2 ≤ lv )if u < q or v < q,otherwise.

where for p ≥ q, lp = L−1
Θ ((1 − p)/(1 − q)). Hence, we get for u, v ≥ q,
DZ1 ,Z2 (u, v) = u + v − 1 + (1 − q)2 LΘ (lu + lv ).
As q tends 0, we get back to the Archimedean continuous copula CY1 ,Y2 of the previous
subsection. Note that there is a jump when u or v equal q. Indeed, we have DZ1 ,Z2 (q, v) = qv
and DZ1 ,Z2 (u, v) ≥ q 2 for u, v ≥ q.
Discrete case
If Y follows an exponential distribution E(θ) and I follows a Bernoulli distribution Ber(1−q), then W = IdY e follows a 0-modified geometric distribution G(q, 1 − e−θ ), where dxe is the202

4.4. Focus on the dependence structure
ceiling function. Indeed, we have
P (W = k) = qP (0 = k) + (1 − q)P (dY e = k) = qδk,0 + (1 − q)(1 − δk,0 )e−θ(k−1) (1 − e−θ ).
Furthermore, P (W ≤ x) = q + (1 − q)P (bY c < bxc) = q + (1 − q)(1 − e−ν(bxc) ) for all x ∈ R+ .
In particular for x = k ∈ N, we have FW (k) = q + (1 − q)(1 − e−θk ).
Let Wi = Ii dYi e with Ii ’s and Yi be conditionnaly independent exponential random variables, i.e. Yi |Θ = θ ∼ E(θ). We have
P (Wi ≤ x) = 1 − (1 − q)LΘ (bxc),
andtel-00703797, version 2 - 7 Jun 2012

P (W1 ≤ x, W2 ≤ y) = 1 − (1 − q)(LΘ (bxc) + LΘ (byc)) + (1 − q)2 LΘ (bxc + byc).
Let DW1 ,W2 be the distribution function of the pair (FW1 (W1 ), FW2 (W2 )). Similarly to theprevious subsection, we get0
if u < q or v < q.
DW1 ,W2 (u, v) =P (W1 ≤ lu , W2 ≤ lv )otherwise
And we get for u, v ≥ q,
DW1 ,W2 (u, v) = 1 − (1 − q)(LΘ (blu c) + LΘ (blv c)) + (1 − q)2 LΘ (blu c + blv c).
There is also a jump when u or v equal q. Indeed, we have DW1 ,W2 (q, v) = q − q(1 − q)LΘ (blv c)
and DW1 ,W2 (u, v) ≥ q 2 for u, v ≥ q.
Differences between DZ1 ,Z2 and DW1 ,W2
Let u ∈ Im(FW1 ) and v ∈ Im(FW2 ), i.e. there exist n, m ∈ N, such that u = FW1 (n) andv = FW2 (m). Firstly, we haveu = FW1 (n) = FZ1 (n + 1) and v = FW2 (m) = FZ2 (m + 1).
Since we have lu = n and lv = m, we have
DZ1 ,Z2 (u, v) = 2 − (1 − q)(LΘ (n) + LΘ (m)) − 1 + (1 − q)2 LΘ (n + m) = DW1 ,W2 (u, v).
Now, we focus on the maximal difference between these two functions, after deriving anelementary lemma.
Lemma 4.4.5. Let f be a completely monotone function, c > 0 and p ∈ [0, 1]. The function
fc,p : x 7→ f (x) − pf (x + c) is also completely monotone.
Proof. As f being completely monotone, (−1)n f (n) is decreasing. Thus, we have(−1)n f (n) (x) ≥ (−1)n f (n) (x + c) ≥ p(−1)n f (n) (x + c),(n)

as x < x + c. Hence, (−1)n fc,p (x) ≥ 0, i.e. fc,p is completely monotone.
Proposition 4.4.6. Let ∆i,j be the difference between functions DW1 ,W2 and DZ1 ,Z2 at the
point u = FW1 (i) and v = FW1 (i). The maximal stairstep is ∆∞,1 = ∆1,∞ .
203

Chapitre 4. Asymptotiques de la probabilité de ruine
Proof. Since for u, v ≤ q = FW1 (0), both functions DW1 ,W2 and DZ1 ,Z2 are zero, we havemax ∆i,j = max ∆i,j .
i,j≥0

i,j≥1

Using Lemma 4.4.5, the function x 7→ LΘ (x) − (1 − q)LΘ (x + c) is convex decreasing. Thus,
x 7→ LΘ (x) − (1 − q)LΘ (x + c) is concave increasing. We get back to the proof of Proposition
4.4.4. So, it follows that the maximal difference is the stairstep ∆∞,1 .
Graphical comparison
As already said, functions DZ1 ,Z2 and DW1 ,W2 are not copula but only distribution functions. We plot on Figure 4.4 these two functions when Θ is gamma distributed Ga(2, 4) withparameter q = 0.3. The appropriate way to deal with non-identifiability issues is again to use
the interpolated copula C z .

1.0

Discrete case - Gamma G(2,4), q=0.3

1.0

0.8

0.8

H(u,v

0.6
0.4

)

0.4

)

C(u,v

0.6

1.0
0.8

0.2
0.0
0.0

0.6
0.2

v

0.4

0.4
u 0.6

0.2
0.8

1.0
0.8

0.2
0.0
0.0

0.6
0.2

0.4

v

tel-00703797, version 2 - 7 Jun 2012

Continuous case - Gamma G(2,4), q=0.3

0.4
u 0.6

0.2
0.8

1.0 0.0(a) Continuous case

1.0 0.0(b) Discrete case

Figure 4.4: Comparison of distribution functions DZ1 ,Z2 and DW1 ,W2

4.5

Conclusion

This paper uses a new class of dependent risk models, where the dependence is based on amixing approach. Emphasis has been put on infinite time ruin probability asymptotics. This
paper validates the A + B/u rule suggested in Example 2.3 of Albrecher et al. (2011). The
asymptotic rule applies both when the claim amounts or the waiting times are correlated. In
the ruin literature, even when some dependence is added in the claim arrival process, e.g.,a Markovian setting or a specific dependence for the claim severity and waiting time, thedecreasing shape of the ruin probability remains unchanged compared to the corresponding
independent case, either exponential e−γu or polynomial u−α . Hence, our particular mixing
approach, leading to A + B/u asymptotics, significantly worsens the situation for the insurer.
2044.6. Appendix

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A large part of this paper focuses on the discrete time framework. We have seen that
discrete time ruin probability asymptotics can be obtained in a similar way as in the continuous
case. However, deriving asymptotics for the claim distribution is far more difficult: complex
analysis is necessary to tackle the binomial alternating sum issue. Furthermore, the nonuniqueness of discrete copulas is also studied. We quantify the maximal difference between
the continuous and the discrete settings. Despite the issues encountered with discrete copula,
the latent variable approach is considered in many other articles, e.g., Joe (1997); Frees andWang (2006); Channouf and L’Ecuyer (2009); Braeken et al. (2007); Leon and Wu (2010).
As mentioned in Albrecher et al. (2011), the approach proposed in this paper to derive
new explicit formula can be used for more general risk models. It could be interesting totest whether the A + B/u still applies for the ruin probability, say, with phase-type claim
distributions. Beyond the study of ruin probability, the mixing approach and the asymptotic
rule might probably be used for finite-time ruin probabilities and the Gerber-Shiu function.

4.6

Appendix

4.6.1

Usual special functions

List of common special functions
Let us recall the definition of the so-called special functions, see Olver et al. (2010) for acomprehensive and updated list. In most cases, the definition does not limit to z ∈ R, but
can be extended to z ∈ C.
R∞
– the gamma function Γ(α) = 0 xα−1 e−x dx,
Rz– the lower incomplete gamma function γ(α, z) = R0 xα−1 e−x dx,∞– the upper incomplete gamma function Γ(α, z) = z xα−1 e−x dx,
R 1 a−1– the beta function β(a, b) = 0 x (1 − x)Rb−1 dx,
z– the incomplete beta function β(a, b, z) = 0 xa−1 (1 − x)b−1 dx,
R1– the complementary incomplete beta function β̄(a, b, z) = z xa−1 (1−x)b−1 dx = β(b, a, 1−z),Rz
2– the error function erf(z) = √2π 0 e−t dt,
R +∞2– the complementary error function erfc(z) = √2π z e−t dt,
n!
– the binomial coefficient nk = k!(n−k)!
.
Known asymptotics of usual special functions
In this subsection, we list asymptotics of the main special functions, see Olver et al. (2010).
Gamma function Known as the Stirling formula, the asymptotic expansion for Γ(z) is
r 

2π
1
gk
−z z
Γ(z) ∼ e z
1++ ··· + k + ... ,
+∞
z
12z
z
for z ∈ C and | arg(z)| < π with g0 = 1, g1 = 1/12, g2 = 1/288, g3 = −139/51480. See page141 of Olver et al. (2010).
We recall that the incomplete gamma functions are defined asZ zZ ∞α−1 −xγ(a, z) =xe dx, and Γ(a, z) =xα−1 e−x dx.
0

z

205

Chapitre 4. Asymptotiques de la probabilité de ruine
for z ∈ C.
For large value of z and fixed a, from page 179 of Olver et al. (2010), we have the following
asymptotics


a−1
un
a−1 −z1+
Γ(a, z) ∼ z e+ ··· + n + ... ,z>>azzwhere un = (a − 1)(a − 2) . . . (a − n) for | arg(z)| < 3π/2. For large value of z, no expansion
is needed for γ(a, z) since we have γ(a, z) ∼ Γ(a).
Beta function Using the Stirling formula, the beta function β(x, y) can be approximatedfor large values x and y fixed. We recall β(x, y) = Γ(x)Γ(y)/Γ(x + y). We haver

tel-00703797, version 2 - 7 Jun 2012

Γ(x)

∼

x→+∞

e

−x x

x

2π X gk
and Γ(x + y) ∼ e−x−y (x + y)x+y
x→+∞
x
xk

r

k≥0

where the term e−x−y (x + y)x+yq

2π∼ e−x xx+yx+y +∞

β(x, y)q

2π X gk
,
x+y
xk
k≥0

2π
x . Therefore we obtain

Γ(y) X dk,
x→+∞ xyxk∼

k≥0

where the coefficients dk are given by
d0 = 1 and dk = gk −k−1
Xdm gk−m .

m=0

From Olver et al. (2010), page 184, we have asymptotics for the incomplete beta ratio
functionX
1β(a, b, x)∼ Γ(a + b)xa (1 − x)b−1
Iβ (a, b, x) =β(a, b) a→+∞Γ(a + k + 1)Γ(b + k)



k≥0

x
1−x

k
,

for large values a and fixed values of b > 0, 0 < x < 1. Multplying by β(a, b), we getβ(a, b, x)

∼

a→+∞

a

b−1

x (1 − x)Xk≥0

Γ(a)Γ(b)Γ(a + k + 1)Γ(b + k)



x
1−x

k
,

with x ≤ 1 and a → +∞.
Finally, to get the asymptotic of β̄(a, b, x) for large values of b, we use β̄(a, b, x) = β(b, a, 1−x). We haveβ̄(a, b, x)

206

∼

b→+∞

b−1 a

(1 − x)xX
k≥0

Γ(b)Γ(a)Γ(b + k + 1)Γ(a + k)



1−x
x

k
,

4.6. Appendix
The Error function From page 164 of Olver et al. (2010), we have2e−zerfc(z) ∼ √
z π

N
−1
X
k=0

!

1.3.5 . . . (2k − 1)
+ RN (z)(−1)k(2z 2 )kfor z → +∞ and | arg(z)| < 3π/4 We can deduce the asymptotic of the error function at −∞or +∞ using erf(x) = 1 − erfc(x), erf(−x) = −erf(x) and erfc(−x) = 2 − erfc(x). We get2e−x
erf(x) ∼ 1 − √ ,
+∞
x π

4.6.2

2

e−x
erfc(x) ∼ √ ,+∞ x π2e−xerf(x) ∼ −1 + √ ,
−∞
x π

2

e−x
erfc(x) ∼ 2 − √ .
−∞x πFor the continuous time model

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The special case of the Lévy distribution
As for the incomplete gamma function, the function Γ(., .; .) satisfies a recurrence on thea parameter,Γ(a + 1, x; b) = aΓ(a, x; b) + bΓ(a − 1, x; b) + xa e−x−b/x ,see Theorem 2.2 of Chaudry and Zubair (2002). Thus we deduceΓ(−3/2, x; b) =1
Γ(1/2, x; b) + 1/2Γ(−1/2, x; b) − x−1/2 e−x−b/x .
bAs reported in Theorem 2.6 and Corollary 2.7 of Chaudry and Zubair (2002), Γ(a, x; b) has asimpler expresssion in terms of the error function when a = 1/2, −1/2, . . . ,
√ !
√ !#
√ " √
√
π 2 b
b
b
−2 b
e erfc x ++eerfc x −Γ(1/2, x; b) =
2
x
x
and

√ !
√ !#
√ "
√
√
π
b
b
2 b
−2 b
Γ(−1/2, x; b) = √ −e erfc x ++eerfc x −.
xx
2 b

Therefore, we have
 √



√ 
√
1
1
2 −x−b/xπ2 b−2 bΓ(−3/2, x; b) =1 − √ e erfc (d+ ) + 1 + √ eerfc (d− ) − √ e
,
2b
πx
2 b
2 b
with
d+ =

√

√

√
√
b
b
x+
and d− = x −.
xx

It yields
 √


√ 
√
2 π
1
1
α u
1− √
erfc (d+ ) + 1 + √Γ(−3/2, θ0 u; α u/4) = 2ee−α u erfc (d− )
α u
α u
α u

2
2
−√
e−uθ0 −α /(4θ0 ) ,
πuθ0
2

207

Chapitre 4. Asymptotiques de la probabilité de ruine√√√uθ0 + α/(2 θ0 ) and d− = uθ0 − α/(2 θ0 ). We deduce that

 √
√
1
θ0 u uθ01− √I(u, θ0 ) =eeα u erfc (d+ )
α
α u



√
1
2
−α u
−uθ0 −α2 /(4θ0 )e.
+ 1+ √
erfc (d− ) − √
e
α u
πuθ0

with d+ =√By reordering the terms, we get the formula of Albrecher et al. (2011), which is
h
√ 2
√ 
I(u, θ0 ) = e−αc/4λ e(cα+2λ u) /4λc −1 + α u erfc (d+ )√(cα−2λ u)2 /4λc

+e

4.6.3

#
√ 
2α
1 + α u erfc (d− ) − p.
πλ/c

For the discrete time model

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Geometric distribution
In this subsection, we study the geometric distribution, its properties and its minor extensions.
Classic geometric distribution The geometric distribution G(p) is characterized by thefollowing probability (mass) function
P (X = k) = p(1 − p)k ,where k ∈ N and 0 ≤ p ≤ 1. Note that it takes values in all integers N. Another definition of
the geometric distribution is p(1 − p)k−1 so that the random variable takes values in strictly
positive integers. In this case, we can interpret this distribution as the distribution of the first
moment we observe a specific event occuring with probability p in a serie of indepedent andidentically distributed Bernoulli trials.
The probability generating function is given byp
GX (z) =.
1 − (1 − p)z
With this characterizition, it is straightforward to see that summing n geometric randomvariates G(p) has a negative binomial distribution N B(n, 1−p), see, e.g., Chapter 5 of Johnson
et al. (2005). The first two moments can be derived E(X) = (1−p)/p and V ar(X) = (1−p)/p2
when p > 0. In the following subsection, a net profit condition will force E(X) < 1 which isquivalent to p > 1/2. Furthermore, we also have P (X > k) = (1 − p)k+1 .
Modified geometric distributions The geometric distribution will be used to model theclaim severity. It is very restrictive to model claims by a one-parameter distribution. Thus, we
introduce two modified versions of the classic geometric distributions: 0-modified geometric
distribution and 0,1-modified geometric distribution.
The principle is simple, we modify respectively the first and the first two probabilities ofthe probability function. Firstly, we introduce the 0-modified geometric distribution. That is
X ∼ G(q, ρ) whenq
if k = 0,
P (X = k) =,(1 − q)ρ(1 − ρ)k−1
otherwise.
208

4.6. Appendix
when k ∈ N. Using the Kronecker product δij , it can be rewritten as
P (X = k) = qδ0,k + (1 − q)(1 − δ0,k )ρ(1 − ρ)k−1 .
The expectation and variance are given by E(X) = (1 − q)/ρ and V ar(X) = (1 − q)(1 − ρ +q)/ρ2 . We also have
P (X > k) = (1 − q)(1 − ρ)k .
We get back to the classic geometric distribution with ρ = q.
Secondly, we present the 0,1-modified geometric distribution: X ∼ G(q, ρ, 1 − α)qif k = 0,(1 − q)ρ
if k = 1,
P (X = k) =,
k−2
(1 − q)(1 − ρ)(1 − α)αotherwise.

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which is stricly equivalent to
P (X = 0) = q = 1 − p and P (X = k/X > 0) = ρδk,1 + (1 − ρ)(1 − α)αk−2 (1 − δk,1 ).
The mean and the variance are given by




1−ρ
3−α
1−ρ 2
2
E(X) = p 1 +and V ar(X) = qρ + (1 − q)(1 − ρ)
−p 1+
.
1−α
(1 − α)21−αWe get back to the 0-modified geometric distribution with α = 1 − ρ.

4.6.4

Error function linked terms

We want to compute the following integral, linked to the error function
Z ∞2
2
J(a, b, x) =e−ay −b/y dy,xwhere a, b, x > 0. The SAGE mathematicalsoftware (Stein et al. (2011)) suggests to do aR −t2change of variable in order to get e dt. Since the equation t2 = ay 2 + b/y 2 does not havea unique solution, we consider√√
t = ± ay + b/y.
This leads to split the integral J(a, b, x). With algebric manipulations, we get
√
√
√
√
√
b
b
2 ady = ady +dy + ady −
dy.
−y 2
−y 2
Therefore,√√
2 aJ(a, b, x) = e2 ab

Z ∞

−t2

e
x̃1

with x̃1 =√√ax + xb and x̃2 =

√

√ Z ∞
−t2
−2 ab

dt + e

e

dt,

x̃2

√

ax − xb . Hence

√ !
√ !#
√ " √
√
√
√
π
b
b
J(a, b, x) = √ e2 ab erfc
ax ++ e−2 ab erfc
ax −
.
x
x
4 a
209

Chapitre 4. Asymptotiques de la probabilité de ruine
This result is in line with Theorem 7 of Chaudry and Zubair (1994) or Theorem 3.1 of Chaudry
and Zubair (2002). It is closely related to the generalized error function
"
√ !
√ !#
√
√
π 2√b 2√bbberfc(x; b) =+ e−2 b erfc x −.
ee erfc x +4xx

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If x = 0, we get√√πJ(a, b, x) = √ e−2 ab = J(a, b).
2 a√
If b = −1, then it is equivalent as changing the occurence of b by the imaginary number i.
We get




√  √
√
√
√
π
i
i
2i a
−2i a
+e
.
J(a, −1, x) = √ e
ax +
ax −
erfc
erfc
x
x
4 a
√

√
This number is of type z1 z2 + z̄1 z̄2 , where z1 = e2i a and z2 = erfc ax + xi . It is easy tocheck thatz1 z2 + z̄1 z̄2 = 2|z1 z2 | cos(arg(z1 ) + arg(z2 )) ∈ R.
Thus, we extend the notation J(a, b, x) for b = −1 by the above expression.

4.6.5

Analysis

Integration by parts
We give in this subsection the integration by part theorem for the Lebesgues and the Stieltjes integrals from Gordon (1994)’s graduate book. Other reference books include Hildebrandt(1971); Stroock (1994).
Lebesgues integral
TheoremR x (Theorem 12.5 of Gordon (1994)). Let f be Lebesgues-integrable on [a, b] andF (x) = a f for each x ∈ [a, b]. If G is absolutely continuous on [a, b], then f G is Lebesguesintegrable and we haveZ b
Z b
f G = F (b)G(b) −F G0 .
aa

If the integral limits as b tends to infinity exist, one can consider the indefinite integral
versionZZ+∞+∞

f G = lim F (b)G(b) −ab→+∞

F G0 .

aStieltjes integral
Theorem (Theorem 12.14 of Gordon (1994)). Let f, g be bounded functions on a closed interval [a, b]. If f is Stieltjes-integrable with respect to g on [a, b], then g is also Stieltjes-integrablewith respect to g on [a, b] and we have
Z b
a

210

g(t)df (t) = [g(t)f (t)]ba −Z bf (t)dg(t).
aBIBLIOGRAPHY
If the integral limits as b tends to infinity exist, one can consider the indefinite integral
versionZZ+∞+∞g(t)df (t) = lim g(b)f (b) − g(a)f (a) −ab→+∞f (t)dg(t).
aRademacher theorem
For a proof of this theorem, see e.g. Clarke and Bessis (1999).
Theorem. Let f : Rn 7→ R be a locally Lipschitz function. Then f is almost everywhere
differentiable.

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tel-00703797, version 2 - 7 Jun 2012

Conclusion et perspectives

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Dans cette thèse, nous nous sommes attachés au problème de modélisation des différentes
composantes d’un marché d’assurance non-vie. Nous avons apporté de nouvelles contributions
aux thèmes de la modélisation des résiliations, des primes et de la probabilité de ruine.
Tout d’abord, nous avons considéré des modèles de résiliation dans le chapitre 1. Les
modèles statistiques de régression permettent de mesurer, à un instant donné, l’impact desévolutions de prix sur la résiliation des contrats d’assurance. Cependant, le chapitre souligne
les déficiences possibles de ces modèles, si l’on ne possède pas les données appropriées. Le rôlecapital dans la calibration d’une estimation de la prime marché (par police) est établi. Mêmedans le cas où les estimations des taux de résiliation sont considérés comme fiables, l’approchepar régression ne permet pas d’expliquer complètement les interactions entre les assurés et lesassureurs sur un marché.
Basé sur ce constat, un jeu non-coopératif est proposé dans le chapitre 2 pour modéliser lemarché d’assurance dans sa globalité. Partant d’une vision sur une période, le modèle est composé de deux niveaux d’agents. D’un côté, les assurés réagissent aux fluctuations de prix par
un modèle multinomial logit, et de l’autre les assureurs maximisent un critère de profitabilité
approché sous contraintes de solvabilité. Nous démontrons l’existence et l’unicité de l’équilibrede Nash et sa sensibilité locale aux paramètres. Une version plus complexe du jeu où l’on prend
mieux en compte la taille de portefeuille espérée par un assureur dans ses fonctions objective
et contrainte est présentée. Mais l’unicité de l’équilibre de Nash est perdue et pose problème
dans son application. La répétition du jeu simple sur plusieurs périodes met en évidence une
corrélation sur la prime moyenne marché. Cette approche apporte un nouveau point de vue
sur la modélisation des cycles de marché. L’évaluation des probabilités de ruine et de domination est rendue possible par une méthode de Monte-Carlo. Les perspectives pour ce chapitre
sont assez nombreuses, nous en donnons deux des plus évidentes. En pratique, les assureurs
différencient leur prime d’assurance en fonction du profil de risque de l’assuré. Dans notre jeu,
nous supposons que tous les assurés ont le même profil de risque. Dans un premier temps, il
serait intéressant de prendre en compte deux classes d’assurés dans le jeu d’assurance. Une
seconde extension tout aussi pertinante serait de rajouter des sinistres catastrophes naturelles
et des réassureurs. Ce troisième type d’agents permettrait de se rapprocher de la réalité desmarchés d’assurance.

217

Conclusion et perspectives

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Le calcul de prime d’équilibre étant rendu nécessaire, le chapitre 3 présente en détails lesméthodes d’optimisation les plus avancées permettant de résoudre les équilibres de Nash généralisé. Les méthodes d’optimisation étudiées réposent sur une reformulation des équationsde Karush-Kuhn-Tucker (KKT) du problème d’équilibre de Nash. Elles permettent d’élargir
le cadre scolaire des jeux simples à deux joueurs aux jeux généralisés à plusieurs joueurs. Un
complément souhaitable serait de fournir un même panorama pour les jeux conjointement
convexes pour lesquelles d’autres reformulations que la reformulation KKT peuvent être utilisés.
Enfin, le chapitre 4 s’intéresse à un tout autre point de vue du marché de l’assurance enétudiant la probabilité de ruine d’un assureur en temps infini. Dans un modèle de risque avec
dépendance entre les montants de sinistre ou les temps d’attente, nous proposons une nouvelleformule asymptotique de la probabilité de ruine en temps continu et discret. La dépendance
entre sinistres, introduite par une variable aléatoire mélange, permet des formules fermées dela probabilité de ruine ultime dans quelques cas particuliers. Mais surtout, une nouvelle formed’asymptotique en A + B/u est démontrée et est à comparer aux décroissances connues, e−γuou 1/uα , pour les sinistres à queues de distribution légères ou lourdes, respectivement. En
dernier lieu, ce chapitre étudie les problèmes liés à l’utilisation des copules pour les variables
aléatoires discrètes. Une quantification de l’écart maximal entre les versions continue et discrètedu modèle est réalisée. Comme souligné dans Albrecher et al. (2011), l’approche par mélange
utilisé dans ce chapitre peut être utilisée pour des modèles de risque plus avancés que le modèlede Cramér-Lundberg. Il serait intéressant de voir si une formule asymptotique de la probabilitéde ruine de ce type peut toujours être obtenue pour d’autres classes de modèles, par exemple,
les modèles phase-type.

218

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