No d’ordre NNT : 2018LYSE1213

THÈSE DE DOCTORAT DE L’UNIVERSITÉ DE LYON
opérée au sein del’Université Claude Bernard Lyon 1
École Doctorale ED 486
Sciences Économiques et de Gestion
Spécialité de doctorat : Sciences Économiques

Soutenue publiquement le 23 Octobre 2018, par :Claire Mouminoux

Behavioral Biases and Strategies of
Insurance Market Players.

Devant le jury composé de :Sara Fisher Ellison, Professeur au Massachussets Institute of Technology
Meglena Jeleva, Professeur à l’Université Paris Nanterre
Arthur J.H.C. Schram, Professeur à Amsterdam School of Economics
Pierre-André Chiappori, Professeur à Columbia University
Mercè Claramunt, Professeur à University of Barcelona

Rapporteure
Rapporteure
Rapporteur
Examinateur
Examinatrice

Stéphane Loisel, Professeur à l’Université Claude Bernard Lyon 1
Directeur de thèse
Christophe Dutang, Maitre de conférences à l’Univ. Paris Dauphine Co-directeur de thèse
Jean-Louis Rullière, Professeur à l’Université Claude Bernard Lyon 1
Suffragant

iiRemerciements
Je tiens tout d’abord à remercier Stéphane Loisel, mon directeur de thèse, sans qui ce projet n’aurait pas été possible. Merci pour son soutien, ses conseils toujours très avisés et pour
avoir été là, toujours au bon moment et dans la bonne humeur. Je tiens également à remercier
Christophe Dutang, mon co-directeur de thèse pour toutes les heures passées à m’aider quel
que soit le motif, merci pour ses conseils et sa disponibilité.
Je suis très honorée que Sarah Fischer-Ellison, Meglena Jeleva et Arthur Schram aient
accepté de rapporter sur mes travaux. Merci à eux pour le temps consacré à parcourir ce manuscrit et leurs remarques pertinentes. Je tiens également à remercier Pierre-André Chiappori
et Mercè Claramunt d’avoir accepté de faire partie de mon jury. Au-delà, je tiens à exprimer
ma gratitude à toutes et tous pour les sources d’inspirations que leurs travaux m’ont apportés.
Je souhaite aussi remercier tout particulièrement Jean-Louis Rullière qui joua un rôle très
important tout au long de ma thèse. Je le remercie pour les travaux que nous avons effectués ensemble, pour nos discussions sans fin sur l’économie comportementale, industrielle etexpérimentale. Merci d’avoir fortement contribué à mon initiation au monde académique etde m’avoir supporté, même lorsque nous n’étions pas d’accord. Un grand merci également à
Hansjoerg Albrecher pour m’avoir aidé et pris le temps de discuter et commenter mes travaux
sur le dernier chapitre de cette thèse.
Je tiens également à remercier l’ensemble des membres du laboratoire SAF et de l’ISFA
pour m’avoir toujours très bien accueilli et de faire du laboratoire un lieu où il est plaisantde travailler. Une pensée toute particulière à Christian Robert, Nicolas Leboisne et Anne
Eyraud-Loisel. Je ne manquerai pas d’adresser un petit clin d’œil aux doctorants et anciens
doctorants : Pierrick, Baptiste, Julien, Steve, Romain, Morgane, Sarah, Junior, Saker, Mabelle, Gladys, Ines,. . . j’espère que la bonne ambiance qui règne à l’ISFA perdurera.
Je souhaiterai également remercier mes anciens collègues d’AXA. Tout d’abord, je remercie Rami Bou Nader mon ancien manager devenu ami. Merci de m’avoir accompagné pendant
toutes ces années, même à l’autre bout du monde. Merci également à Quiqui, Cucu, Robinou, Momo, Gragra, Gab, Eelaman, merci pour tous les bons moments partagés chez AXA etailleurs et merci d’avoir adhéré au concept du surnom, souvent ridicule.
Je voudrais également remercier mes amis proches qui se reconnaitront et tout particulièrement merci à Marie, qui depuis 27 ans me supporte et me soutient dans les bons commeles mauvais moments. Je ne manquerai pas de mentionner dans ces remerciements les personnes
avec qui j’ai sans doute passé le plus de temps ces dernières années de thèse, les loulous. Merci
Thomas, Romain, Olive, Sébou pour avoir partagé la passion du rock, de la bière et des vagues,
tout en écoutant mes monologues sans fin sur le comportement humain.
Pour finir je voudrai dire un très grand merci à ma famille, à commencer par mon frère.
Merci pour toute l’aide qu’il m’a apportée dans cette thèse et notamment dans l’élaboration
des expérimentations. Merci également à lui d’avoir toujours été présent pour m’écouter, meloger, discuter, surfer, manger (particulièrement des fondus très réconfortantes). Il serait difficile de lister toutes les choses pour lesquelles il a fait de ces années de thèse de très belles
années.
Enfin, merci à mes parents, sans qui rien de tout cela n’aurait été possible.
iii

iv

RésuméCette thèse a pour objectif d’analyser les interactions entre les agents économiques opérant sur
le marché de l’assurance de détail. D’un côté, les assurés souhaitant se couvrir contre un risquede perte doivent explorer le marché afin de souscrire un contrat en ligne avec leur perception durisque. D’un autre côté, les assureurs se font concurrence sur un marché régulé, leur imposant
un certain niveau de capital afin de garantir leur solvabilité dans un contexte d’incertitude surles risques souscrits. D’autre part, des intermédiaires proposent leurs services afin de faciliter
l’interaction entre les consommateurs, averses aux risques, et les firmes, preneuses de risques.
C’est donc dans ce contexte que nous analysons les comportements des acteurs de l’assurance
à travers différentes perspectives. Les Chapitre 1 et 2 de cette thèse résultent d’expérimentations en laboratoire, effectuées à l’aide d’une interface web conçue spécifiquement pour cesétudes. Les résultats du Chapitre 3, quant à eux, sont basés sur un modèle théorique et dessimulations numériques.
Le Chapitre 1 se concentre sur la relation entre l’honnêteté et les croyances en l’honnêteté des agents économiques. À l’aide des données collectées en laboratoire, nous montronscomment l’incertitude et le sentiment de se trouver dans des conditions plus ou moins avantageuses impactent à la fois le niveau d’honnêteté mais aussi la croyance en l’honnêteté envers lesautres. En règle générale, les consommateurs surestiment l’honnêteté des intermédiaires. Ainsi,
ce résultat justifie leur présence sur le marché de l’assurance. D’autre part, nous montrons
aussi que les incitations financières proposées aux intermédiaires sont sources de distorsion
des croyances en l’honnêteté. Plus le niveau d’incitation est faible, plus les consommateurs
anticipent un comportement malhonnête.
Dans le Chapitre 2, nous mettons en évidence le dilemme dont fait face le consommateur
sur un marché comprenant une multitude de canaux de distribution. Doit-il explorer par luimême et choisir parmi un large ensemble de contrats ou bien déléguer une partie de sa décisionà un intermédiaire plus ou moins honnête ? En utilisant une expérimentation en laboratoire,comportant des coûts de recherche, nous montrons que l’obfuscation liée à une importantequantité d’information et les croyances en l’honnêteté des intermédiaires sont les principauxdéterminants des décisions de recherche et d’achat. Nous montrons également que l’obfuscation et l’attitude des intermédiaires sont sources d’inefficience dans les prises de décisions, enparticulier vis-à-vis des caractéristiques des contrats d’assurance souscrits par les consommateurs. Dans ce sens, l’identification d’un effet de focalisation appuie l’importance du niveau
des prix dans les prises de décision au détriment de l’environnement de risque et du niveau decouverture. L’introduction des coûts de recherche dans le processus d’exploration, ainsi quel’hétérogénéité des croyances en l’honnêteté justifient les stratégies de distribution multicanal
adoptées par les assureurs.
Une analyse d’un jeu non coopératif répété est exposée dans le Chapitre 3 de cette thèseoù les pertes et le comportement des consommateurs sont stochastiques et les assureurs se
font une concurrence en prix. Afin d’intégrer les contraintes des régulateurs, nous déterminonsles équilibres de Nash sous contrainte de solvabilité. Nous analysons également la sensibilitédes primes d’équilibre en fonction des paramètres du jeu, en particulier lorsque les firmes
ne bénéficient pas des mêmes avantages comparatifs (i.e. réputation conduisant à différents
niveaux de rétention des clients, ancienneté des assureurs conduisant à différents stocks encapital).
Mots-clés: expérimentation, économie comportementale, intermédiation, canaux de distribution, honnêteté, obfuscation, théorie des jeux.

Abstract

This thesis aims at explaining interactions among economic agents operating in the retailinsurance market. On the one hand, the policyholder is willing to be covered against a risk.
To do so, they have to explore the insurance market to purchase a contract in line with theirrisk perception. On the other hand, insurers compete in a regulated market which imposes
capital constraints for shock loss absorption purposes. In between, intermediaries may provide services in order to facilitate interaction between risk-adverse consumers and risk-takerfirms. In this context, we analyze economic behaviors of insurance actors through different
perspectives. Chapter 1 and 2 both result from original laboratory experiments, conducted
through a web-interface especially designed for these studies. Results in Chapter 3 rely on atheoretical model and numerical simulations.
Chapter 1 emphasizes on the relationship between honesty and beliefs about honesty of
economic agents. According to laboratory results, we show how the uncertainty and the perception of advantageous conditions impact the level of honesty and beliefs about honesty. In
general, consumers estimate that intermediaries are more honest than they really are, hence
supporting their physical presence in the insurance market. However, intermediary financial
incentives are a source of distortion of honesty beliefs: the weaker the level of the incentive,
the stronger the deviation anticipations.
In Chapter 2, we shed light on the dilemma faced by insurance purchasers under a multichannel distribution. Should the consumer, themselves, choose from a large set of insurance
policies, or rather delegate a part of their decision to a more or less honest intermediary?
Using experimental approaches, including exogenous search costs, we show that obfuscation
and beliefs about intermediary honesty are the main determinants of individual choices. We
also find that obfuscation and intermediaries’ deviation are the main sources of inefficiency indecision-making, especially regarding the features of the insurance contracts chosen by consumers. Our identification of the focal point effect supports the importance of the price levelon purchasing decisions rather than the risk environment or the coverage level. The introduction of search costs in the exploration process, as well as the heterogeneity of beliefs about
honesty, justify multichannel distribution strategies adopted by insurers.
An analysis of insurer price competition with a repeated one-period non-cooperative gameis conducted in Chapter 3, where both insurer losses and consumer behaviors are stochastic. Because of regulatory obligations, we consider a solvency constraint when computing
Nash-Equilibrium. We determine the sensitivity of the premium equilibrium with respect tothe parameters, especially when firms do not benefit from same competitive advantages (i.e.
reputation effect leading to differences in consumers inertia or market seniority leading todifferences in capital stock). We also study insurers’ market share in response to the entry of
new insurer undercutting prices but dealing with binding solvency constraints.
Keywords: experimentation, customer behaviors, intermediation, distribution channel, honesty, obfuscation, game theory.

Contents
Remerciementsiii

Résumé

v

Abstractvii

Tables des matières

ix

Introduction générale

Introduction

3

1

Introduction Générale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2

Incitations Financières et Honnêteté . . . . . . . . . . . . . . . . . . . . . .

5

3

Informations, Intermédiaires et Choix d’Assurance . . . . . . . . . . . . . .

94Stratégies des Assureurs, Avantages Comparatifs et Contraintes . . . . . . . 14

Contributions et principaux résultats

23

Contributions and main results

27

Contributions en économie comportementaleix

Contents
Chapter 1 Are we more honest than others think we are?

33

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.2

Experiment Design and Procedure . . . . . . . . . . . . . . . . . . . . . . . 37

1.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.4

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.5

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 2 Obfuscation and Honesty: Insurance Demand with Multiple
Distribution Channels

49

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.2

Experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3

Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4

Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.5

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

2.6

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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

Contributions en sciences actuarielles

Chapter 3 A game-theoretic analysis of insurers pricing strategies

91

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.2

Description of the repeated game . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3

Properties of policyholder behaviors and consequences on insurer portfolio
sizes and losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.4

Properties of insurer underwriting strategies. . . . . . . . . . . . . . . . . 112

3.5

Theoretical properties of the repeated game . . . . . . . . . . . . . . . . . . 115

3.6

Empirical evidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.7

Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.8

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
x

Conclusion

Conclusion et perspectives

195

Bibliographie

197

xi

Contentsxii

Introduction générale

1

Introduction"You can’t simply choose what you want, you have to be chosen.” A. Roth1

Introduction Générale

Bien souvent le tout premier réflexe en économie, consiste à raisonner en termes de marché.
L’assurance n’échappe à cette règle, à tel point que la notion de marché de l’assurance est aussi
souvent évoquée qu’elle reste en revanche peu définie. A priori ce besoin de définition n’apparaît pas primordial : la possibilité de survenance d’un événement indésirable pourrait a priori
constituer un objet d’échange sur le marché. C’est ce que nous appellerons plus communément
le risque. Bien que le risque peut aussi être défini par un aléa sur un évènement favorable,
c’est bien dans le domaine des pertes que le concept d’assurance prend tout son sens. En effet,d’un côté certains agents économiques seraient désireux de se débarrasser d’un risque, motivés
entre autres par leur aversion pour le risque, tandis que d’autres seraient prêts à supporterce risque moyennant une rémunération, appelée prime d’assurance. Dès lors, ce raisonnement
pourrait laisser penser que le risque représente un bien comme un autre. Cependant, l’aspectd’incertitude qui l’entoure complexifie le processus d’échange. En effet, le transfert de risqued’un agent à un autre est conditionné par la perception qu’ils en ont.
Ce phénomène n’est pas nouveau, dès l’antiquité les sociétés voient apparaître les prémicesde ce qui sera communément appelé “assurance”. En effet, les expéditions maritimes, coûteuses
et risquées, furent les premières à justifier et illustrer la nécessité d’un marché de l’assurance,
fondée sur l’échange de risque. D’un côté, les marchands effectuent un prêt afin de financer
la marchandise transportée et paient une somme supplémentaire aux prêteurs correspondantà une partie de la vente de marchandise. Cet échange est cependant conditionné par le bon
déroulement de l’expédition. En effet le prêt n’a pas à être remboursé si la marchandise est
volée ou perdue suite à un naufrage. D’un autre côté, les prêteurs ou investisseurs sont prêts
à supporter le risque de perte d’un navire, parmi une flotte, en échange des montants supplémentaires reçus pour l’ensemble des expéditions se déroulant sans aucune perte (référerà Martin (1876), pour une revue de l’histoire de l’assurance maritime). C’est le principe demutualisation du risque, pilier du marché de l’assurance. Il faudra cependant attendre le XVIIème siècle, et notamment le grand incendie de Londres de 1666, pour voir apparaître un cadre
légal et une généralisation de la souscription de contrat d’assurance dite “moderne” .
La littérature moderne en économie de l’assurance est cependant là pour attester que cettereprésentation marchande est pour le moins sommaire. Deux éléments fondamentaux sont à
prendre en compte, qui rend le marché de l’assurance singulier.
3

Introduction
D’une part, le risque qui serait donc l’objet de l’échange est rarement exogène, dans le sens
où bien souvent il dépend du comportement (ex ante ou ex post) de l’agent économique.
Qui plus est, la nature de cette dépendance reste le plus souvent incertaine. Ainsi, l’asymétrie d’information entre l’assureur et l’assuré est l’une des principales explications des biais
comportementaux inhérents au marché de l’assurance. Elle est, de manière plus ou moinsconsciente, un phénomène connu des praticiens de l’assurance depuis ses origines. Son étude
par les économistes est plus récente et a permis de réelles contributions en matière de compréhension des mécanismes des marchés d’assurance. Cette asymétrie d’information donne lieu
à deux phénomènes bien connus en économie comportementale : l’anti sélection et l’aléa moral.
L’un des premiers économistes à avoir mis en évidence l’importance et les conséquences
de l’asymétrie d’information en matière d’assurance est le Prix Nobel K. J. Arrow (1963).
En prenant en compte l’aversion aux risques, aspect élémentaire du comportement humain
dans la prise de décision dans un domaine incertain, il apporte une première contribution
fondamentale à la compréhension du partage de risque optimal entre l’assureur et l’assuré. Il
souligne aussi les problèmes rencontrés par les assureurs faisant face à de l’aléa moral, basés
sur une modification des comportements conduisant à l’augmentation du risque a postériori(i.e. après la souscription d’un contrat d’assurance). Ses travaux sur l’anti sélection sont moinsconnus mais il est le premier à déclarer que “l’assurance nécessite, pour avoir un effet social
pleinement bénéfique, une discrimination des risques qui soit la plus grande possible”. Une
deuxième contribution fondamentale à la compréhension des mécanismes de marché et de rencontre de l’offre et de la demande en assurance est celle apportée par Rothschild et Stiglitz(1976), notamment grâce à leur modélisation de l’anti sélection. Ainsi, ils montrent qu’il est
possible de mettre en place des mécanismes de sélection des risques même si l’on ne connaît
pas, a priori, le degré de risque de chaque individu.
D’autre part, traiter du marché de l’assurance suppose que la rencontre entre les deuxcôtés du marché se fait spontanément et sans coût. Or les défauts d’information subis parl’offreur d’assurance, le pousse à offrir un service bien plus complexe qu’un simple transfert
de risque. En effet, tout vendeur de risques (le consommateur dans la version de base) n’est
pas prêt à rencontrer tout acheteur de risque (l’assureur). Inversement, tout assureur chercheà sélectionner, trier, “screener” (Spence (1973)) les consommateurs. A tout cela s’ajoute le fait
que l’offre se constitue sous le régime concurrentiel, qui conditionne à son tour les stratégies
de prix et d’intermédiation des assureurs.
Ainsi, et de manière générale, le fonctionnement d’un marché concurrentiel ne se caractérise
pas nécessairement par le fait que les consommateurs ont la possibilité d’acheter une quantité
voulue à un prix donné. Les assureurs vont à la fois spécifier une quantité et un prix afin deprocéder à la sélection du risque. En effet, les contrats qui ne discriminent pas suffisamment
sont susceptibles d’attirer principalement les mauvais risques présents sur le marché. De fait,ce mécanisme de sélection conduisant à des offres d’assurance hétérogènes rend, au passage, lacompréhension des choix d’assurances plus complexes pour le consommateur. Par conséquent,avant même le partage du risque en tant que tel, le marché de l’assurance et l’ensemble desstratégies opérées par ses acteurs sont avant tout conditionnés par la problématique de rencontre et le design de ce processus de rencontre, “matching”. Comme souligné par Roth (2015),“il ne suffit pas de choisir ce que l’on veut, il faut aussi être choisi”.

4

2. Incitations Financières et Honnêteté
Le marché de l’assurance se révèle donc comme un terrain d’étude des comportements àla fois complexe et varié. L’incertitude omniprésente sur le marché de l’assurance, conduisantindéniablement à l’asymétrie d’information, rend les mécanismes de choix et les stratégies
des agents économiques bien plus complexes. D’un côté, les consommateurs souhaitent maximiser leur utilité, celle-ci conditionnée en partie par leur aversion au risque mais aussi parleurs capacités et possibilités de choix. De l’autre, les assureurs cherchent à maximiser leurprofit dans un marché compétitif et fortement contraint par les régulateurs, de par la naturemême de leurs produits basés sur : le risque. La complexité de la rencontre entre l’offre etla demande fait naturellement émerger une multitude d’intermédiaires facilitant le transfert
de risque et, comportant des caractéristiques diverses elles-mêmes révélatrices d’informations
et de biais comportementaux. La démocratisation de l’utilisation d’internet dans le processus
et les stratégies d’achat et de vente sont des facteurs supplémentaires de multiplicité et decomplexification des comportements. Cela bouleverse l’accès à l’information pour les consommateurs mais aussi les stratégies de ventes et la compétition entre assureurs, notamment grâceà l’émergence de nouveaux intermédiaires d’assurance dans un marché toujours plus concurrentiel (Brown et Goolsbee (2002)).
La singularité du marché de l’assurance a essentiellement été étudiée selon la première
dimension au travers bien évidemment de la modélisation de la sélection contraire et de l’aléa
moral. En revanche la prise en compte de la deuxième dimension est bien plus récente et fait
l’objet précisément des essais contenus dans cette thèse.
Cette thèse se propose donc d’étudier les biais comportementaux et des stratégies desacteurs de l’assurance non-vie. L’objectif de mes recherches consiste en la compréhension
des stratégies des assureurs souhaitant acquérir de l’information sur les consommateurs, tout
en répondant aux exigences des régulateurs, afin de maximiser leur profit. Je m’intéresseégalement au processus de décision des consommateurs, dans un environnement incertain,
ou l’accès à l’information tend à s’accroître mais aussi à se complexifier. Ainsi, et commementionné précédemment, le marché de l’assurance est avant tout un “ matching market”
où la rencontre entre les assureurs et les futurs assurés se fait à l’aide d’intermédiation. Je
consacre donc une partie de cette thèse à l’étude des choix des intermédiaires, en prenant
notamment en compte l’émergence de nouveaux acteurs tel que les comparateurs de prix sur
internet. D’autre part, les intermédiaires ont également leurs propres intérêts économiques etfinanciers dans ce jeu de “matching”. Ces intérêts structurent le marché de l’assurance, tantl’offre que la demande, et révèlent de nouveaux biais comportementaux.

2

Incitations Financières et Honnêteté

Dans le premier chapitre de cette thèse “Sommes-nous plus honnêtes que l’on croit ? ” (“Are we more honest than others think we are ?”), je me suis donc intéressée au comportementd’honnêteté mais aussi aux croyances dans l’honnêteté des autres. Cette étude des comportements d’honnêteté est basée sur une approche expérimentale et l’analyse est effectuée sur lesdonnées collectées en laboratoire. L’objectif est de mesurer l’adéquation entre l’honnêteté desuns et la croyance en l’honnêteté des autres.
En effet, comme mentionné précédemment, le processus d’achat en assurance repose sur5Introduction
un large réseau d’intermédiaire. Ces mêmes intermédiaires ont leurs propres incitations financières pouvant les amener à ne pas proposer le meilleur contrat mais le plus rentable. Avantmême d’analyser les choix de délégation en assurance, il est donc important de disposer d’une
mesure permettant de comprendre le niveau de confiance des consommateurs, et si celle-ci est
en adéquation avec le comportement d’honnêteté observé. En effet, les économistes sont d’accord pour dire que les comportements malhonnêtes sont source d’inefficience dans les échanges
entre agents économiques. Cependant, tout ne repose pas sur l’honnêteté des uns. Même dans
un monde des plus honnêtes, il est primordial que les agents croient en cette honnêteté poureffectivement réduire les coûts de transactions.
De nombreuses recherches se sont déjà intéressées au sujet, notamment en analysant demanière agrégée les impacts des caractéristiques sociales des individus et de leur relation surleur niveau de confiance et d’honnêteté. Berg et al. (1995) furent les premiers à mettre en évidence l’importance du contexte social sur la réciprocité de l’honnêteté. En effet, ils montrent
que la présence d’interaction entre les sujets à tendance a accroître le niveau de confiance etd’honnêteté. Dans le même sens, Glaeser et al. (2000) montrent que les individus ont tendance
à avoir plus confiance aux personnes ayant les mêmes caractéristiques sociales. Plus récemment, Ermisch et al. (2009), analysent plus précisément les caractéristiques des populations,et mettent en évidence, par exemple, que les individus ayant des situations financières “confortables” ont tendance à plus faire confiance. Quant à Hugh-Jones (2016), il démontre une réelle
hétérogénéité des croyances en l’honnêteté à travers les pays.
Ces études sont des apports essentiels à la compréhension des croyances et de l’honnêteté mais aussi à la compréhension des interactions, plus ou moins facile, entre des groupesd’individus. En effet, l’étude des caractéristiques sociales permet aux économistes de mieux
comprendre comment les échanges, basés sur la confiance, se forment et qui sont les plus enclins à y participer. Cependant, dans cette thèse, je me suis concentrée sur l’aspect financier
des incitations à l’honnêteté. Il est vrai que déléguer une partie de sa décision peut aussiêtre étroitement liée à la relation sociale que le consommateur a avec ses intermédiaires. Par
exemple, un individu demandera plus facilement un conseil au courtier l’ayant déjà conseillé,
et ce, à confiance égale. Or, le but n’est pas d’analyser les effets de réputation et de vécu sur lechoix de délégation. En effet, je souhaite ici comprendre à quel point les incitations financières
à la malhonnêteté affectent le niveau de confiance et donc le choix de délégation. Ainsi, l’étude
des incitations financières sur l’honnêteté dans un contexte anonyme paraît la plus adéquate
pour répondre à cette question.
C’est donc dans la lignée de Mazar et al. (2008), Fischbacher et Follmi-Heusi (2013), Houser et al. (2012), et Galeotti et al. (2017) que s’inscrivent les études menées dans ce premierchapitre. Par exemple, Mazar et al. (2008), Fischbacher et Follmi-Heusi (2013), Abeler et al.
(2016) montrent que l’accroissement des incitations financières n’impacte pas significativement
le niveau d’honnêteté. Néanmoins, Kajackaite et Gneezy (2017) ne trouvent pas les mêmes
résultats en considérant un “jeu de déception” où la possibilité de mentir est explicitement
exposée aux sujets. Le fait de faire du mensonge une règle explicite du jeu semble conduire lessujets à apporter plus d’importance au risque d’être pris entrain de mentir. Cela les conduit
donc à une plus grande analyse du compromis coûts-bénéfices. Pour appuyer ce résultat, Yaniv et Siniver (2016) montrent que dans un environnement sans danger d’être pris en train detricher, les individus trichent bien plus. Ainsi, ces résultats semblent démontrer que l’honnê6

2. Incitations Financières et Honnêteté
teté des individus est largement affectée par les conséquences de leurs actes sur leur éthiquesociale. D’autre part, Houser et al. (2012) ainsi que Galeotti et al. (2017) mettent en évidence l’importance de la perception de l’environnement sur le comportement d’honnêteté. En
effet, les sujets sont moins honnêtes dans des conditions injustes. L’inégalité des situtations
diminue la considération de l’éthique sociale dans la prise de décision. Toutefois, à ma connaissance, aucune étude n’a confronté, d’une part le comportement d’honnêteté et d’autre partles croyances en l’honnêteté, étant donné les incitations financières et l’environnement plus ou
moins injuste du sujet, tenté à adopter un comportement malhonnête.
J’ai donc élaboré une nouvelle méthode expérimentale afin de créer à la fois, une mesured’honnêteté et une mesure de croyance en l’honnêteté. Afin que ces deux mesures soient parfaitement comparables et contrairement au jeu standard de Berg et al. (1995), j’ai introduit
une règle implicite permettant de mesurer l’écart entre la règle et le comportement observé
ou estimé. En effet, le jeu de Berg et al. (1995) implique des effets d’aversion à l’inégalité etd’altruisme dus au manque d’une règle objective. À aucun moment une règle spécifie aux sujets un devoir de transfert d’argent ou de récompense du transfert d’argent. Chaque sujet a sa
propre interprétation du jeu. Ainsi, il aurait été difficile de comparer l’honnêteté d’un côté etles croyances de l’autre. L’intérêt de mes recherches est de comprendre la délégation de choix
en assurance et le comportement des intermédiaires incités financièrement. Il existe bien unerègle explicite, déontologique, demandant aux intermédiaires de rendre un service de conseil
adéquat, autrement dit de proposer le contrat correspondant le mieux au consommateur etce, au meilleur prix.
D’autre part, et comme mentionné précédemment, je souhaite comprendre dans quelle mesure le niveau d’incitation financière influe sur le comportement d’honnêteté et des croyances
en l’honnêteté. C’est donc avec cet objectif que j’ai introduit dans l’expérimentation des conditions plus ou moins favorables lors de la prise de décision. En effet, tous les sujets font face àla même règle mais celle-ci est plus ou moins sévère. Si ils suivent cette règle, certains sujets
se retrouvent très peu rémunérés par rapport à la moyenne tandis que d’autres le sont beaucoup plus, affectant donc leurs décisions d’être honnête mais aussi les croyances en l’honnêteté.
Cette nouvelle expérience a quatre principaux avantages. Tout d’abord, je m’assure quela malhonnêteté ne puisse pas être détectée, évitant ainsi un potentiel effet de demande. De
plus, après avoir contrôlé l’anonymat des choix, je collecte les données sur le comportementd’honnêteté au niveau individuel contrairement à Fischbacher et Follmi-Heusi (2013) ou encore Hugh-Jones (2016). Enfin, j’introduis différentes conditions plus ou moins favorables afinde comprendre l’impact des incitations financières sur les croyances et l’honnêteté. Afin, cetteméthode est facile à mettre en place en laboratoire et relativement rapide.
Les résultats montrent que les individus sont en moyenne moins honnêtes que ce que l’on
croit, en particulier quand les conditions sont favorables. Les résultats m’ont permis de modéliser un effet de distorsion des comportements, à la fois de l’honnêteté et des croyances, enfonction des différentes conditions plus ou moins favorables.
Tout d’abord, les individus sont en moyenne malhonnêtes (i.e. ils ne respectent pas larègle). Cependant, cette malhonnêteté est d’autant plus importante lorsque qu’ils font face àdes conditions peu favorables. Ainsi, le sentiment d’injustice semble justifier leurs décisions
7

Introduction
malhonnêtes. Ce qui est d’autant plus intéressant est le fait que les sujets, estimant le comportement d’honnêteté, anticipent ce phénomène. Ils ont moins confiance face à un individu
aux conditions peu favorables, tandis qu’ils surestiment significativement l’honnêteté dans desconditions favorables. Cependant, ils ont une meilleure estimation de la malhonnêteté lorsqueles conditions se détériorent.

Les implications pour comprendre les comportements d’achat en assurance sont grandes,
d’autant plus que la nouvelle Directive de Délégation en Assurance renforce les contraintes
sur la transparence des conditions de rémunération des intermédiaires. Les consommateurs
pourront alors connaitre dans quelle mesure et à quel niveau leurs intermédiaires sont rémunérés. Cela implique une grande différence de choix de distribution, étant donné que les“bons” risques peuvent être assimilés à de bonnes conditions. Ainsi et si ils sont conscientsde leurs caractéristiques avantageuses, ils auront donc tendance à déléguer d’avantage, malgré
une forte déviation des intermédiaires.

Dans un second temps, je me suis intéressée à la corrélation entre la confiance en l’honnêteté et l’aversion au risque. Tandis que la littérature est mitigée sur ce point, les résultats decette expérimentation ont permis de concilier les différents résultats sur le sujet. En accord
avec Corcos et al. (2012), je trouve que le niveau moyen de confiance en l’honnêteté n’est
pas corrélé à l’aversion au risque. Ainsi, faire confiance n’est pas une décision risquée. Cependant, mes résultats supportent aussi une éventuelle corrélation, comme mentionné par Naefet Schupp (2009). En effet, bien que l’aversion au risque n’explique pas le niveau de confiance
des individus, elle explique cependant la distorsion des croyances lorsque les conditions deviennent peu favorables. Ainsi, les sujets averses aux risques réduisent significativement plus
leur confiance lorsque la situation est défavorable.

D’autre part, tandis que Grolleau et al. (2016) montrent que les hommes ont d’avantage
tendance à tricher (i.e. être malhonnête), je montre qu’ils ont aussi moins confiance aux autres.
Leur malhonnêteté les incite eux-mêmes à ne pas croire en l’honnêteté des autres.

Le premier chapitre de cette thèse m’a donc permis de mieux comprendre le comportementd’honnêteté et de croyances en l’honnêteté suivant différentes conditions financières. Ainsi,
nous voyons bien que malgré une règle explicite, les individus sont en moyenne malhonnêtes.
De plus, cette malhonnêteté est globalement sous-estimée par la population de notre analyse.
Ces résultats justifient la présence d’intermédiaires plus ou moins honnêtes et montrent uneréelle hétérogénéité des croyances en l’honnêteté. C’est donc une première explication desdifférences de comportements d’achat en assurance et des choix de délégation. Cette expérimentation, relativement simple à mettre en place, m’a également permis de créer une mesurecontrôlant les croyances en l’honnêteté. Cette mesure est utilisée dans le deuxième chapitrede cette thèse afin de mieux comprendre l’ensemble du processus d’achat en assurance, de larecherche d’information au choix du contrat.

8

3. Informations, Intermédiaires et Choix d’Assurance

3

Informations, Intermédiaires et Choix d’Assurance

Après avoir créé et analysé une nouvelle mesure d’honnêteté et de croyance en l’honnêteté, je me suis intéressée dans le deuxième chapitre à l’ensemble du processus d’achat d’uncontrat d’assurance. Le chapitre central de cette thèse intitulé “Obfuscation et honnêteté : uneétude expérimentale de la demande d’assurance sur un marché sous intermédiation” (“Obfuscation and Honesty : Experimental Evidence on Insurance Demand with Multiple Distribution
Channel”) se concentre sur la demande d’assurance. Autrement dit, il est consacré à l’étude
du processus d’achat composé à la fois de recherche de l’information mais aussi du choix decouverture d’assurance, étant donné les caractéristiques de l’assuré et l’information disponible.
Le but de ce chapitre est de mettre en évidence le dilemme dont fait face le consommateur
entre, d’une part, explorer le marché par lui-même au risque de se perdre dans une quantité
trop importante d’information, ou bien déléguer une partie du processus d’achat au risque detomber sur un intermédiaire malhonnête. Ainsi, les analyses menées dans ce chapitre m’ont
permis de mettre en évidence les effets de focalisation ou d’ancrage dus à une quantité trop
importante d’information, mais aussi les effets de déception conduisant à des changements
important de stratégies d’exploration. De plus, j’ai aussi étudié l’optimalité des choix afind’analyser dans quelles mesures la sur-confiance en l’honnêteté des intermédiaires et l’obfuscation sont sources d’inefficience dans les choix.
Le deuxième chapitre de ma thèse est donc essentiellement tourné vers l’étude de la demande d’assurance, bien qu’il permette de mieux appréhender les différentes stratégies possibles et optimales des assureurs suivant les canaux de distribution, mais aussi le comportement
des intermédiaires face aux incitations financières des assureurs. Les résultats sont tirés d’une
expérimentation en laboratoire dans laquelle une partie des sujets a dû choisir un contrat
d’assurance parmi un choix limité, dépendant de leur stratégie de recherche via plusieurs intermédiaires : un comparateur de prix, un courtier et un agent général d’assurance. Les sujets
restants ont joué le rôle d’intermédiaires “physiques” ayant leurs propres incitations financières. J’ai donc utilisé dans ce chapitre la mesure de confiance et d’honnêteté détaillée dansle premier chapitre de ma thèse. Celle-ci permettant de contrôler le rôle de la confiance dansle processus d’achat, notamment dans les décisions de délégation, mais aussi dans l’efficiencedes choix. D’autre part, comme les choix d’assurance, basés sur la théorie de l’utilité espérée(Rothschild et Stiglitz (1976)), sont fortement liés à l’aversion au risque des agents, il a été
nécessaire de la contrôler pour l’ensemble des sujets.
Avant de résumer l’ensemble des résultats du chapitre deux de cette thèse et leur implication, il est important de rappeler les éléments théoriques clés qui ont permis la constructionde cette expérimentation. Ces éléments sont basés essentiellement sur la théorie de l’economie
de l’assurance, tel que la théorie de l’utilité espérée mais aussi sur la théorie des coûts derecherches. En effet, il est important qu’une experimentation comportementale ne comporte
pas de choix strictement optimaux. En d’autres termes, il ne doit pas exister de "bonne réponse". Ainsi, chaque décision est révélatrice des préférences et donc des biais cognitifs dessujets étudiés.
Ainsi, afin de révéler au mieux les stratégies de recherche des consommateurs, j’ai intégré
dans l’expérimentation des coûts de recherche conditionnant l’accès à l’information. Ces coûts
9

Introduction
sont donc la monétisation du temps, effort ou argent dépensé par un consommateur recherchant un produit ou un service. Contrairement à Brynjolfsson et Smith (2000), l’objectif demes recherches n’est pas de mesurer les coûts de recherche individuels. Inspiré de la méthodologie utilisée par Schram et Sonnemans (2011) , ces coûts de recherche sont donc exogènes.
Cependant, et encore une fois dans un souci de neutralité des choix, ils ont été défini sous deux
différentes contraintes. La première contrainte consiste à définir pour chaque canal de distribution un coût d’accès à l’information, tel que, en espérance le coût pour atteindre le contrat
optimal du marché soit identique. En effet, supposons qu’il soit moins couteux de découvrir
l’ensemble des contrats en visitant un a un chaque assureur que de demander un conseil à uncourtier, dans ce cas, le choix du sujets ne révèle pas ses préférences de canaux de distribution
mais uniquement le fait qu’il minimise bien ses coûts de recherches. La deuxième contrainte
m’a permis de définir le montant appelé "crédit de recherche" qui permet au sujet d’explorer
le marché. Chaque coût de recherche correspondant à une révélation et, est débité sur ce crédit. Ainsi, le montant non consommé par le sujet lui est payé à la fin de l’expérimentation.
Ce crédit d’information est donc défini de manière à ce que le sujet puisse être pleinement
informé, c’est à dire puisse explorer à l’aide de l’ensemble des canaux de distribution. Le choix
du canal n’est donc pas unique.
Du côté de l’offre, elle est également définie de façon exogène et ne tient pas compte deseffets des coûts de recherche sur les équilibres de marché. Cependant, il est vrai que les coûts
de recherches sont essentiels à la compréhension des équilibres de marché. Le premier à avoir
étudié de manière théorique l’impact des coûts de recherches sur les primes d’équilibres est
Diamond (1971). Dans son papier de référence, il montre pourquoi le consommateur est prêtà payer plus pour éviter d’engendrer un nouveau coût de recherche, expliquant la relation
croissante entre coûts de recherche et prime de marché. Plus tard, Brown et Goolsbee (2002)
démontreront de nouveau cette relation sur le marché de l’assurance vie. En utilisant des données empiriques ainsi que des données sur l’utilisation d’internet dans le processus d’achat,ils montrent que l’apparition des cyber-canaux au cours du XXème siècle à significativement
diminuée les prix des assurances vie. D’autre part, Baye et al. (2004) montrent une forte corrélation entre le nombre de produit listé sur un site de comparaison et le niveau moyen des prix.
Ici encore, les cyber-canaux ont pour effet d’augmenter la competition en diminuant les coûts
de recherches. Autrement dit, en facilitant l’accès à l’information. Ainsi, leur résultats supportent la loi du prix unique. Enoncée par Bertrand et donnant lieu au paradoxe de Bertrand,elle supporte le fait que, sur un marché totalement compétitif, le prix d’équilibre est unique etest égal au coût marginal du produit vendu. Plus récemment, Branco et al. (2012) analysent
les prix d’équilibres tout en tenant compte du signal reçu à chaque nouvelle information dévoilée. Bien que les résultats aillent dans le même sens que ceux présentés précédemment, defaible coûts de recherche dans un contexte de recherche séquentielle (i.e. visiter un à un lesassureurs par exemple) peuvent augmenter les prix d’équilibres. En effet, les consommateurs
utilisent toute information additionnelle comme un signal permettant d’ajuster leurs croyances
sur les informations non révélées. Ainsi, les firmes peuvent à l’équilibre proposer une prime
plus élevée afin de décourager la recherche, pourtant peu coûteuse.
Ainsi, bien qu’il soit vrai que la stratégie de distribution conditionne également la stratégiede prix et donc que chaque canal ne propose pas nécessairement les mêmes type de contrat,
le but de ce chapitre est de révéler les biais comportementaux conditionnant la demanded’assurance. Afin de faciliter l’analyse des choix et leur efficience je me suis donc largement103. Informations, Intermédiaires et Choix d’Assurance
inspirée de Schlesinger (2013) dans la construction des contrats proposés, qui sont donc lesmêmes pour chaque canal de distribution. Ces contrats se composent donc de deux éléments :une prime commerciale CP et une franchise D tel que,
CP = p × (R − D) × (1 + λ),où p est la probabilité de perdre le montant R, et λ est le taux de chargement de l’assureur. La
définition d’un seul et même contrat à l’aide de deux éléments distincts m’a permis d’incluredans l’expérience l’aspect d’hétérogénéité et de complexité des contrats d’assurance.
Au total, le “marché” d’assurance proposé dans cette expérimentation comporte quatre
assureurs. Puisque dans la réalité les assureurs ne proposent jamais un contrat unique, chaqueassureur propose donc deux contrats. Ainsi, l’offre d’assurance disponible se résume à huitcontrats. Le sujet “consommateurs” peut donc explorer le marché par lui même en dévoilant un à un les "menus" de contrat de chaque assureur et en demandant éventuellement unconseil à un agent général, au sein même d’un assureur. Il peut également explorer le marché
à l’aide du comparateur de prix. Dans ce cas, trois assureurs sur quatre sont présents sur lecomparateur. Ce choix est justifié par le fait que, en réalité, la présence des assureurs sur uncomparateur et le fruit d’une négociation. Ainsi, tout les assureurs n’y sont pas présents. Le
raisonnement est similaire lorsque le sujet choisit de demander un conseil à un courtier (i.e.
un sujet jouant le rôle d’intermédiaire). Une information privée est donnée au courtier sousla forme d’un classement représentant l’optimalité des contrats (nous revenons ultérieurement
sur le calcul de l’optimalité des contrats). Six contrats (correspondant à trois assureurs) sontaffichés et le courtier renvoie un classement de seulement trois de ces six contrats.
Un autre élément, indispensable à l’étude sur l’efficience et l’optimalité des choix, est basé
sur la définition de l’aversion au risque permettant le classement des contrats en termes d’optimalité. En effet, bien que les contrats n’ont pas le même taux de chargement, ou autrement dit
marge ; cette marge n’est pas le seul élément permettant de définir l’optimalité d’un contrat.
Etant donné l’aspect de risque inhérent au marché de l’assurance, est en ligne avec la théorie
de l’utilité espérée, il est important de tenir compte de l’aversion au risque des individus.
Ainsi, un contrat i est strictement optimal par rapport à un contrat j pour l’individu k si etseulement si(1−p)×U (Wk −CPi , rk )+p×U (Wk −CPi −Di , rk ) > (1−p)×U (Wk −Cj, rk )+p×U (Wk −CP j−Dj , rk ),où Wk est la richesse de l’individu k, U (.) est la fonction d’utilité des individus et rk est leparamètre d’aversion au risque pour l’individu k, utilisé dans la fonction d’utilité.
De nombreux économistes de Rothschild et Stiglitz (1976) en passant par Schlesinger(2013), pour ne citer qu’eux, se sont intéressés aux choix de couverture des consommateursd’assurance en fonction de leur aversion aux risque. Ainsi, la théorie économique de l’assurance
montre que plus un agent est averse au risque plus il est prêt à payer pour se débarrasser dece même risque, autrement dit de l’aléa inhérent à ce risque.
Chaque individu k souhaite maximiser son espérance utilité tel que
max(CP,D)∈M(1 − p) × U (Wk − CP, rk ) + p × U (Wk − CP − D, rk ),11

Introduction
où (CP, D) sont les éléments d’un seul contrat appartenant à l’ensemble M de police d’assurance disponible sur le marché. De plus, chaque assureur propose un prime commerciale telque
CP (α) = E(X) × (1 + λ) × α,
où α est la couverture (i.e. R−DR ), E(X) l’espérance du coût (i.e. p × R) et λ le facteur dechargement. Etant donné que la probabilité et le montant de la perte sont de connaissancecommune, les individus sont capables de déterminer le facteur de chargement pour chaquecontrat. Ainsi, la police d’assurance optimale peut-être définie comme(CP (αk∗ ), D(αk∗ )) = (p × R × (1 + λ) × αk∗ , (1 − αk∗ ) × R),
avec αk∗ donné pararg max(1−p)×U (Wk −p×R×(1+λ)×αk , rk )+p×U (Wk −p×R×(1+λ)×αk −R×(1−αk ), rk ).
αkAinsi, à l’aide des conditions de premier et second ordres suivantes
CP O =∂E(U (., αk , rk ))= p(−pR(1 + λ) + R)U 0 (wk+ , rk ) + (1 − p)(−pR(1 + λ))U 0 (wk− , rk ),
∂αk

et
CSO =

∂ 2 E(U (., αk , rk ))= p(−pR(1+λ)+R)2 U 00 (wk+ , rk )+(1−p)(−pR(1+λ))2 U 00 (wk− , rk ),∂αk2

où wk+ = Wk − pR(1 + λ)αk − R(1 − αk ) et wk− = Wk − pR(1 + λ)αk . il est facile de montrer
que pour un agent averse au risque (i.e. U 00 < 0) , le programme de maximisation admet une
unique solution avec αk∗ > 0. Pour un agent aimant le risque ou neutre vis à vis du risque, lecontrat optimal ne comportera aucune couverture.
Partant de ces résultats théoriques, je définis un paramètre d’aversion au risque rk pourchaque sujet k de l’expérimentation. Il est calculé pour chaque individu au début de l’expérimentation à l’aide de la méthode de liste multiple (MPL) de Holt et Laury (2002). Cela
consiste à proposer au sujet une liste de choix entre deux loteries plus ou moins risquées.
L’espérance mathématique de la loterie la plus risquée des deux augmente petit à petit par
rapport à celle la moins risquée. Le sujet va donc, à un moment basculer de la loterie la moins
risquée à la loterie la plus risquée. Le niveau de changement du sujet permet donc de mesurer son aversion au risque et donc son paramètre d’aversion, utilisé dans la fonction d’utilité
espérée.
Ainsi, ces éléments me permettent de classer les contrats proposés pour chaque individu.
Le classement des contrats est réellement important puisque cette information est donnée aux
sujets jouant les intermédiaires “physiques” d’assurance , qui sont potentiellement amenés àdonner des conseils. La construction du classement des contrats pour chaque individu en accord avec la théorie de l’utilité espérée est aussi importante afin de mesurer l’efficience deschoix des sujets tout en contrôlant l’impact de la quantité d’information M (i.e. le nombre decontrat disponible lors du choix), celle-ci largement dépendante de la stratégie d’exploration.

12

3. Informations, Intermédiaires et Choix d’Assurance
D’autre part, la mesure construite dans le premier chapitre de cette thèse m’a permis demontrer l’importance des croyances en l’honnêteté dans le choix des canaux de distribution.
Les individus croyant à l’honnêteté des intermédiaires demandent plus de conseils, en particulier lorsque le risque encouru est important. Cependant, la malhonnêteté des intermédiaires
conduit les sujets à changer de canaux, en particulier lorsque leurs attentes sont importantes,
mettant en évidence un effet de déception.
Les observations indiquent des stratégies d’exploration très différentes d’un individu àl’autre. Certains sujets économisent l’ensemble de leurs coûts de recherche en allant directement sur le comparateur de prix et en prenant le contrat le moins cher en termes de primecommerciale, ce qui ne signifie par nécessairement celui ayant la marge la plus élevée pourl’assureur. D’autres, quant à eux, préfèrent dépenser en coûts de recherche afin de bénéficier,au moment du choix, de l’information la plus complète possible. Les sujets les plus averses
au risque privilégient la seconde stratégie, en particulier lorsque que la probabilité de perteaugmente.
Concernant le choix du contrat d’assurance, seule la probabilité de perte influe sur le niveau
de couverture choisi. Cependant, les résultats mettent en évidence des effets de focalisation
et d’ancrage liés à une importante quantité d’information. L’obfuscation conduit les sujets àchoisir leur contrat uniquement sur une caractéristique limitée : la prime commerciale. Cet
effet est semblable à celui identifié par Schram et Sonnemans (2011). Cependant, les sujets
tiennent aussi compte de la différence de prix entre les contrats disponibles, autrement dit ladistance entre chaque contrat, mettant en évidence ici un second effet, celui d’ancrage.
D’autre part, l’effet de focalisation sur les prix, dû à l’obfuscation est une source d’inefficience des choix. Une autre source d’inefficience est liée à la sur-confiance portée aux intermédiaires “physiques”. La complexité des contrats et les coûts de recherche conduisent certains
sujets, confiants, à déléguer une partie de leur choix à un intermédiaire. Cependant, celui-ci
s’avère dans une majorité des cas peu honnête et fortement influencé par ses propres incitations
financières. En outre, l’expérimentation m’a permis de montrer que la décision de déviation
des intermédiaires est considérée comme une décision risquée. En effet, les intermédiaires lesplus averses au risque sont aussi les plus honnêtes.
Ainsi, je montre dans ce chapitre que bien que les intermédiaires ne proposent pas lescontrats les plus optimaux, l’obfuscation justifie leur présence sur le marché de l’assurance.
Une partie des consommateurs, assez confiants, continue de valoriser les courtiers et agentsd’assurance, en particulier lorsque le risque encouru est important. La complexité de l’offred’assurance, comportant différents niveaux de couverture, de prime et de franchises donnelieu à l’obfuscation. Celle-ci apparaît donc comme un potentiel moyen marketing et est sourced’inefficience.
L’ensemble des stratégies est donc source d’inefficience dans les choix de contrat. D’unepart l’exploration par soi-même conduit à l’obfuscation et d’autre part la délégation montre
une sur-confiance dans les intermédiaires. Ainsi, le développement de stratégies multicanauxde la part des assureurs est essentiel pour leur garantir une bonne couverture du marché. Cependant, cela implique de tenir compte des spécificités de la demande rencontrée sur chaquecanal d’offre.
13

Introduction

En effet, certains consommateurs ont suffisamment confiance en l’honnêteté des intermédiaires pour déléguer une partie de leur décision afin d’éviter l’obfuscation, tandis que d’autres
préfèrent explorer par eux même, conduisant à une quantité et qualité différente de l’information disponible. Tandis que la théorie économique classique en assurance met l’accent sur
l’importance de l’aversion au risque et de la nature du risque dans les prises de décision, je
montre dans ce chapitre que la quantité et la qualité de l’information sont tout aussi importantes dans le processus de décision.
Les biais identifiés dans ce deuxième chapitre justifient aussi la différence d’inertie desconsommateurs, autrement dit d’élasticité-prix. Tandis que certains consommateurs comparent peu ou délèguent la recherche à un intermédiaire, d’autres préfèrent explorer, conduisantà l’obfuscation et justifiant une importante compétition en prix pour les assureurs, notamment
à cause des effets d’ancrage et de focalisation.
La stratégie de distribution choisie par les assureurs va donc conditionner leur stratégied’offre de contrat, puisqu’ils ne feront pas face aux mêmes consommateurs. En effet, le choix
du canal de distribution est révélateur des caractéristiques des consommateurs, notamment enterme d’élasticité-prix. Ces résultats m’ont donc poussé à analyser, dans le troisième et dernier chapitre de cette thèse, comment se définissent les prix d’équilibre des assureurs dans un
marché concurrentiel ou chaque assureur ne bénéficie pas des mêmes avantages comparatifs,
notamment lorsque l’inertie des clients est différente d’un assureur à un autre.

4

Stratégies des Assureurs, Avantages Comparatifs et Contraintes

Le troisième et dernier chapitre de cette thèse, intitulé : “Analyses des stratégies de prix desassureurs dans un jeu non-coopératif ” (“ A game-theoretic analysis of insurers pricing strategies”), constitue une analyse détaillée des équilibres de marché émergeants d’une competition,
modélisée à l’aide de la théorie des jeux. Ainsi, après s’être intéressé aux comportements desintermédiaires et à celui des consommateurs, je propose ici une approche plus théorique afinde comprendre comment se forme et se comporte l’offre d’assurance. En effet, même dans un
marché fortement compétitif comme celui de l’assurance non-vie, l’inertie des consommateurs(différente d’un assureur à l’autre) et les contraintes de solvabilité spécifique au marché del’assurance, assurent des primes au delà du coût marginal (en l’occurence de l’espérance descoûts lorsque l’on parle d’évènement aléatoire).
Concernant l’inertie des consommateurs, celle-ci peut s’expliquer par les nombreux biais
cognitifs exposés précédemment. Tout d’abord, les coûts de recherche tendent à décourager
les consommateurs à changer d’assurance. Dans ce sens, les assureurs observent un taux derésiliation décroissant en fonction de l’ancienneté du client. Ceci est particulièrement vrai en
France, où le consommateur bénéficie d’une reconduction tacite. D’autre part, cette inertie
n’est pas la même pour tous les assureurs. En effet, comme vu précédemment, le choix du canal de distribution joue un rôle déterminant sur le niveau d’élasticité-prix des consommateursciblés. Les agrégateurs de prix ont tendance à renforcer la compétition. Les assureurs distribuant donc leur offre d’assurance sur ce canal de distribution feront face à des consommateurs
14

4. Stratégies des Assureurs, Avantages Comparatifs et Contraintes
plus élastiques que la moyenne. Au contraire, les assureurs distribuant leurs produits via unréseau d’agents généraux ou de courtiers pourront bénéficier de cette intermédiation “physique” pour proposer des primes plus élevées, s’appuyant sur la confiance que peut apporter lesconsommateurs aux intermédiaires.
Cependant, le dénominateur commun à tout acte d’achat reste le niveau de prix. Il paraît
en effet naturel que l’assuré soit tenté de résilier son contrat d’assurance s’il peut trouver
moins cher chez un autre assureur. C’est donc pour cette raison que le modèle de compétition proposé dans ce chapitre se concentre sur le niveau de prix proposé par chaque assureur.
Le prix est donc la variable de décision de chaque assureur. Bien que le comportement duconsommateur ne s’explique pas seulement par le niveau de prix, la detection d’un effet defocalisation et d’ancrage sur les prix me rassure quant à la pertinence de l’utiliser comme lepremier déterminant du choix de l’assureur. Toutefois, afin de prendre en compte les autreséléments et biais cognitifs expliquant également la demande d’assurance, le jeu proposé intègre des paramètres de sensibilité de prix. Ces paramètres sont différents d’un assureur àl’autre et permettent d’analyser leur impact sur les decisions des assureurs ainsi que la répartition des consommateurs sur le marché, autrement dit les parts de marché de chaque assureur.
De ce fait, la demande d’assurance est intégrée à l’aide d’un modèle stochastique défini à
l’aide du modèle logistique multinomial suivant :
1
if j = k,P 1+ efj (xj ,xl )l6=j
pj→k (x) =.
efj (xj ,xk ) 1+ P efj (xj ,xl ) if j 6= kl6=jpj→k (x) correspond à la probabilité qu’un consommateur étant assuré chez l’assureur j, changed’assurance au profit de l’assureur k. x est le vecteur des prix proposés par l’ensemble desassureurs sur le marché. La sensibilité-prix des consommateurs est modélisée à l’aide de lafonction fj définie parxf¯j (x, y) = µj + αj ou f˜j (x, y) = µ̃j + α̃j (x − y).
yDans la littérature économique, pj→k est considéré comme un modèle de choix discret. Ici,
nous considérons la probabilité de survenance de l’évènement, celle-ci comprise entre 0 et 1.
McFadden (1981) ou encore Anderson et al. (1989) proposent un modèle de choix logistique ou
probit multinomial. Tout comme dans Dutang et al. (2013), je considère ici uniquement le cas
logistique. Une fonction lien probit n’améliore pas nécessairement la modélisation des choix
malgré sa complexité additionnelle. De plus, l’objectif central de ce chapitre est de comprendre
quels sont les effets de la sensibilité des consommateurs sur les prix d’équilibres. C’est donc laraison pour laquelle, j’intègre également deux types de fonctions de sensibilité, l’une en ratio
et l’autre en différence.
Il faut noter tout de même que, bien que la prime d’assurance soit calculée en fonction desconditions de marché, elle dépend tout autant des caractéristiques propres à l’assuré. Ici, etpour raison de simplicité nous considérons tous les assurés identiques. La probabilité de survenance d’un sinistre ainsi que le montant de ce sinistre est donc la même pour chaque assuré15

Introduction(je présente plus loin dans cette section le modèle de perte considéré). De plus, la sensibilité
au prix est également la même pour chaque assuré et dépend uniquement de l’assureur chezlequel il se trouve. Ainsi, cela permet une analyse plus global des comportements du marché
sachant les différences d’inertie entre les assureurs.
Bien entendu, il est aussi intéressant d’inclure dans l’analyse d’équilibre les deux éléments
présentés ci-dessus. C’est ainsi que Albrecher et Daily-Amir (2017) reprennent le modèle deDutang et al. (2013), en y incluant différents types de consommateurs ainsi que de l’asymétrie
d’information. Cela permet de prendre en compte les problèmes de selection adverse rencontrés
par les assureurs sur un marché où les informations ne sont pas de connaissances communes.
Leurs résultats montrent que les assurés de “bon” type (i.e. ayant le moins de sinistres) ont intérêt à se révéler afin de diminuer cette asymétrie d’information. En effet, la théorie économiquemontre que l’asymétrie d’information a pour conséquence d’augmenter les prix d’équilibres(Rothschild et Stiglitz (1976)). D’un autre coté, Barsotti et al. (2016) considèrent quant à
eux une éventuelle contagion des résiliations à travers le temps. Leur modélisation permet deprendre en compte d’autres effets comme la mise en vigueur d’une nouvelle loi ou encore laperte de réputation d’un assureur amplifiée par un phénomène de bouche à oreille par exemple.
Ainsi, la modélisation stochastique de la demande considérée dans le chapitre trois de cettethèse me permet d’intégrer sans trop de complexité les différences de sensibilité aux prix desconsommateurs d’un assureur à un autre. L’objectif est de se concentrer sur l’offre d’assurance
en tenant compte des éléments essentiels, non négligeables et spécifiques au marché assurantiel.
C’est ainsi que j’en viens au second élément caractéristique du marché de l’assurance : l’aléa
sur les coûts, conduisant à une forte régulation sous forme de contrainte de solvabilité. En effet,
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 contre-partie decette assurance, l’assuré paie une prime d’assurance au début de la période de couverture, parexemple un an en assurance automobile. L’assureur s’engage donc, en cas de survenance d’une
perte, à indemniser l’assuré. Cependant, au moment de la souscription, l’assureur ne connait
pas le montant total réel qu’il aura à rembourser : c’est ce que l’on appelle communément lecycle de production inversé.
Par conséquent, chaque assureur se doit de modéliser les éventuels sinistres qu’il aura à
rembourser afin d’ajuster au mieux la prime d’assurance. Cette modélisation donne lieu à laprime technique. Elle est égale au montant moyen espéré des coûts lié aux sinistres, chargéeéventuellement des coûts de gestion. Cette modélisation est donc essentielle pour chaque assureur afin de se garantir des profits et d’être dans la capacité de répondre à ses engagements
vis-à-vis de ses assurés. Elle se base sur les observations passées et peut également intégrer desdonnées externes (données météo, données de géolocalisation,...). L’usage des données externes
est d’ailleurs de plus en plus utilisé afin d’affiner la mesure du risque. En effet, grâce aux nouvelles technologies “big data”, de nombreuses et nouvelles données sont facilement accessibles.
La méthode la plus communément admise pour la modélisation de cette prime techniqueest appelée la modélisation fréquence-sévérité (Bowers et al. (1997)). Celle-ci est utilisée dansle chapitre 3 de cette thèse dans le but de tenir compte de l’aspect stochastique des éventuelles
pertes encourues par les assureurs.
16

4. Stratégies des Assureurs, Avantages Comparatifs et Contraintes

Cette méthode consiste tout d’abord à modéliser le nombre de sinistres espérés pour chaqueassuré i au temps t. Dans ce chapitre et comme mentionné précédemment je ne considère pasde sélection adverse. Autrement dit, le nombre de sinistres moyen par assuré est une variable
aléatoire indépendante et identiquement distribuée. Cette variable, Mi,t , suit une loi de Poisson
Mi,t ∼ P(λ) ou bien une loi binomiale-négative Mi,t ∼ N B(r, p). Elle représente la fréquence
des sinistres.
D’un autre côté, l’assureur doit aussi estimer le coût moyen de chaque sinistre. C’est ce
que l’on appelle la sévérité. Je considère également le coût moyen des sinistres indépendant et
identiquement distribué entre les assurés et indépendant de la fréquence de sinistres.
Ainsi, Zi,l,t , le montant du sinistre l de l’assuré i à la période t suit une loi Gamma
i.i.d.

i.i.d.

Zi,l,t ∼ G(µ1 , σ1 ) ou une loi lognormal Zi,l,t ∼ LN (µ2 , σ22 ).
Les hypothèses i.i.d me permettent de simplifier le processus de simulation. En effet, lemontant total de sinistres pour l’assureur j à la période t, Sj,t , est égal à :Nj,t (xt )Sj,t (xt ) =

X

Mi,t

Yi,t , où Yi,t =i=1

X

Zi,l,t ,l=1

et Nj,t (xt ) est la taille du portefeuille de l’assureur j à la période t étant donnée le vecteur deprix xt , résultant des stratégies prix des assureurs.
C’est donc dans ce contexte d’aléa sur les coûts que les assureurs ont à définir au mieux lesprix proposés sur le marché. Pour maximiser son profit, l’assureur va donc essayer d’estimer
au mieux la sinistralité tout en tenant compte de l’élasticité-prix des consommateurs afin des’assurer un volume suffisamment important. Au delà du montant total de primes encaissé,l’augmentation du volume de business, grâce à un prix plus faible que ses concurrent, permet à un assureur de diminuer la volatilité de ses pertes, et donc de réduire le niveau de lacontrainte. C’est la loi des grands nombres et le principe de mutualisation inhérent au marchéde l’assurance. En effet, à l’aide la théorie des ordres stochastiques, je démontre dans ce chapitre qu’il est possible d’ordonner stochastiquement la distribution des pertes encourues parles assureurs, sachant leurs stratégies de prix.
Pour rappel, le but des notions d’ordres stochastiques est de comparer des variables aléatoires. Les ordres se regroupent en deux grandes classes : i) la dominance stochastique d’ordre
1, permettant notamment d’ordonner l’espérance et, ii) la dominance stochastique d’ordre 2(dont fait partie l’ordre convexe ≤cx utilisé dans les résultats de cette thèse) permettant en
particulier d’ordonner la variance. L’ordre convexe a la particularité d’ordonner les variances
des lois ayant une espérance égale. Ainsi, on définit la relation de préférence suivante F ≤cx G
lorsque pour toute fonction convexe u : R 7→ R, on a
Z +∞
Z +∞
u(x)dF (x) ≤u(x)dG(x).
00Notons X et Y deux variables aléatoires ayant les fonctions de répartition F et G. Si X ≤cx Y
17

Introduction
alors cela implique E(X) = E(Y ), V ar(X) ≤ V ar(Y ) et E((X − a)+ ) ≤ E((Y − a)+ ).
Ce dernier point est d’autant plus important puisque la régulation en assurance est trèscontraignante. En effet, pour garantir la solvabilité des assureurs et que ceux-ci soient bien
en mesure de rembourser tout sinistre, même en cas de chocs, le régulateur impose un capital
minimum. Ce capital est fonction de la volatilité des risques encourus. C’est la réglementation
Solvency II qui définit ce montant en capital appelé le SCR : Solvency Capital Requierement.
Cette contrainte est donc aussi impactante sur les équilibres de marché. En effet, un assureur
subissant un choc sur les pertes va devoir diminuer son capital. Bien qu’il souhaite être compétitif et proposer une prime relativement basse afin de se garantir un volume de business,
il sera contraint de proposer une prime plus haute que souhaité afin de garantir le niveau decapital demandé par le régulateur.
Dans le troisième chapitre de cette thèse, j’inclus donc dans mon analyse d’équilibre cetteforte contrainte qui prend la forme suivante :gj,t (xj,t ) =Kj,t−1 + nj,t−1 (xj,t − πj,t )− 1,
SCR(nj,t−1 )où Kj,t−1 est le capital de l’assureur j en début de période, πj,t est coût moyen de chaquepolice, intégrant à la fois la sinistralité mais aussi les coûts de gestion. nj,t−1 (xj,t − πj,t ) correspond donc au futur capital estimé par l’assureur j étant donné sa stratégie de prix xj,t et√
SCR(nj,t−1 ) = k995 σ(Y ) nj,t−1 et correspond au montant nécéssaire dont doit disposer unassureur pour faire face à un choc bicentenaire. Autrement dit, c’est une approximation duquantile à 99,5% de la distribution du montant total des sinistres.
Notons tout de même que la contrainte de solvabilité est évaluée en nj,t−1 ce qui correspond au nombre de contrat déjà en portefeuille au début de la période. Cette simplification
permet de s’affranchir des problèmes de concavité lors de la définition de la fonction objectif
maximisée par chaque assureur j. D’autre part, les modèles internes des assureurs permettant
de répondre à la contrainte de solvabilité sont calculés en début d’année et donc sur les polices
déjà présentes en portefeuille.
Les éléments détaillés ci-dessus sont donc partie intégrante du modèle de théorie des jeux
présenté dans le chapitre trois. D’une part, je tiens compte de l’inertie plus ou moins importante des consommateurs face aux prix proposés par chaque assureur. Cette différence desensibilité peut constituer un avantage comparatif considérable pour les assureurs. Elle est
donc déterminante dans la définition du niveau d’équilibre et des parts de marché des assureurs sur le court-terme comme sur le long-terme. D’autre part, l’aléa sur les coûts lié aucycle de production inversé et la régulation en résultant ne peuvent pas être ignoré dans lamodélisation d’un marché assurantiel. De plus, chaque assureur se fait concurrence en tenant
compte de la stratégie des autres assureurs. Ainsi, chaque assureur a également conscience quesa propre stratégie va influencer celle des concurrents. C’est la raison pour laquelle la théorie
des jeux est pertinente pour modéliser des prix d’équilibre d’un marché concurrentiel.
Elle se définit comme 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
18

4. Stratégies des Assureurs, Avantages Comparatifs et Contraintes
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.
Elle 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 lesbases 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. Il a d’ailleurs reçu le prix
Nobel d’économie en 1994 pour ces travaux.
C’est sur la base des travaux de Nash que je determine dans ce chapitre les équilibres demarché. Les études de la compétition, à l’aide de la théorie des jeux, s’intéressent généralement à deux principaux types d’équilibres : l’équilibre de Nash et l’équilibre de Stackelberg.
L’équilibre de Nash suppose que les joueurs prennent leurs décisions simultanément tandis quel’équilibre de Stackelberg suppose que les actions sont prises séquentiellement (voir Fudenberg
et Tirole (1991), Osborne et Rubinstein (2006) pour une description plus complète).
Dans ce chapitre, je considère uniquement l’équilibre de Nash. En effet, sur le marché del’assurance non-vie, en particulier pour le marché de masse, il est difficile de considérer qu’unassureur prend sa décision en premier. Les stratégies de prix sont généralement définies au début de l’année en fonction des observations de l’année passée telles que les pertes, les parts demarchés, les caractéristiques financières (inflation, valeur temporelle,...) ainsi que des attentes
des actionnaires pour l’année suivante.
L’équilibre de Nash x? = (x?1 , . . . , x?J ) est défini tel que, dans un jeu avec J joueurs, une
fonction de gain Oj et une contrainte gj , tout j = 1, . . . , J résout le problème suivant :max Oj (xj , x?−j ) s.t. gj (xj ) ≥ 0,xj

où xj est l’action du joueur j et x−j les actions des autres joueurs. De ce fait, quand le marché
atteint un équilibre de Nash, aucun joueur, ici les assureurs, ne peut augmenter son profit enmodifiant sa stratégie, étant donnée les stratégies optimales des autres joueurs.
L’utilisation de théorie des jeux dans la science actuarielle n’est pas nouvelle. Les premiers
travaux remontent à Borch (1962, 1974), Bühlmann (1980, 1984), et Lemaire (1984, 1991),
qui ont appliqué des modèles de jeux coopératifs afin de comprendre le transfert de risqueentre les assureurs et les réassureurs (voir aussi Aase (1993), Brockett et Xia (1997), Tsanakas
et Christofides (2006), Boonen (2016)). Concernant les jeux non-coopératifs, deux types demodèles ont été considérés sur le marché de l’assurance non-vie : a) l’oligople de Cournot, où
les assureurs basent leurs stratégies sur le choix d’un volume de contrat (voir Powers et Shubik(1998), Powers et al. (1998)) et b) l’oligopole de Bertrand, où les assureurs choisissent le prixdes contrats d’assurance (voir Polborn (1998), Rees et al. (1999) et Dutang et al. (2013)).
Comme mentionné précédemment, le chapitre trois de cette thèse est une extension dumodèle proposé par Dutang et al. (2013). Ce modèle semble le plus approprié à l’analyse de la19

Introduction
sensibilité-prix et des contraintes de solvabilité sur les prix d’équilibres. Il inclut un modèle derésiliation, un modèle de perte ainsi que des contraintes de solvabilité. De plus, je m’intéresseégalement à des trajectoires de long-terme à l’aide de simulation numérique. Afin de modéliser ces trajectoires de long-terme, je répète le jeu d’une période et réajuste les paramètres audébut de chaque période. En effet, bien que ce chapitre n’a pas pour objectif d’analyser lescycles de marché en assurance, de nombreuses recherches ont démontré la présence d’un tel
cycle (Rantala (1988), Malinovskii (2010), Emms (2012), Boonen et al. (2018)).
Concernant l’analyse théorique temporelle présentée dans ce chapitre, j’ai utilisé la théorie
des chaînes de Markov afin de définir les matrices de transitions des assurés d’un assureurà un autre. De plus, dans le cadre d’un marché fortement régulé, les chaînes de Markov me
permettent de définir des mesures explicites de convergence. Cela permet d’analyser les probabilités de distribution des assurés au sein d’un marché, autrement dit les parts de marchédes assureurs, et les conséquences des prix et de la sensibilité au prix sur cette distribution.
En effet, pour toute période t le jeu présenté dans ce chapitre ne dépend que de l’information disponible au début de la période. Pour rappel, une chaîne de Markov est un processus
stochastique où l’information utile pour la prédiction du futur est entièrement contenue dansl’état présent du processus et n’est pas dépendante des états antérieurs.
Autrement dit, un processus aléatoire (Xn )n à temps discret est une chaine de Markov
homogène si pour tout n ∈ N, pour tout i0 , . . . , in−2 , i, j, ∈ E, on a :P (Xn = j|Xn−1 = i, Xn−2 = in−2 , . . . , X0 = i0 ) = P (Xn = j|Xn−1 = i) = pij .
La probabilité pij représente la probabilité de transition du processus vers l’état j sachantqu’il est dans l’état i.
Ainsi, la théorie des chaînes de Markov permet de calculer relativement simplement les
probabilités de transition des assurés d’un assureur à un autre (Figure 1a). De plus, sur unmarché fortement régulé, où les prix sont définis par le régulateur, la chaîne de Markov, illustrant les transitions des assurés, devient homogène dans le temps. Dans ce cas, la théorie
Markovienne permet également de définir la mesure invariante, autrement dit, pour un assuré,
sa probabilité d’appartenir à chaque assureur (Figure 1b).
La mesure invariante µ est définie telle que ∀n , on note µn = (P (Xn = x))x∈E la loi deXn , où (Xn )n≥0 est une chaîne de Markov homogène. On a :µn+1 = µn P.
Ainsi si Xn admet une limite quand n est grand, alors µn → µ et µ vérifie :µ = µP.
Si de plusPx∈E µx = 1, alors µ est une mesure de probabilité invariante.

Je considère, par exemple, dans la Figure 1, un marché fortement régulé (i.e. la chaîne deMarkov est homogène) de trois assureurs où l’Assureur 1 bénéficie d’un paramètre plus faible20

4. Stratégies des Assureurs, Avantages Comparatifs et Contraintes(a) Matrice de transition d’un assuré(b) Mesure invariante

Figure 1 – Chaine de Markov d’un assuréde sensibilité aux prix par rapport à ses concurrents. En considérant le modèle de résiliationprésenté précédemment, un consommateur présent chez l’Assureur 1 a une probabilité de 90%
de rester chez ce même assureur, tandis qu’il a une probabilité de 5% d’aller à la période
suivante chez l’Assureur 2 ou 3 (Figure 1a). Je définis également la mesure invariante. Ainsi,
sur le long-terme un assuré a donc une probabilité de 40% d’être chez l’Assureur 1 (Figure 1b).
La définition de mesures explicites permet de mieux comprendre comment les assureurs
définissent leur prix en fonction des avantages comparatifs qu’ils peuvent avoir. D’une part,certains assureurs bénéficient d’une plus grande inertie de leurs consommateurs, pouvant s’expliquer par leur choix de distribution ou encore leur réputation. Ils ont donc la capacité deproposer des prix plus élevés tout en conservant, sur le long-terme, une place de leader. D’autrepart, et étant données les contraintes de solvabilité imposées par les régulateurs, les assureurs
ayant un stock de capital plus important, seront moins impactés par la contrainte, leur permettant ainsi d’être plus compétitif. Par exemple, les assureurs historiques sur un marché ont
tendance à avoir un niveau de capital bien supérieur au SCR demandé. Ce stock de capital,
accumulé au fils des années, leur permet de rester compétitif en cas de choc sur la sinistralité. D’autres en revanche seront contraints d’augmenter leur prix au détriment du volume,
essentiel pour la mutualisation des coûts mais aussi pour s’assurer un profit de long-terme.

21Introduction

22

Contributions et principaux résultats
Cette thèse est composée de trois chapitres indépendants et a pour objectif d’analyser lescomportements des différents acteurs du marché de l’assurance non-vie. Chaque chapitre est
tiré d’articles, chacun abordant une composante du marché : la demande, l’offre et l’intermédiation.
Les deux premiers chapitres reposent sur une approche expérimentale. J’ai construit et développé une interface web à l’aide des languages HTML et JavaScript. Concernant la collectedes données via mon interface, l’architecture a été développée en Java et PostgreSQL. J’aiégalement organisé et encadré les sessions expérimentales en laboratoire dans les locaux del’ISFA (l’Institut de Science Fiancière et d’Assurances), à Lyon. Les données collectées ont été
analysées à l’aide du logiciel R. Pour une totale transparence, ces données ainsi que le code R
sont disponibles sur demande.
Le dernier chapitre de cette thèse inclut une approche plus théorique, reposant sur les outils
mathématiques tel que l’existence d’équilibres en théorie des jeux, les principes d’optimisation
ainsi que les processus Markovien. J’ai aussi réalisé des simulations numériques, également àl’aide du logiciel R et de paquets R spécifiques, developpés avec Christophe Dutang.
J’ai réalisé cette thèse lorsque je travaillais chez AXA Global France, une compagnie d’assurance. Ainsi, certaines hypothèses et choix de modélisation ont été pris en adéquation avec
mon expérience de terrain et mes recherches.

Chapitre 1 : Sommes-nous plus honnêtes que l’on croit ?
Ce chapitre a pour objectif d’analyser les comportements d’honnêteté ainsi que les croyances
en l’honnêteté des autres. J’ai construit une nouvelle expérimentation dans le but de mesurer
les deux. Cette expérimentation permet de comparer l’adequation entre les croyances et l’honnêteté réellement observée. J’ai inclus différents environnements, plus ou moins avantageux,
dans le but d’analyser les distortions de comportement en fonction des incitations financières.
J’ai pour cela créé une nouvelle métrique, utilisée dans le Chapitre 2, afin de controler l’effet
des croyances sur les décisions de délégation ainsi que l’impact du niveau global d’honnêteté
sur la pertinence des conseils des intermédiaires d’assurance.
Globalement, j’observe que nous sommes moins honnêtes que ce que l’on croit. Cependant, la relation entre l’honnêteté et les croyances en l’honnêteté est fortement dépendante
des incitations financières et du sentiment d’injustice. Quand l’environnement devient moinsfavorable, les sujets sont moins honnêtes. De plus, ce comportement est anticipé par les autres,23

Contributions et principaux résultats
en particulier lorsqu’ils sont averses aux risques. J’observe également que les individus surestiment significativement l’honnêteté lorsque les conditions sont favorables. Cependant, lorsquecelles-ci se détériorent (i.e. augmentation des incitations financières à être malhonnête), lessujets donnent une meilleure estimation du niveau d’honnêteté.

Chapitre 2 : Obfuscation et honnêteté : une étude expérimentale
de la demande d’assurance sur un marché sous intermédiation.
Dans ce chapitre, je construis une expérimentation reflétant les différents canaux de distribution présents sur le marché de l’assurance non-vie. Le but est de révéler les déterminants
du processus d’achat d’assurance, de la recherche d’information au choix du contrat. J’analysed’abord les stratégies d’exploration des consommateurs, qui peuvent, soit déléguer une partde leur décision à un agent général ou un courtier, soit rechercher par eux-même au risquede faire face à de l’obfuscation. J’étudie ensuite l’impact de la qualité et la quantité d’information sur le choix final du contrat d’assurance. J’explique également, dans quelles mesures,les stratégies de recherche peuvent conduire à des choix sous optimaux, selon la théorie del’utilité espérée. Pour finir, j’analyse le comportement des intermédiaires qui sont amenés àconseiller les consommateurs, étant données leurs propres incitations financières.
Tout d’abord, j’observe que les croyances en l’honnêteté (mesurées à l’aide de l’expérimentation présentée dans le Chapitre 1) sont déterminantes dans les décisions de délégation.
Cependant, les sujets réalisant que les intermédiaires suivent principalement leurs propres incitations changent définitivement de stratégie d’exploration : c’est l’effet de déception. Dans
un second temps, je démontre un effet d’obfuscation. J’observe notamment des effets de focalisation et d’ancrage. Lorsque la quantité d’information augmente, les consommateurs basentleurs choix sur un critère limité : le prix. Les variables classiques, telles que la probabilitéde perte ou encore l’aversion aux risques, jouent dès lors, un rôle secondaire dans le choix
final du contrat. Ainsi, je montre que l’obfuscation est une source importante d’inefficiencedes choix. Une autre source d’inefficience tient au fait que les intermédiaires ne proposent pas
les contrats les plus optimaux mais les plus rémunérateurs. Puisque certains consommateurssont assez confiants envers les intermédiaires, et ceux pour éviter l’obfuscation, les comportements des intermédiaires les conduisent à choisir des contrats sous-optimaux. Pour conclure,ces résultats me permettent de mieux comprendre pourquoi le processus d’achat est une sourced’information pour les assureurs. En effet, il révèle leurs propres caractéristiques, telles que,leur élasticité-prix, leur croyance en l’honnêteté et leur niveau d’aversion aux risques.

Chapitre 3 : Analyses des stratégies de prix des assureurs dans
un jeu non-coopératif.
Dans ce chapitre, la théorie des jeux est utilisée afin de comprendre comment les assureurs
définissent leurs stratégies de prix. Je considère que chaque assureur maximise sa propre fonction objectif (i.e. le profit sous contrainte de solvabilité) tout en tenant compte que leur stratégie affecte la stratégie des autres assureurs. Le modèle considéré est un jeu non-coopératif,
intégrant un comportement stochastique de la part des consommateurs et un aléa sur lescoûts. L’objectif est de mesurer l’impact de la différence de sensibilité des consommateurs24

et des chocs sur les coûts sur les équilibres de marché. J’analyse également les trajectoiresde long-terme à l’aide des chaînes de Markov et simule numériquement les interactions desassureurs en fonctions des différents paramètres du jeu.
Je montre que la sensibilité des consommateurs aux prix a un effet majeur sur les prixd’équilibres. Un assureur bénéficiant d’un avantage comparatif sur l’inertie de ses clients a uneplus forte probabilité d’être le leader du marché sur le long-terme. De plus, je montre également
que plus la taille de portefeuille est importante, plus la volatilité des coûts est faible. Ainsi, cela
réduit le niveau de la contrainte de solvabilité. Ce dernier point est en ligne avec le premier. En
effet, sur un marché régulé tel que celui de l’assurance, un assureur subissant un choc sur sescoûts lors des périodes précédentes pourra être contraint d’augmenter ses prix au détrimentde la taille de son portefeuille. Ainsi, un assureur dont les consommateurs sont peu sensibles
aux prix, sera moins impacté par un tel choc. Il gardera donc une part de marché relativement
importante, lui garantissant des profits de long-terme. Au delà du niveau de prix, l’inertie
des consommateurs peut se justifier par les différents biais étudiés dans les chapitres 1 et 2.
Par exemple, la stratégie de distribution ou encore la réputation peut expliquer les différencesd’élasticité-prix. Ces résultats soulignent donc l’importance, pour les assureurs d’élaborer desstratégies de distribution et de communication, tout en s’assurant de bien mesurer les risquesqu’ils souscrivent.

25

Contributions et principaux résultats

26

Contributions and main results
This thesis is composed of three independent chapters and aims at analyzing behaviors ofthe different players of the non-life insurance market. Each chapter is based on a article andaddresses specificities of each market component: the supply, the demand and the intermediation.
The two first chapters are based on an experimental approach. I designed experiments anddeveloped a web interface with HTML and JavaScript, the backend with Java and PostgreSQL
as the database. I also organized and monitored experimental sessions in laboratory at thepremises of ISFA (Graduate School of Actuariat Studies) in Lyon. The collected data havebeen analyzed with the R software. For full transparency, I make data and R code available
on request.
The third and last chapter of this thesis includes a more theoretical approach using mathematical tools such as equilibrium existence properties in game theory, optimization principles
and Markov chains. I also made some numerical simulations, using the R software and packages that I developed with Christophe Dutang.
I realized this thesis while I was working at AXA Global France, an insurance compagny.
Hence, some hypothesis and choices modelling have been taken according to my field experience.

Chapter 1: Are we more honest than others think we are?
This chapter aims at analyzing honesty behaviors and beliefs about others honesty. I build
an original experiment to measure both. This experiment permits me to compare adequacy
between the beliefs about honesty and the effective honesty observed. I also include different environments, more or less lucky, in order to analyze behavioral distortion according tofinancial incentives. I create a new metrics that I use in Chapter 2 to control honesty beliefs
impact on delegation decisions and global honesty effect on advice decisions of insurance intermediaries.
I find that, globally, we are less honest than others think we are. However, the relationshipbetween honesty and beliefs is highly dependent on financial incentives and unfair feeling.
When environment becomes less favorable, subjects are less honest. In addition, this behavior
is expected by the others subjects, in particular those who are risk-averse. I also find that whenconditions are favorable, people significantly overestimate honesty. However, when conditions
27

Contributions and main results
become less favorable (i.e. increase of incentives to be dishonest), subjects give a better
estimate of others honesty level.

Chapter 2: Obfuscation and honesty, experimental evidence on
insurance demand with multiple distribution channels.
In this chapter, I design an experiment reflecting the different distribution channels present
in the non-life insurance market. The goal is to determine what the determinants of insurance
purchasing process are, from the research of information to the choice of contract. Thus, I
first analyze the exploration strategies of consumers, who can either delegate a part of theirdecisions by requesting advice to a broker or a tied-agent, or research by themselves and facepotential obfuscation. Second, I analyze the impact of the quality and quantity of informationon the final insurance contract choices. I also explain to which extent each research strategy
could lead to inefficient choices according to the expected utility theory. Last, I analyze intermediaries behaviors, who should advice consumers under their own financial incentives.
At first, I find that the beliefs in others honesty (elicited according to the experiment
presented in Chapter 1) is a determinant of delegation decision. However, subjects who realize that intermediaries follow only their own incentives change definitively their exploration
strategies: this is the deception effect. Second, I also demonstrate obfuscation effect. Thisobfuscation effect leads to focal point and anchoring effects. Indeed, when the quantity of information increases, consumers base their decisions on a single criterium: the price. Classicalvariables, such as the probability of loss or the risk aversion, play therefore a minor role on
the final contract decision. Hence, I demonstrate that obfuscation is an important source of
inefficiency of choices. Another source of inefficiency emerges from the intermediairies behaviors, who recommend non-optimal but more remunerated contracts. Since some consumers
are confident enough and prefer to avoid obfusctation, intermediaries behaviors conduct tothe subscription of non-optimal contracts. To conclude, all these results allow me to better
understand why purchasing process decision is a source of information for the insurers. Indeed,
consumers exploration choice reveals their own characteristics such as price-elasticity, honesty
beliefs and risk-aversion level.

Chapter 3: A game-theoretic analysis of insurers pricing strategies.
In this chapter, I use a game-theoretic approach in order to understand how insurers build
their pricing strategies. I consider that each insurer maximizes her own objective function(i.e. the profit under solvency constraint) while taking into account that her strategy willaffect others insurers’ strategies. The game-theoretic model considered is a non-cooperative
game which includes stochastic consumers behaviors and stochastic losses. The objective isto analyze the impact of price-sensitivity of consumers, and the impact of loss shocks on market prices equilibrium. I also analyze long-run patterns with Markov chains and numericallysimulate market interactions under different settings.

28

I demonstrate that price-sensitivity of consumers has a major effect on prices equilibrium.
An insurer has a higher probability to be a leader in the long-run if she benefits from acomparative advantage on consumers inertia. In addition, I show that a larger portfolio size
implies a lower volatility of losses. Therefore, it reduces the level solvency constraint. This
result points in the same direction as the previous one. Indeed, in a regulated market, insurers
suffering from sever losses on the previous period could be obliged to increase their market
premiums to the detriment of portfolio size. Therefore, insurers benefiting from important
inertia of their consumers are less impacted by a shock and are able to keep a relatively
important size, which guarantees long-run profit. In addition to price level, consumers inertia
could also be explained by the different biases studied in Chapters 1 and 2. For instance, thedistribution strategies as well as the reputation can explain the difference in consumers priceelasticity. These results underline the importance for insurers to take care of their distribution
and communication strategies while insuring an appropriate estimation of their underwriting
risks.

29

Contributions and main results

30

Contributions en économie
comportementale

31Chapter 1

Are we more honest than others think
we are?
—Sommes-nous plus honnêtes que l’on
croit?
This chapter is based on an article co-written with Jean-Louis Rullière.

33

Chapter 1. Are we more honest than others think we are?

Abstract
While the laws are justified on the basis of the efficiency they provide to society, policy
makers and researchers focus on the reasons why people violate the law. Crimes and violations
induce directly costs. But there is another indirect costs that is generally ignored: the fact thata person can violate the law (whether it does or not) can reduce trust in one’s honesty. Thus,
even if the economic agent is honest and respects the law, this loss of confidence, which could
be unfounded, is also a source of inefficiency. We introduce in an experiment, a normative
rule of “decision” in order to elicit both honesty and beliefs about honesty from subjects in thelab. There is no direct transfer of money between both part to avoid any inequality aversion
or altruism aversion. The main question remains how individuals trust in the honesty of ananonymous group. Subjects are split into two groups: those who are subject to the temptationof (unverifiable) dishonesty and those who value the dishonesty of others. We inform eachparticipant that we cannot identify defection. We find an important heterogeneity of trustin honesty through subjects. On average, subjects A suggests that participants B are morehonest than they are. Moreover, we identify distortion of effective honesty and beliefs aboutother honesty when the environment of players A is unfavorable.
Keywords: behavioral economics, trust measurement, honesty, experiment.

Résumé
Tandis que les lois sont un vecteur d’efficience pour la société, les responsables politiques
et chercheurs se concentrent principalement sur les raisons pour lesquelles les individus transgressent les règles. Il est vrai que les crimes et délits impliquent des coûts directs pour la société,
cependant, il existe un autre coût indirect généralement ignoré : le fait qu’un personne puissetransgresser les règles (qu’il le fasse ou non) peut réduire la confiance portée par les autres.
Ainsi, même si un agent économique est honnête et respecte la loi, la perte de confiance, mêmeinfondée, est aussi une source d’inefficience. Nous introduisons dans une expérimentation enlaboratoire une règle de décision normative afin de révéler à la fois l’honnêteté des uns et lescroyances en l’honnêteté des autres. Afin d’éviter toute aversion à l’inégalité ou effets d’altruisme, notre expérimentation ne comprend aucun transfert monétaire entre les deux parties.
La principale question reste “À quel point les individus font-ils confiance en l’honnêteté d’ungroupe d’anonyme ?”. Les sujets sont répartis en deux groupes : ceux sujets à la tentation (nonvérifiable) d’être malhonnête et ceux estimant la malhonnêteté des autres. Nous les informons
que nous ne sommes pas en mesure d’identifier les éventuelles transgressions. Les résultats nous
montrent une importante hétérogénéité vis-à-vis des croyances en l’honnêteté. En moyenne,
les individus suggèrent une plus grande honneteté que celle observée. De plus, nous identifions
une importante distorsion des comportements quand l’environnement des joueurs devient peu
favorable.
Mots-clés: économie comportementale, mesure d’honnêteté, croyances, expérimentation.

34

1.1. Introduction

1.1

Introduction

Honesty and beliefs about others honesty are central in many economic and social interactions. Above the fact that dishonesty leads to direct cost, inadequacy of honesty on the
one hand and beliefs in others honesty on the other hand, is also a source of indirect costand inefficiency. For instance, under-estimating people’s honesty leads to useless and costly
control of population, or self-privation of value added services. Why hire ticket inspector ifeveryone is traveling with valid ticket? Why do not rent our houses if they remains intact? At
contrary, over-estimating people’s honesty can lead to the creation of non-efficient exchange,
as investment based on dishonest information. It can also support existence of non-efficient
actors such insurance broker advising for most profitable instead of most appropriate contracts.
An important number of researches are devoted to the understanding of social backgroundeffects on honesty and confidence. Berg et al. (1995) were the first to shed on light importanceof social background into trust reciprocity in particular when subjects already get social history. Along the same lines, Glaeser et al. (2000) show how common social characteristics raise
trust and trustworthiness among individuals. More recently, Ermisch et al. (2009) analyze
population characteristics impact on trust and trustworthiness by showing, for instance, that
individual with “comfortable” situation are more likely to trust. In another study, Hugh-Jones(2016) show in what extent honesty and beliefs about others honesty is significantly different
across countries.
Previous results demonstrate the lively interest of the scientific community in understanding relationship between honesty behaviors and beliefs about others honesty. Above the analysis of social interaction, background and individuals characteristics effects, a large part ofstudies are also dedicate to the analysis of incentive and environment effects. For instance,
Mazar et al. (2008), Fischbacher and Follmi-Heusi (2013), Abeler et al. (2016) find that theincrease of financial incentive does not significantly impact honesty level. However, Kajackaite
and Gneezy (2017) show that this result does not hold when we considered “deception game”,
where lying possibility is explicitly exposed to subjects. The fact that lying is an explicit ruleof the game may conduct subjects to put more importance of the possibility to be caught as
a liar, and can lead subjects to deeper analyze cost-benefits trade off. It can thus explainincrease in cheating when an incentive rises. To support this result, Yaniv and Siniver (2016)
show that in a safe environment people would cheat in a large extent. Hence, previous results
seem to indicate that people honesty is mainly driven by the consequences of their acts on
their social ethics. In addition, Houser et al. (2012) as well as Galeotti et al. (2017) highlightthe importance of perception of a given environment on honesty behaviors. Indeed, subjects
tend to be less honest in unfair condition. Unequal situations reduce the social ethics consideration of subject in their honesty decision-making.
The objective of this paper is to deeper understand relationship between effective honesty and people beliefs about others honesty, depending on different conditions more or lessfavorable. We do not address concern about the relationship between social interaction of agroup and degree of honesty in relationship, but we rather focus our analyses on honesty inanonymous subjects. Indeed, as previously mentioned, honesty behaviors (i.e. honesty actions
and expectations) depend on feeling to belong to a social group. However, many economic
relationships and decisions, based on honesty, take place in anonymous environment without
35

Chapter 1. Are we more honest than others think we are?
social knowledge or history between interested parts.
We thus design an original experiment in laboratory in order to elicit honesty and beliefs
about others honesty, including different level of favorable context. While standard trust gameof Berg et al. (1995) is mainly used for studying trust relationship, the lack of normative rulefor both sides does not allow us to answer our questions. Indeed, in the standard trust game,
there is no rule asking people to transfer money or to reward transfer. Hence, game condition
implies subjective rule, leading to inequality aversion and altruism effect. However, the objective here, is to reveal honesty and beliefs in others honesty in an exogenous context (i.e. moreor less favorable). In other words, by implicitly adding a normative rule to the game we areable to define a similar measure for those who are subject to the temptation of (unverifiable)
dishonesty and those who value the dishonesty of others.
Our experiment also differs from others honesty and cheating game in two ways. First, we
inspire our game from repeated coin flip experiment (Cohn et al. (2015)), but we introduce diversity on players’ environment. Players who are subject to the temptation of rule violation do
not face same “severity” conditions. Last and contrary to others honesty games (Fischbacher
and Follmi-Heusi (2013), Hugh-Jones (2016)) we collect data at the individual level for bothsides, while ensuring anonymity of choices. This last point allows us to measure effective
deviation with respect to the rule, allowing us to introduce different dishonesty temptation at
the individual level.
We find that people are less honest than others think they are, particularly when ruleconditions are favorable. We also catch a distortion effect of honesty behaviors, that is people
honesty decreases when they face unfavorable conditions. Interestingly, this distortion effectis anticipated by subjects valuating honesty of an anonymous group. While literature results
remain mixed about the correlation between honesty beliefs and risk aversion of subjects, wefind that the average level of honesty expectation is not correlated to risk aversion level. In
lines with Corcos et al. (2012), we find that believe in others honesty is not assimilated toa risky decision. However, our results can also support the existence of correlation between
trust and risk aversion (Naef and Schupp (2009)). Indeed, we show that risk aversion of subjects in the gain domain is a determinant of honesty expectations distortion. When subjects
are risk-averse, the beliefs about others honesty decrease more significantly when conditions
become unfavorable.
We also investigate impact of subject’s characteristics (such as gender and age) on beliefs
in others honesty. While Grolleau et al. (2016) show that men are more likely to cheat, wefind that they are also more likely not to believe in others honesty. Finally, we find that theage is not a determinant of honesty beliefs level. However, oldest subjects are more likely tochange their expectation when conditions become favorable and, they tend to belong to more
extreme profile (i.e. those who do not believe at all or those who entirely believe in others
honesty).
The remainder of the paper is organized as follows. We set out the experimental design andprocedure in Section 2. Section 3 then describes the detailed results, and Section 4 concludes.
36

1.2. Experiment Design and Procedure

1.2

Experiment Design and Procedure

We design an experiment to identify honesty and beliefs about other’s honesty. This design
has four main advantages. First, dishonest behaviors cannot be detected at the individuallevel, which reduces potential demand effect. Second, even after controlling anonymity ofhonesty choices, we collect data at the individual level. Third, we introduce different favorable
condition levels to analyze responding deviation. Finally, the method is easy to implement ina laboratory. We also control for subject’s attitude towards risk according to MPL (Multiple
Price List) method of Holt and Laury (2002).

1.2.1

Elicitation of honesty and beliefs in others’ honesty

During this first experimental part, each subject is assigned to a group, either A or B.
This part involves a potential exchange of coins between a wallet and a padded envelope.
Before the session starts, a wallet is put on the table of each participant. This wallet either
contains 10 coins of 0.50e and a small card showing the result of 10 independent draws, or
nothing. The 10 draws are of a red or green ball without replacement from a bag containing
7 red balls and 7 green balls (so that there are five possible draws, corresponding to 3, 4, 5, 6
or 7 red balls).
The content of wallets defines the group of participants. If the wallet is not empty, subject
belongs to group B. We ask them to discretely apply the following rule. For each green balldraw on the small cardboard, they can collect 0.50e in the wallet and put these 0.50e in thepadded envelope. Coins put on the envelope represent their gain for this part. The remaining
euros in the wallet should correspond to 0.50e times the number of red balls. Nevertheless, we
inform them that no one in the room, including the other participants and the experimenter,are able to know if they have applied the rule or not.
To guarantee anonymity as well as to collect data according to more or less favorable situation, we also ask subject B to let into the wallet the small cardboard. Then, we scrutinize
indistinguishable wallets after putting them in the same bag at the end of this part.
If the wallet is empty, subject belongs to group A. We display on their screen the different
series distributed to subjects B. Series displayed order is random for each participant. They
have to indicate in front of each of them how much, in euros, do they think that subject B
have let in her wallet. For this part, to calculate the gain of participant A, we randomly selectone of these sequences (i.e. to avoid portfolio effect) and participants A receive 5 euros minus
their estimation error.
Our honesty measure is defined as the difference between the amount left in the wallet andthe amount which should be left according to the rule.

1.2.2

Elicitation of risk aversion

In this part, all gains are expressed in ECU (Experimental Currency Unit). ECUs areconverted at the end of the session according to the following rate: 50 ECU = 1e. Risk
37

Chapter 1. Are we more honest than others think we are?
preferences have been measured for all subjects in the gain and loss domain. Subjects haveto make two series of 10 decisions between an alternative “A” and an alternative “B”. First
10 questions concerns risk attitude towards gain. For instance, they have to choose between
receiving 50 ECU with a probability of 10% and receiving 20 ECU with a probability of 90%(alternative “A”) or receiving 85 ECU with a probability of 10% and receiving 5 ECU with aprobability of 90% (alternative “B”).
The next questions concern risk attitude towards loss. We attribute to each subject andfor each question an initial endowment. Then, they have to choose one alternative more or lessrisky. For instance, they should choose between the two following alternatives: loss 50 ECU
over their 100 ECU (i.e. initial endowment) with a probability of 10% and loss 80 ECU witha probability of 90% (alternative “A”) or loss 15 ECU over their 100 ECU with a probabilityof 10% and loss 80 ECU with a probability of 95% (alternative “B”).
In addition, to compare risk aversion in loss and gain domain we fit payoff such that the
expected payoff for each level of question in the gain and loss are equal. In other words, if thealternative “A” of the first questionnaire is the following: win 10 ECU with a probability of
10% then alternative “A “of the second one is: over 100 ECU losses 90 ECU with a probabilityof 10%.
We measure risk aversion among participants with respect to the first switch from a safe toa risky option, for both series of questions. A subject who switches at Question 5 is considered
to be more risk-averse than a subject switching at the Question 2.

1.2.3

Procedure

An HTML web interface and a server database were designed especially for this experiment. Subjects were students from the University of Lyon 1 – Claude Bernard. 217 subjects
participated in the experiment, 27 in average for each session including 5 participants of TypeB for a total of 8 different sessions.
The honesty game was played first; all subjects received identical instructions, including
comprehension questions. Then, they made their decision for this part. Afterwards, subjects
received written instructions for the risk elicitation task and made their choices. Finally, we
asked them to answer some general questions about age, gender and education degree. All
treatments were framed in a neutral manner. The sessions lasted about 35 minutes. The
average payoff was about 8 euros including a show up fee of 3e.

1.3

Results

1.3.1

Descriptive Statistics

We define here the deviation of B-type subjects receiving draw t asdevi|t = RBt − Li ,where RBt corresponds to the number of red ball of the draw t times 0.50e and Li corresponds to the observe amount left into the wallet by subject i.

38

1.3. ResultsThus, devi|t = 0 if B-type subject i is fully honest and perfectly follow the rule whiledevi|t > 0 if we observe dishonesty.
We detail in Table 1.1 some general descriptive statistics. The average deviation of B is
0.99e (std. dev. 1.26). Over the 40 B-type subjects, 19, or 47% respect the rule and 14, or35%, fully deviate. Others 18% (i.e. 7 subjects) take in average 0.71e.
Table 1.1 – Descriptive Statistics% of subjects% of subjects
who respect
who fully
the rule
deviateA-type (Expectation)
0.49e
15%
33%
B-type
0.99e
47%
35%
∗
Question number of the first switch from a safe to a risky option
Participant Type

Deviation

Others% of men

Average Age

52%
18%

72%
70%

21.1
20.6

Risk Aversion Levelin the gain (loss)
domain∗5.6 (4.9)
5.5 (4.4)

We also define beliefs in others honesty HBi,t (i.e. deviation expectation) of a A-type
subject i for draw t as
HBi,t = RBt − Ai,t
, where Ai,t corresponds to the answer of subject i (i.e. “How much do you think that B haveleft into the wallet?”) for draw t.
Thus, HBi,t = 0 when A-type subject i believes in others honesty, otherwise beliefs in
HBi,t > 0 when A-type subjects i do not trust other’s honesty.
The average deviation expectation of A is 0.49e (std. dev. 1.29). Over the 177 A subjects
27, or 15%, believe in other’s honesty and 104, or 59%, have positive value of deviation expectation. Using a non-parametric signed-rank Wilcoxon test (p − value < 2.09−9 ), we showthat subjects are significantly less honest than others think they are.

Table 1.2 – Correlation Analysis under Draw Condition
Average

Averagep-value - H0:Expected% of fully% of fully% of full% of full

Draw

Expected

Deviation

H0:Expected Deviation

Confident

HonestExpected

Deviation

Condition

Deviation

Type B< Deviation

Type A

Type B

Deviation

Type B

Type A
Favorable
Unfavorable
H0:Favorable< Unfavorable(signed ranked Wilcoxon Test)

Type A

0.14
1.01

0.73
1.32

< 0.001
0.17

36.7%
28.5%

46.5%
50.3%

15.3%
10.7%

31.6%
36.2%

< 0.001< 0.001

-

-

-

-

-

Interestingly, results seem to be highly dependent on draws’ conditions. It reminds us
Fehr et al. (1993) and more recently Galeotti et al. (2017) results, where subjects who arepaid more are more likely to reciprocate by shirking less. Indeed, in Figure 1.1, the difference
between the amount which should be left according to the rule and the amount estimated by
A-type subjects increases with unfavorable draw condition. We observe the same effect withrespect to the effective amount left by B-type subjects.

39Chapter 1. Are we more honest than others think we are?
Figure 1.1 – Average behavior w.r.t the rule

We thus classify draw condition into 2 types: favorable (i.e. amount RBt ≤ 2.5) vsunfavorable (i.e. amount RBt > 2.5). In Table 1.2, we find significant positive difference
in deviation and expected deviation under favorable with respect to unfavorable conditions.
A-type subjects anticipate the strongest deviation of B-type subjects in front of unfavorable
conditions. However, while under favorable conditions A-type subjects over estimate honesty(p − value < 0.001), this result does not hold anymore for unfavorable draw (p − value > 0.17). These results suggest that subjects are able to anticipate distortion of honesty with respect
to incentive conditions but overestimate honesty when incentives to be dishonest are lowest.

1.3.2

Econometrics Modelling Specification

1.3.2.1

Honesty Beliefs Distortion

We previously see how draw condition affect both honesty and beliefs about others honesty.
We thus propose in this section some econometric regressions in order to better understandthe distortion effects, and in what extent subjects anticipate dishonest behavior. We focus
our analysis on beliefs in others honesty by performing a linear regression.
We specify the following relationship, where Ai|t is answer of subject i for draw t:
Ai,t = f (Xi , Xt ) × RBt + i,t ,and f (Xi , Xt ) is a linear combination of individual (Xi ) and draw (Xt ) explanatory variables.
Hence, E(f (Xi , Xt )) = 1 means that subject i fully believes in others honesty. However, whenE(f (Xi , Xt )) decreases, it means that subject i has lowest expectations with respect to others
honesty.
We show in Model 1 of Table 1.3 that, in average, subjects do not believe in others honesty.
Indeed, the average level of honesty beliefs is significantly lower than 1. A believe that B aregoing to take 23% more than according to the rule.

40

1.3. Results
Table 1.3 – Ordinary Least Squares Regression
Explanatory Variable

A’s Expectation

Coefficients

Model 1Model 2Model 3Model 4Model 50.77∗∗∗

0.84∗∗∗

1.47∗∗∗

1.48∗∗∗

1.49∗∗∗

0.01

0.05

0.08

0.15

0.15

−0.02∗∗−0.01∗∗−0.01∗∗−0.01∗∗

0.00

0.00−0.21∗∗∗

0.00−0.21∗∗∗−0.20∗∗∗

Std. error

Should be left

0.08

Should be left × Risk Aversion (loss)

0.08

Should be left × Risk Aversion (gain)
Should be left × Should be left

0.02

Should be left × Age
Should be left × Gender (Ref. level: Female)

0.00

0.02

0.02

−0.01

−0.01

0.01

0.01

0.04

0.04

0.03

0.03
−0.02∗∗

Should be left × B’s Deviation

0.01

Nb. Observations
Nb. Subjects
R2
Adjusted R2

885
177
0.721
0.721

885
177
0.723
0.722

885
177
0.744
0.743

885
177
0.744
0.742

885
177
0.745
0.744

Signif. codes for p-values: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’

In Model 2 (Table 1.3), we find that risk-averse subjects in the gain domain tend to haveless honesty expectations when the draw conditions of B-type subjects are unfavorable. This
result is particularly interesting since we find in Model 6 of Table 1.4 that risk aversion in theloss and gain domain are not significant on the average level of honesty beliefs. Indeed, we
model the Average Honesty Beliefs (i.e. deviation expectation) of each subject i, defined asfollowing
AHBi = T1PT

t=1 HBi,t ,where t is the different draw distributed during a session to each B-type participants (i.e.
T = 5). Thus and in lines with Eckel and Wilson (2004), we do not find correlation between
the level of trust (in our case Average Honesty Beliefs) and risk aversion. Hence, confidence
in others honesty is not a risky decision. While risk aversion level (i.e. in the gain domain)
does not explain the base level of subjects honesty beliefs, it explains why some people change
their expectations when incentives to be dishonest are highest. Risk-averse participants reduce
more their honesty expectations when B-type participants face unfavorable conditions.
Because we are interesting on the shape of honesty beliefs given draw conditions, we add
in Model 3 (Table 1.3) the square of the amount who should be left according to the rule. As
previously seen, A’s expectations significantly depend on B draws: favorable condition reduces
rule violation expectation.
Elsewhere, we check in Model 4 of Table 1.3 and Model 6 of Table 1.4 eventual subjects’characteristics effects (i.e. Age and Gender). We want to explain both: the average levelof honesty beliefs and the distortion effect with respect to draw conditions. We do not findsignificant effect of such variables in both models and others results remain true.

41

Chapter 1. Are we more honest than others think we are?
However and because of the weak dispersion of subjects age into our sample, we decideto create a categorical variable with few values. Thus, we split our sample into three classes:
subjects being the same age as the median (i.e. 21 years old) who represent 20% of our sample
and subjects being a higher (resp. lower) age than the median, representing 41% (resp. 39%)of our sample. We find in Model 5 bis of Table 3 bis of Appendix that youngest subjects havelowest distortion beliefs effect. While this method allows us to catch information from tails
distribution of age, we find in model 6 bis of Table 4 bis (Appendix) that age does not explain
global honesty beliefs level of subjects (in lines with Model 6 of Table 1.4).

Table 1.4 – Ordinary Least Squares Regression
Explanatory
Variable
Coefficients

A’s Average
Honesty Beliefs
Model 6

Std. error

Risk Aversion (loss)
Risk Aversion (gain)
Age
Gender (Ref. level: Female)
B’s Average Deviation
Constant
Nb. Observations
Nb. Subjects
R2
Adjusted R2

−0.02
0.04

0.06
0.03

0.03
0.02

0.14
0.16

0.13
0.11

−0.04
0.66

177
177
0.02
-0.04

Signif. codes for p-values: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’Last, we add in Model 5 (Table 1.3) the effective deviation observed for each draw, inorder to better catch anticipation effects. While the rule enforced to B remains the more
important effects on expectation, we show that A-type subjects significantly anticipate B’s
deviation distortion.

1.3.2.2

Honesty Beliefs Profile

After analyzing distortion of honesty beliefs, we now focus our study on more global honesty belief profiles. In Figure 1.2, we split our subjects into three different profiles. First,
subjects who do not expect any deviation from B-type subjects, named the “Believers”. “Believers” profile represent in average 33% of our sample, and Figure 1.2 displays a decreasing
percentage when draw conditions becomes less favorable. Second, we classify subjects with nohonesty beliefs into a “Not at All” profile, they represents around 13%. We regroup all others
subjects into a last group called “Others”, representing in average 53% of subjects.

421.3. Results
Figure 1.2

We carry out a multinomial logistic regression including random effects (Table 1.5, Model
7 and 8) with the honesty beliefs group as the dependent variable (i.e. “Believers”, ”Not at
All” and “Others”). Random effects allow us to control for the relationship between errors in
our panel data (Croissant (2018)).
Table 1.5 – Multinomial Logistic Regression including Random EffectExplanatory Variable
Coefficients

Honesty Beliefs Profile
Model 7
Model 8
“Not at all”
“Others”“Not at all”
“Others”

Std. error

Should be left

−0.64∗∗

0.65∗∗∗−0.62∗∗

0.66∗∗∗

0.27

0.16

0.27
0.16∗

−0.03

Risk Aversion (gain)
Age
Gender (Ref. level: Female)
Constant (Ref. level: “Believers”)

0.17

0.09

0.06

−0.26

−0.08∗

0.18

0.04

1.90∗∗−0.75∗∗

0.59

0.25

−3.15∗∗∗−0.95∗∗∗

0.25

1.48

0.99

0.42

2.35

1.10

Nb. Observations
Nb. Subjects
R2
Adjusted R2

885
177
0.259
0.254

885
177
0.263
0.253

Signif. codes for p-values: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’

As previously seen in Figure 1.2, we show in Model 7 (Table 1.5) that the percentage of“Believers” is decreasing when conditions become unfavorable. Under stable coefficients withrespect to Model 7, we catch in Model 8 additional information on subjects honesty beliefs
profile by adding some subjects variables (i.e. risk aversion level, age and gender).
According to Models 2, 3, 4 and 5 of Table 1.3, subjects more risk-averse in the gain domain get higher probability to belong to “Not at All” profile.
Gender of subjects do not appear as a significant determinant of distortion in honesty
beliefs (Models 4 and 5 of Table 1.3). It does not explain global honesty beliefs level either(see Model 6 of Table 1.4). However, by analyzing more global profile in Model 8 (Table 1.5),
43Chapter 1. Are we more honest than others think we are?
we find that men have highest probability to belong to extreme profile (i.e. “Believers” or “Notat All” profile). Last, using the extent of coefficient of gender variable, men have significantly
lower expectation than women about others honesty.
As a reminder, we find in Model 6 (6 bis) of Table 1.4 (4 bis) that age does not explain
global level of honesty beliefs but explains expectation distortion beliefs(Model 5 bis of Table
3 bis). We find here, in Model 8 (8 bis) of Table 1.5 (Table 5 bis) that oldest subjects tend tobelong to more extreme profile. However and according to Table 1.4, results do not allow us
to determine difference in the honesty beliefs level.

1.4

Conclusion

We detail in this paper the design of a new honesty elicitation game. The objective consistson the measure of adequacy between honesty and honesty beliefs in an anonymous context.
We introduce different level more or less favorable and avoid any inequality aversion or altruism effect by defining a normative rule to determine honest behavior. These new metrics
defined by devi|t for B-type subject and HBi,t for A-type subjects is relatively quick and easy
to implement.
We distinguish the honesty concept from that of trust, which is widely explored in the
behavioral economics. We are interested not only in honesty and its measure, but especiallyin the question of beliefs about the supposed honesty of others. Economists are agree tosay that rise of economic agents honesty lead to an increase to social welfare. For instance,
Fetchenhauer and Van der Vegt (2001) argue that honesty lead to reduce transaction cost andthus stimulates economics growth.
However, this approach do not take into account adequacy between honesty and beliefs.
Indeed, even in a fully honest world, transaction costs could no be reduced if no one believes
in others honesty. Hence, the level of honesty remains important to explain sources of inefficiency of economic relationship, yet an interaction decision is also mainly determined bybeliefs about others honesty.
We globally find that people are less honest than other think they are. However, this measure should be used with caution. Indeed, we show an important increasing pattern regarding
honesty (i.e. and beliefs) and favorable condition. Peoples defect more when they are in an
unfavorable environment and others anticipate. However, in a “safety” context, expectation
are lower than effective dishonesty.
Second, we find that beliefs in others honesty is not a risky decision. Indeed, risk aversion(in the gain and loss domain) does not explain average level of honesty beliefs. However andregarding to the distortion of beliefs in unfavorable condition, we show that risk-averse subjects in the gain domain downgrade more significantly their expectations about other honesty.
Last, while individuals characteristics do not allow us to determine average beliefs aboutothers honesty, they explain more general profile. Indeed, we find that oldest subjects are
more likely to change their expectations depending on condition and belong to extreme pro44

1.4. Conclusion
file. Besides, a larger part of men than women do not believe at all in others honesty andexpect a full deviation (i.e. no remaining euros in the wallet).
The relevance of our measure has been verified in Mouminoux et al. (2018) to explaindelegation decisions in insurance markets. We find that beliefs in other honesty is a major determinant of intermediaries choices. Indeed, even under an high degree of brokers dishonesty,
because of their own financial incentives, (i.e they do not propose the most optimal contract),some consumers prefer to ask for advice. This is a good example of consequences of gap between honesty and beliefs about others honesty.
Elsewhere people seems to allow themselves to do something bad (i.e. dishonest) because
of injustice feeling. The fact that expectations are relatively closed to effective behaviors inbad conditions reinforced the idea that its cognitive bias is globally anchored in decision making. This results may have important consequences in fraud behaviors for instance. We canimagine that an insurance policyholder who never declared claims while paying their insurance
premium for many years will miss-declared her first claims.
We do not adress here issues about repetitive interactions and consequences of the observation on honesty behaviors. It could be interesting to add a dynamic part into our honesty
elicitation gain in order to better understand how individuals adjust their beliefs. We canimagine that dishonesty observation in favorable condition may have significant effect on honesty expectation in unfavorable one.
Another interesting approach, and in lines with Yaniv and Siniver (2016), could consist in
adding in our experiment potential punishment of dishonest behaviors. For instance, we can
add a probability to be detected and may expect difference in both: honesty behavior andbeliefs about honesty. It could also be interesting to analyze, under punishment condition, if
distortion of honesty resulting from eventual feelings of injustice keeps hold.

45

Chapter 1. Are we more honest than others think we are?

1.5

Appendix
Table 3 bis – Ordinary Least Squares Regression
Explanatory Variable

A’s Expectation

Coefficients

Model 5Model 5 bis

1.49∗∗∗

1.42∗∗∗

Std. error

Should be left
Should be left × Risk Aversion (gain)
Should be left × Should be left

0.15

0.09

−0.01∗∗−0.02∗∗

0.00

0.01

−0.20∗∗∗−0.20∗∗∗

0.02

0.03

−0.01

Should be left × Age

0.01

0.09∗∗

Should be left × Age<Median Agei

0.04

−0.01

Should be left × Age>Median Agei

0.04

Should be left × Gender (Ref. level: Female)
Should be left × B’s Deviation
Nb. Observations
Nb. Subjects
R2
Adjusted R2

0.04

0.04

0.03
−0.02∗∗

0.03−0.03∗∗

0.01

0.01

885
177
0.745
0.744

885
177
0.748
0.746

Signif. codes for p-values: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’iref. level: Age = Median Age

Table 4 bis – Ordinary Least Squares Regression
Explanatory
Variable
Coefficients

A’s Average
Honesty Beliefs
Model 6 bis

Std. error

Risk Aversion (loss)
Risk Aversion (gain)
Age<Median Agei
Age>Median Agei
Gender (Ref. level: Female)
B’s Average Deviation
Constant
Nb. Observations
Nb. Subjects
R2
Adjusted R2

−0.02
0.04

0.06
0.04

−0.26
0.20

0.03
0.21

0.14
0.17

0.12
0.11

−0.04
0.66

177
177
0.043
-0.003Signif. codes for p-values: 0 ‘*** ’ 0.001 ‘** ’ 0.05 ‘* ’ 0.1 ‘ ’ 1
i46

ref. level: Age = Median Age

BIBLIOGRAPHY
Table 5 bis – Multinomial Logistic Regression including Random EffectExplanatory Variable
Coefficients

Honesty Beliefs Profile
Model 8Model 8 bis“Not at all”
“Others”“Not at all”
“Others”

Std. error−0.62∗∗

0.66∗∗∗−0.62∗∗

0.66∗∗∗

0.27
0.16∗

0.17

−0.03

0.27
0.37∗∗∗

−0.05

0.01

0.07

Age<Median Agei

−0.52

−0.25

0.54

0.29

Age>Median Agei

0.72

−0.74∗∗

Should be left
Risk Aversion (gain)
Age

Gender (Ref. level: Female)
Constant (Ref. level: “Believers”)

0.09

0.06

−0.26

−0.08∗

0.18

0.04

0.54

0.29

1.35∗∗−0.75∗∗

0.25

0.54

0.25

1.48

−6.42∗∗∗

0.22

1.10

1.70

0.62

1.90∗∗−0.75∗∗

0.59

0.25
2.35

Nb. Observations
Nb. Subjects
R2
Adjusted R2

885
177
0.259
0.254

0.17

885
177
0.263
0.253

Signif. codes for p-values: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’iref. level: Age = Median Age

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48

Chapter 2

Obfuscation and Honesty:
Experimental Evidence on Insurance
Demand with Multiple Distribution
Channels—
Obfuscation et honnêteté : une étude
expérimentale de la demanded’assurance sur un marché sousintermédiation
This chapter is based on an article co-written with Jean-Louis Rullière and Stéphane Loisel.

49

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels

Abstract
This paper aims to shed light on the dilemma faced by insurance purchasers faced with multiple distribution channels. Should the consumer herself choose from a large set of insurance
policies or rather delegate a part her decision to an intermediary who is more or less honest?
We consider decisions based on a number of real-world insurance distribution channels withdifferent information frames. Beliefs about intermediary honesty are the main determinants ofindividual choice. In addition, the obfuscation of information is the main source of inefficiency
in decision-making, particularly regarding the characteristics of the insurance contracts chosen
by consumers.
Keywords: behavioral economics, distribution channels, honesty, insurance, intermediation, obfuscation, search costs

Résumé
Cet article a pour objectif d’étudier le comportement d’achat d’assurance sur un marché
comprenant différents types de canaux de distribution. Le consommateur choisit-il par luimême parmi un large choix de police d’assurance ou délègue-t-il une partie de sa decision à unintermédiaire plus ou moins honnête ? Nous étudions ici le processus d’achat d’assurance dans
un cadre inspiré des canaux de distribution d’assurance du marché réel conditionnant l’accès
à l’information. La croyance en l’honnêteté des intermédiaires est le principal déterminant deschoix de canaux de distribution. De plus, l’obfuscation liée à une quantité importante d’information conduit à des choix d’assurances sous-optimaux.
Mots-clés: économie comportementale, canaux de distribution, honnêteté, assurance , intermédiation ,obfuscation, coûts de recherche.

502.1. Introduction

2.1

Introduction

The internet has significantly affected purchasing behavior by changing access to information. No industry has been immuned to this considerable change, including more recently the
insurance sector, see e.g. Cappiello (2018). In principle, greater access to market information
allowing the better comparison of offers is a good news for consumers. In the past ten years,
we have seen the erosion of the traditional distribution channels of brokers and tied-agents,
see e.g. technical reports of FFA (French Federation of Insurance). However, these channels
continue to co-exist with new channels such as cyber-brokers and online insurance websites.
Consumers therefore continue to value physical intermediation.
This is partly explained by the complexity of insurance products, leading to obfuscation.
Obfuscation here reflects the limited discernment of agents due to excess information or the
introduction of a great deal of irrelevant information. Physical intermediaries reduce thisobfuscation by offering comparisons and advice. However, physical intermediaries also havetheir own financial incentives from insurers and consumers demand for their services. The
extent to which intermediaries are honest and how consumers perceive this honesty are therefore central in the delegation decision. Consumers will delegate only if they believe others
to be sufficiently honest. The consumer decision is thus based on a trade-off between finding
the appropriate information in a context of obfuscation and obtaining good information from
more or less honest intermediaries.
The aim of this paper is to understand how consumers choose their insurance contracts inthis kind of multichannel environment. Consumer decision-making is complex as, in addition
to insurance-contract choice, they have to choose their search strategy. We here provide someinsights into how this choice reveals information about the customer profile (price elasticity,
risk profile etc.). The results underline the importance for firms of managing their distribution
strategy and contract offers among heterogeneous consumers in order to screen between them.
We take into account consumers potential search costs to access information when considering delegation and information-gathering. These costs are defined as the time, effort andmoney expended by a consumer who searches for a product or service. Our goal here is not
to measure these individual search costs, contrary to Brynjolfsson and Smith (2000). Searchcosts are considered to be neutral and exogenous, such that the expected search cost to findthe optimal policy is the same for each distribution channel. The presence of search costs
ensures that individual exploration choices are taken conscientiously. We also do not consider
the effect of search costs on the market price equilibrium, contrary to Diamond (1971), ?,Baye et al. (2004) and Branco et al. (2012)), and focus on the demand side of the market withexogenously-defined offers.
Social psychology theory suggests that greater available information and choice should
improve the quality of decisions by informing consumers of all of the possibilities. However,S. Iyengar and Lepper (2001) carry out a number of different behavioral experiments andshow that this assumption does not necessarily hold: the desire for choice is not unlimited.
This conclusion also pertains in the experimental analysis in Schram and Sonnemans (2011)
regarding the choice of health insurance. They find that when there are many alternatives,
subjects only consider a smallest part of the available information and carry out a process of51

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
elimination based on limited characteristics. Hence, search costs and delegation choice affect
the quantity of information available for choice, which in turn might lead to focal point andanchoring effects due to obfuscation (see e.g. Gabaix et al. (2006), Ellison and Ellison (2009),
and Ke et al. (2016)).
Search costs and obfuscation can lead to delegation by consumers. Insurance intermediaries are meant to follow deontological rules and propose optimal contracts, as they are paid to
do so. However, the relationship between the consumer and physical intermediaries underlines
the role of honesty. This exchange game affects trust, which might at first lead us to think oftrust-game analysis (Berg et al. (1995)). However, their exchange game contains no normative
rules (i.e. there is no rule that individuals should transfer money or reward transfers). Thus,
this approach does not allow for objective deviation to be controlled for, as each individual
will have a subjective view of the behavior to be adopted, which could depend on inequality
aversion (Glaeser et al. (2000)).
The insurance market, as mentioned above, has intermediaries who sell advice, implying anormative rule. The presence of this implicit rule therefore leads us to appeal to more recent
concepts of honesty (Fischbacher and Föllmi-Heusi (2013)) and beliefs about others honesty(Hugh-Jones (2016)). Although there are normative rules for insurance intermediation, Cummins and Doherty (2006) note the importance of insurers’ financial incentives on their advice.
Houser et al. (2012) find greater dishonesty when people have been treated unfairly. Inequality
observed by the intermediary between different bonus levels can produce feelings of unfairness
as from her point of view these inequalities are groundless. This result is confirmed by Galeotti
et al. (2017), who show that financial incentives influence intermediaries’ honesty. Intermediary behavior is a concrete issue and regulatory concern, as the IDD (Insurance Distribution
Directive) illustrates. This new European directive comes into force in October 2018, andaims to monitor insurers financial incentives more strictly.
As honesty is difficult to observe in the field, we here elicit honesty and belief about others
honesty in a laboratory experiment. The experimental approach allows us to measure individual risk aversion, which affects decisions in uncertain contexts such as insurance.
All of the above concerns appear in our experiment. We find that honesty beliefs areimportant in the choice of distribution channel. Subjects who expect honesty ask for more
advice, in particular when the probability of loss is higher. Risk-averse individuals do notdirectly visit insurers and prefer to explore the market via brokers or cyber-brokers. We also
see that intermediary dishonesty leads subjects to change channel, in particular when they
initially expect honesty.
We identify two opposing search strategies: “saving search (costs)” and “deep search”.
Risk-averse subjects avoid “saving search”, especially as the probability of loss rises. Regarding contract choice, we find that only the probability of loss has a significant positive effecton coverage choice. However, we also find focal-point and anchoring effects with respect tothe quantity of information. Obfuscation leads subjects to choose their contract by comparing
prices (i.e. the focal-point effect). Subjects also take into account the gaps between contract
prices, with the average contract price acting as a benchmark in purchasing decisions (i.e. theanchoring effect).
52

2.2. Experimental design

We also find that obfuscation leading to focal-point effects produces inefficient choice.
As the quantity of information rises, so does the probability of choosing the lowest price,which leads to inefficient choices. Intermediary deviation is also a source of inefficient choice,although the detection of deviation leads to more efficient choice. The trade-off between delegation and self-exploration thus essentially depends on beliefs in others’ honesty when market
exploration is costly and information is not easy to compare.
Last, as intermediaries have their own financial incentives, we find considerable deviation(i.e. they do not propose the best contract), especially when it is profitable to do so. Wealthand the nature of risk do not affect deviation. However, deviation is considered to be risky, as
risk-averse intermediaries deviate less. We also find a correlation between honesty and deviation in advice in the insurance market at the session level, where more honest intermediaries
are more likely to suggest the optimal contract.
The remainder of the paper is organized as follows. We set out the experimental designin Section 2, and Section 3 briefly summarizes the procedure. Section 4 then describes thedetailed results, and Section 5 concludes.

2.2

Experimental design

We designed an experiment to identify the determinants of insurance choice, includingintermediaries and search costs (the cost of exploration). After controlling for individual
risk attitudes and beliefs about other’s honesty, the experiment allows us to investigate theconsumer’s exploration decision and contract choice. The elicitation of risk aversion is used toparameterize the design of the contract choice. The instructions for each part of the experiment
as well as screenshots of the interface can be found in Appendix. Subjects are divided into
two groups, A and B, and remain in the same group throughout the experiment.

2.2.1

The elicitation of honesty and beliefs about others’ honesty

In this first part, we propose an original experiment in order to elicit individual honesty
and beliefs about other’s honesty. We do so via the exchange of coins between a wallet and apadded envelope.
Before the session starts, a wallet is put on the table of each participant. This wallet either
contains 10 coins of 0.50e and a small card showing the result of 10 independent draws, or
nothing. The 10 draws are of a red or green ball without replacement from a bag containing
7 red balls and 7 green balls (so that there are five possible draws, corresponding to 3, 4, 5, 6
or 7 red balls).
The content of each wallet establishes the participant’s group. If the wallet is not empty,
the subject is in group B. They are asked to discreetly apply the following rule: for eachgreen ball drawn on the card they can take 0.50e from the wallet and put it in the paddedenvelope: the coins in the envelope represent their gains for this part of the experiment. The
Euros remaining in the wallet should then correspond to 0.50e times the number of red balls.
53

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
However, participants are informed that no-one in the room, including other participants andthe experimenter, will check if they follow this rule. The wallets cannot be identified within
individuals, and are all put in the same bag at the end of this part of the experiment.
If the wallet is empty, the subject is in group A. We show on their computer screen the
different draws that were distributed to the B subjects. They then indicate for each draw how
many Euros they think that the B subjects left in their wallets. The gain of the A participantsin this part comes from the random selection of one of these draws, with their earnings being
5 Euros minus the estimation error.

2.2.2

The elicitation of risk aversion

In this part, all gains are expressed in ECU (Experimental Currency Units), which areconverted at the end of the session using the following rate: 50 ECU = 1e. Risk preferences
are measured for all subjects in the gain and loss domains. We use the Multi Price List (MPL)
method suggested by Holt and Laury (2002) and convert dollars into ECU.
Subjects make two series of 10 decisions between alternatives “A” and “B”. The first 10
questions concern risk attitudes towards gains. For example, subjects have to choose between
receiving 50 ECU with a probability of 10% and 20 ECU with a probability of 90% (alternative“A”) and 85 ECU with a probability of 10% and 5 ECU with a probability of 90% (alternative“B”).
There are then questions about risk attitudes towards losses. We attribute to each subject
and for each question an initial endowment. They then have to choose one more or less risky
alternative, for instance between losing 50 ECU of their 100 ECU (the initial endowment) witha probability of 10% and losing 80 ECU with a probability of 90% (alternative “A”) and losing
15 ECU of their 100 ECU with a probability of 10% and losing 80 ECU with a probability of
95% (alternative “B”).
It can be argued that the house-money effect (Thaler and Johnson (1990)) makes participants more willing to take risks. However, Etchart-Vincent and l’Haridon (2011) compare
subjects’ risk attitudes in three payment conditions: a real loss condition with a randomlottery, “losses-from-an-initial-endowment” and a hypothetical-loss condition. They find no
significant difference between these three payment conditions in the loss domain, supporting
our approach.
In addition to comparing risk aversion in the gain and loss domains, we choose payoffs
such that the expected payoff for each level of the gain and loss question are the same. In
other words, if alternative “A” in the first questionnaire is “win 10 ECU with a probability of10%”, then alternative “A” in the second is “from 100 ECU lose 90 ECU with a probability of
10%”.
We measure participants’ risk aversion by their first switch from a safe to a risky option,
for both series of questions. For instance, a subject who switches at Question 5 is considered
to be more risk-averse than one who switches at Question 2.
54

2.2. Experimental design

2.2.3

Main gameIn the main game, participants retain their type from the first part of the experiment (Aor B): A-type participants now play the role of insurance customers while B-type participants
are “human” intermediaries. The game is repeated for eight periods (called rounds), including
two trial rounds.
A participants select an insurance contract in each round that protects against a known
loss with known risk (probability). Each contract includes a fixed premium and a deductible
paid in the case of loss. B-type participants can provide advice to A-type participants. The
A’s can call on human intermediaries to obtain information about the contracts. In this case,one of the B’s is chosen randomly to provide advice. We explain below the precise nature ofthis interaction.

2.2.3.1

The design of the exploration game

The A’s choose one contract in each round: they cannot remain uninsured. In each round,they have an initial level of wealth and a probability and amount of loss. These elements
can vary by period. There are eight different available contracts grouped into four menus of
two contracts each (A, B, C and D). These different menus represent the insurance companies
on the market. These contracts are hidden at the beginning of each round, and to obtain
information on them A’s have to explore the market. To do so, they have an exploration
allowance that is equally re-endowed at the beginning of each round. This credit is run down
for each exploration action, and the unused exploration credit forms part of A’s earnings. A
screenshot of the instructions at the beginning of each round appears in Appendix.
Participants can explore the market directly by choosing one of the four “company” menus.
This corresponds to individual sequential research, such as visiting insurance vendors or websites one by one. To do so, the A’s pay 12 ECU (debited from their exploration credit). Oncethey access the information about the two contracts on the menu, they can ask for a recommendation for 4 ECU (also deducted from the exploration credit) corresponding to time spent
with a tied-agent or a call-center for advice about the products from one unique company. In
this case, one of the B’s is selected to give advice.
This B-type participant receives additional information about the contracts: their screen
shows a ranking of the two contracts depending on the policy features and A’s risk attitude.
We explain below how this ranking is produced. In addition, there is a bonus for each contract that is paid to B if A chooses this contract via B’s intermediation. The bonuses are notpaid by A. B decides which contract to suggest according to this information. The A-typeparticipants do not know the bonus information.
Another possibility for A-type subjects is to look for an advisor (i.e. a B-type) corresponding to a broker. To do so, A has to pay 12 ECU (from their exploration credit). One
B-type participant is then randomly selected. This B’s screen shows the same information as
discussed above, but now for six different contracts (from three of the four companies). B heresuggests three ranked contracts from the six to participant A. In addition to the potential
55

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
bonus, B receives a fixed fee of 14 ECU (debited from A’s initial wealth). A screenshot of theinteraction with B as broker or tied-agent appears in Appendix.
Note that the difference between tied-agents and brokers comes from the nature of theirrelationship with the insurance companies: brokers can sell insurance from different insurers,
while tied-agents are constrained by an exclusivity contract to offer insurance from only one
company.
Last, A can explore the market using a comparator, corresponding to a cyber-broker (alsocalled an aggregator). Using the comparator costs 12 ECU (debited from the explorationcredit). Six of the eight possible contracts then appear on A’s screen (from three of the four
companies). Only premium information is shown: to find out the associated deductible of acontract, A has to pay 3 ECU (i.e. it is possible to choose a contract without knowing thecorresponding deductible). The comparator does not rank these contracts.
Figure 2.1

Subjects can continue to explore as long as they have exploration credit. We explain below
how we define the search costs. To sum up, Figure 2.1 sets out the different decisions andpotential interactions between players.
The final gain for this part of the experiment is that from one of the last six rounds, randomly selected (the first two rounds are trials). The loss event comes about according to theprobability in this selected round. In addition to the potential gains from player interaction,
B’s receive a fixed endowment in this part.

56

2.2. Experimental design
2.2.3.2
2.2.3.2.1

Theoretical Framework
The definition of contracts

For each round, we generate eight different contracts, defined by a premium CP and adeductible D. The contracts are designed as in Schlesinger (2013) such that
CP = (1 + λ)p × R × α,

(2.1)

where λ is the loading factor of the insurer, p the probability of loss, R the amount of theloss and α the coverage rate. In order to offer contracts similar to those in the real world, we
define a deductible D from (2.1) such thatα=(R − D)CP⇔D =R−.
R(1 + λ) × p

We generate for each round a set of contracts G1 such that
CPG1
G1 = CPG1 ∈ [0, W ]; DG1 = R −, λ ∈ [−0.2, 1] ,(1 + λ) × p

(2.2)

(2.3)

where W is the initial wealth allocates to subjects. For simplicity, we round each element to
the nearest integer.
We then define a new subset G2 ⊂ G1 such that all contracts are unique, possible andnon-dominated. We define a possible contract k such
W − CPk − Dk − 14 > 0.

(2.4)

We cannot offer contracts that produce a negative final gain in a given period. As a reminder,
14 ECUs correspond to the fixed fee paid to B when A chooses a contract via B’s intermediation: this is the fee paid for broker advice.
A contract i is dominated by contract j if CPi > CPj and Di > Dj . We exclude dominatedcontracts as these could affect purchasing decisions and, in the real world, contracts include somany different elements that it is difficult to say that one contract clearly dominates another.
We finally randomly draw eight contracts from G2 in each period.

2.2.3.2.2

The definition of ranking

In order for intermediaries to provide advice in insurance markets, we give them private
information via a ranking of contracts. Contracts are ranked by expected utility based onrisk aversion. For simplicity, and because wealth changes from one round to another, we use
a CRRA utility function (Constant Relative Risk Aversion). The use of IRRA or DRRA
(Increasing/Decreasing relative risk aversion) (Saha (1993)) requires the estimation of two
parameters, and therefore multiple applications of the Holt and Laury test with different levels of wealth. We therefore assume that consumers display the same risk-aversion for any
risk-wealth ratio:( 1−rx, if r 6= 1
U (x) = 1−r,(2.5)
ln(x), otherwise
57

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
where r is the aversion parameter with r = 0 for risk-neutral subjects, and r > 0 (resp. r < 0)for the risk-averse (resp. risk lovers). A contract i is optimal with respect to another contract
j at round t if
pt ×U (Wt −CPi,t −Di,t )+(1−pt )×U (Wt −CPi,t ) > pt ×U (Wt −CPj,t −Dj,t )+(1−pt )×U (Wt −CPj,t ).
(2.6)
The next step consists in the definition of the subject’s risk-aversion parameter from the answers
to the MPL (Multiple Price List). As risk aversion is different in the gain and loss domains (Tversky
and Kahneman (1981)) and insurance consumers face risks in the loss domain, we use only the answers
from the loss domain.

Table 2.1 – Risk Aversion Parameters (Holt and Laury (2002))
Questions of firstswitch from Safe
to Risky Option
0-1
2
3
4
5
6
7
8
9-10

Range ofRelative Risk
Aversionr ≤ −0.95−0.95 < r < −0.49−0.49 < r < −0.15−0.15 < r < 0.15
0.15 < r < 0.41
0.41 < r < 0.68
0.68 < r < 0.97
0.97 < r < 1.37
1.37 ≤ r

Risk
Preference
Classification
highly risk lovingvery risk loving
risk loving
risk neutralslightly risk averse
risk averse
very risk averse
highly risk averse
stay in bed

Parameters Used
−0.95
−0.72
−0.32
0
0.28
0.55
0.83
1.17
1.37

To do so, we take the mean of the Holt and Laury intervals (see Table 2.1), which gives us acoefficient according to the first switch from a safe to a risky option. It can be argued that our ranking
definition is based on a strong hypothesis. As suggested in Kobberling and Wakker (2005), the reference point (here initial wealth) may play an important role in the risky decision. However, parameter
estimation at the individual level would require multiple decisions for the same subject, which, for time
reasons, is unrealistic in our experiment. Another criticism refers to our definition of risk aversion.
A Fechner specification (popularized by Hey and Orme (1994)) or a Luce (1959) specification, which
consists in the estimation of parameters by log-likelihood including all the answers to the MPL, could
allow us to relax the hypothesis of expected utility. However, it also introduces assumptions about
the distribution of the probability of choices. For a complete review of risk aversion in the laboratory,
we refer to Harrison and Rutström (2008).
We do not suppose that we perfectly know participants’ optimal policies. In reality, intermediaries
provide advice with respect to their own knowledge and interpretation of the risk. Potential errors are
therefore an integral part of the distribution process. In addition, the ranking does not much dependon risk aversion, with the margin defined as λ ∈ [0.8, 2] playing a major role in the definition of optimality. Thus, rankings are almost the same for different levels of risk aversion. When a contract is
optimal, it is so for about 30% of the risk-aversion profile defined as below. This reduces any problemsof errors in ranking for individuals based on their risk aversion from the MPL.

58

2.2. Experimental design
2.2.3.2.3

The definition of search costs and the fixed fee

We include search costs defined as the cost of each exploration action. To encourage consumers toreveal their exploration preferences, in other words their distribution-channel preferences, we include
a fixed search cost of x ECU to access each kind of exploration.
We add an additional search cost u for asking for a recommendation within a specific menu. This
cost represents the time spent with a tied-agent in a traditional insurance context, as well as the timespent on the phone with an insurance advisor. We also add an additional search cost y for the revelation of deductible information by the comparator. In reality, price comparators display a ranking ofcontracts depending mainly on the premium: consumers have to click on a particular contract to findout the details, which is costly.
Last, consumers pay an additional fixed fee k to choose a contract that is suggested by the advisor.
This cost represents the fees paid to a broker in exchange for their services. We do not consider thisfee as a search cost, and it is therefore deducted from initial wealth.
These costs mentioned above are constrained in two ways. First, one channel should not dominate
another. For instance, imagine that it is more expensive to seek out an advisor and underwrite throughher than to consult four menus sequentially and ask for a recommendation for each of them. This
will affect our results due to the presence of an optimal exploration strategy whatever the subjects’
attributes and the round parameters.
We hence set search costs and the advisor fixed fee so that the expected costs of underwriting
the optimal contract in the market are the same for each intermediary. We of course assume at thisstage that there is no problem of obfuscation (the identification of optimal contracts) and beliefs about
honesty. We thus define the expected matching cost (EMC) for each decision design as follows:8
EM C(Advisor) = k + ,
6
8
5
4
3
2
1
EM C(Comparator) = (x + y + y + y + y + y + y),
6
6
6
6
6
6
3
2
1
EM C(M enu) = x + x + x + x,4
4
4
Using (2.7),(2.8) and (2.9), we have

(2.7)
(2.8)
(2.9)

EM C(Advisor) = EM C(Comparator) = EM C(M enu) ⇔ 43 x + k = 4x = 43 (x + 6y).
We in addition do not want to limit subjects in their exploration and let them explore the entire
market. We assume that it is always possible to be fully informed but that consumers do not benefit
from any time saved. Thus, C = 7x + 6y + 4u, where C is the exploration credit.

2.2.3.2.4

The definition of the bonus

As mentioned below, intermediaries are financially motivated by insurers in order to build up profitable portfolios. We thus introduce bonuses for each contract proposed. These bonuses represent thecommission paid by insurers to their “human” intermediaries (brokers or tied agents). In our game,we randomly define bonuses such that
Bi = max((CPi − p × (R − Di )) × U[0.2; 0.4], 0).

(2.10)The bonus is therefore proportional to the profit generated by the contract. Nevertheless, the bonus
rate is defined as a uniform random variable over the interval (0.2, 0.4). The reason for doing so is59

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
that the most profitable contract should not be the most incentivized: insurers may wish to incentivize
less-profitable contracts for reasons such as loyalty goals or brand image. We do however assume that
non-profitable contracts are never incentivized.

2.3

Procedure

A web interface and server database were designed specifically for this experiment. The interface
was developed with HTML and JavaScript, the backend with Java and PostgreSQL as the database.
The subjects were students from the University of Lyon 1 – Claude Bernard, France. 217 subjects
participated in the experiment, 27 in average for each session including 5 participants of type B.
The honesty game was played first. All subjects received identical instructions, including the comprehension questions. Subjects were assigned to a group for the rest of the session. Then, they madetheir decisions for this part. Afterwards, subjects received written instructions for the risk-elicitation
task and made their choices. Finally, they received instructions for the main part of the experimentand a comprehension questionnaire that we corrected with them. Before leaving the room to privately
receive their payments, we asked them to answer some general questions about age, gender and education. The different periods of the exploration game were displayed randomly for each session. All
treatments were framed in a neutral manner.
We decided to play the honesty game at the beginning of the session to avoid learning effects frompotential interactions during the main part of the experiment. The risk-elicitation task was played
just before the main game in order to be able to rank contracts for each A subject. The payoffs in
the different tasks were revealed at the end of the entire experiment. The sessions lasted about 105
minutes. The average payoff was about 16 Euros including a show up fee of 3 Euros.
For full transparency, we will make available the R code (RCore-Team (2018)) and experimental
data used in this paper upon request. Please note that authors should be referenced in the case of theuse of these data or this code for further research.

2.4

Results and Analysis

2.4.1

Subject variables: distribution and impact on the main game

2.4.1.1

Risk Aversion

We first construct, for each individual, two variables representing their risk aversion level in thegain and loss domain resulting from the Holt and Laury (2002) elicitation game. We look at risk
aversion in both domains for two reasons. In insurance, there is a probability of wealth loss. However,while searching information, individuals are uncertain about the value of the information gain.
Risk aversion is defined by the number of the question of the first switch from the safe to therisky option. Neutral subjects will switch at question 5, while the risk-averse will switch later. The
average question switch number is 5.56 in the gain domain and 4.82 in the loss domain. These figures
are significantly different using a non-parametric two-sided Wilcoxon test (p − value < 1.7e−6 ). This
result is in line with Chakravarty and Roy (2009), who found significantly less risk-aversion in the loss
than in the gain domain (Figure 2.2).
We find that risk-averse individuals avoid search on their own, and prefer to compare offers via
a cyber-broker or broker (Table 2.2). Risk aversion in the loss domain has a more significant impact
than that in the gain domain for distribution-channel choice. The risk averse in the gain domain
also collect more information in the exploration process (Table 2.5). However, contrary to standard
expected-utility theory, neither risk aversion in the loss or gain domains affect contract choice (Table60

2.4. Results and Analysis
Figure 2.2 – Empirical cumulative distribution functions from 1st switch

2.6). Regarding intermediaries, we find a strong correlation between risk aversion in the loss domain
and deviation behaviors (i.e. the suggestion of a non-optimal contract) (Table 2.9). Deviation istherefore considered to be a risky decision.

2.4.1.2

Honesty and beliefs about others’ honesty

As we here analyze distribution-channel choice in insurance including human intermediaries, oursecond key variable is honesty. Honesty beliefs are individual, but we only observe actual honesty at
the session level.
As a reminder, we distributed different draws to the five B participants in each session including
3, 4, 5, 6 or 7 red balls. We find interesting patterns of honesty and beliefs about others’ honesty inthe different draws (Chapter 1: Mouminoux and Rulliere (2018)). The favorable draw produces morehonesty and greater honesty expectations, as in Houser et al. (2012) and Galeotti et al. (2017). For
simplicity we here only use the answer of A faced with the least-fortunate draw given to the B’s ineach session (the draw with 7 red balls, where B should leave 3.5e in the wallet according to therule). This draw corresponds to the situation where honesty and beliefs about honesty are the mostheterogeneous and the lowest. Honesty beliefs for subject i (i.e. HBi ) are defined as follows:
HBi = answeri − 3.5e .
On average, A’s think that B’s will take 1.23e more than the rule. Over the 177 A subjects,
41 trust the B’s, 8 are over trusting (they think that the B’s take less than the rule) and 128 are
untrusting (they think that B’s take more than allowed by the rule).
We define dishonesty at the session level as the average amount taken above the rule by the five
B participants in each session. On average, the B’s take 0.97e more than the rule: 47% respect therule and 35% fully deviate.
As risk aversion and honesty beliefs will be introduced together, we check the correlation between
them: this is insignificant (Kendall, Pearson and Spearman correlation tests), as in the seminal paperof Eckel and Wilson (2004) who find no correlation between trust (in our case honesty beliefs) andrisk aversion.

61

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
Honesty beliefs help explain insurance purchase. Those with high honesty expectations avoid cyberbrokers (Table 2.2) and are more sensitive to broker deviation, suggesting disappointment (Table 2.4).
Regarding intermediary behavior, we find a correlation between honesty and the deviation rate at thesession level. The more honest B players are less likely they deviate in their suggestions (Table 2.9).

2.4.2

Purchasing Process

As noted above, we break the purchasing process down into two parts: subjects explore the market
to uncover alternatives before choosing an insurance contract. We are first interested in the “searchstrategy”, i.e. the determinants of distribution-channel choices. To do so, we distinct two different
independent variables called “Rounds’ First Choice Channel” and “Rounds’ Underwriter Channel”
corresponding respectively to the first channel chosen for exploration and the last channel chosen forunderwriting. We then investigate extreme search strategies: subjects who adopt a “saving search cost
strategy” (i.e. take the lowest price on the comparator without uncovering the deductible) and the“deep search cost strategy” (i.e. subjects who uncover at least one contract from all insurers). Then,
given the information gleaned from these search strategies, we consider the final insurance-contract
choice.

2.4.2.1

Search Strategies

Figure 2.3 shows final channel decision of type A individuals. Globally over the entire experiment,
we note that 57.4% (std. dev. 0.045) of subjects chose their insurance via the comparator, while 16.0%(std. dev.0.029) chose via a broker and 26.6% (std. dev.0.029) directly from an insurer. However,from Figure 2.3, we see that channel choice changes over round, with a rise in the use of comparators
over time. We first focus on first channel choices and inter-period learning effects. We then investigate
the final underwriting choices and intra-period channel decisions.

Figure 2.3 – Percentage of Choices by Underwritting Distribution Channel

2.4.2.1.1

First Channel Choices and inter-period effects

In the following, we consider a series of three statistical models to analyze first channel choices
and inter-period effects, where the reference level is the broker channel. We first carry out a multinomial logistic regression with random effects (Table 2.2, Model 1) with the first channel choices as
the dependent variable (i.e. Broker, Comparator or Insurer). Random effects allow us to control forthe relationship between errors in our panel data (see e.g. Croissant (2013)). We also consider p-value

62

2.4. Results and Analysis
under Holm-Bonferroni multiple correction in all presented models (?). In the first two regressions,
we consider only non-lagged explanatory variables.
Honesty beliefs are a significant determinant of first channel choices. The probability that subjects
first choose the comparator falls with honesty beliefs (p−value < 0.001). Risk averse individuals avoid
insurance as a first choice (p − value < 0.001). Regarding the “round” parameters, only the probabilityof loss has a significant effect on first choices: a higher probability leads subjects to avoid comparator(p − value < 0.001).
Model 2 of Table 2.2 does not demonstrate learning effects in the significance of the round number.
Previous results on honesty beliefs, risk aversion and probability of loss continue to hold.
Finally, by incorporating dummies of lagged variable (i.e. explanatory variables followed by t − 1)in Model 3 of Table 2.2 we try to capture potential hysteresis in choice. We add the subject’s previous
round’s first choice and do not find any significant effects. We also add a dummy variable for the subject
having asked for advice (though a broker or insurer) in the previous period and having received suboptimal advice. We find no effect here: subjects do not seem to detect or to take into account deviationin previous rounds.
We also control the effect of the number of revealed contracts in the previous period. Again, thiscontrol does not reveal learning effects. The other estimated coefficients in Model 3 are similar to
those in Models 1 & 2.

Table 2.2 – Multinomial Logistic Regression with Random EffectExplanatory Variable
Model 1
Coefficients

Comparator

Rounds’ First Choice Channel
Model 2

Insurance

Comparator−0.67∗∗∗

0.26

0.12

−0.01

0.14
−0.40∗∗∗

0.07

0.09

0.00

Model 3

Insurance

Comparator

Insurance−0.69∗∗∗

0.29−0.79∗∗∗

0.24

0.12

0.16

0.00

1.99
−0.41∗∗∗

0.04

0.17
−0.35∗∗∗

0.07

0.09

0.08

0.11

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00
−4.42∗∗∗

0.00

0.00

−3.44

0.00
−4.32∗∗∗−3.67−3.78∗∗∗

−3.03

1.29

1.63

1.30

1.65

1.04

1.30

0.14

−0.22

0.23

−0.26

0.07

0.10

0.11

0.26

Std. error

Honesty Beliefs
Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
Round numberFirst choice t − 1: Comparator
First choice t − 1: Insurance
Deviation t − 1
Contract no. discovered t − 1
Constant (ref. level: Broker)
Nb. Observations
Nb. Subjects
R2
Adjusted R2

−0.23

0.60

0.46

0.58

0.47

1.20

0.47

0.60

−0.08

0.31

0.42

0.53

−0.18

−0.06

0.08

0.10

2.07∗∗∗

2.24∗∗∗

1.25∗

3.44∗∗∗

1.08

2.70

0.53

0.65

0.67

0.81

0.98

1.08

1062
177
0.336
0.327

1062
177
0.346
0.336

865
177
0.340
0.325

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’63

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
2.4.2.1.2

Intra-Period switching behavior

Now, we consider a series of four statistical models to analyze intra-period switching behavior,
still with the reference level being the broker channel. Over the entire experiment, we note that 21.0%(std. dev. 0.047) of subjects decide to switch channel type at least once during a round. Figure 2.4
shows the percentage of switches with respect to the first channel choice. On average, 41.7% (std.
dev. 0.058) of subjects who first chose a broker switch, with the analogous figures for Comparator andInsurance being 14.1% (std. dev. 0.030) and 17.6% (std. dev. 0.061). Among the switchers, 15.2%(std. dev. 0.032) finally return to their first choice.

Figure 2.4 – Percentage of Switch by First Choice Channel

In Model 4 of Table 2.3, we show that these switches are not explained by the round parameters or
subject variables. Only first channel choice in the period is significant (p−value < 0.001) in explaining
the final underwriting channel choice.
We find intra-period inertia in choice and, when subjects switch, they prefer to switch between thecomparator and an insurer. Switching might depend on first channel choice and we do not capture thiseffect in Model 4. We thus separate our sample into three parts conditional on first channel choice, andrun for each a logistic regression where the dependent variable is a dummy for the subject switching
channel type during the round (Models 5, 6 and 7 of Table 2.4).
Model 5 focuses the switching analysis on subjects who first chose a Broker. The round parameters and risk aversion do not affect switching. We then consider the deviation degree, defined as theaverage gap between the ranks of the three contracts proposed compared to the optimal ranking thatappears on the brokers’ screen. This deviation degree is computed for each interaction between A
and B. If the broker proposes the ranking that appears on the screen, then the deviation degree iszero.The maximum deviation degree would come from proposing the worst contract (ranked 6) first,
then the second worst (ranked 5) second, and the third-worst third: this produces an average gap of1
3 ((6 − 1) + (5 − 2) + (4 − 3)) = 3. The probability of switching rises with deviation degree, in particularfor subjects with high honesty expectations.
Contrary to the lack of an inter-period deviation learning effect (Model 3 of Table 2.2), subjects
do take into account intra-period deviation. They react more strongly when they are disappointed bytheir choices (i.e. “Deviation Deg. x Honesty Belief” significant at 10%). Since subjects can choose
different channels during a same round, they can collect additional information allowing us to detect
important intermediaries’ deviation. We also control in Model 5 bis of Table 4 bis in Appendix where642.4. Results and Analysis
Table 2.3 – Multinomial Logistic Regression with Random EffectExplanatory Variable
Coefficients

Rounds’ Underwriter Channel
Model 4Comparator
Insurance

Std. error

Honesty Beliefs
Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
Round numberFirst choice t − 1: Comparator
First choice t − 1: Insurance
Constant (ref. level: Broker)
Nb. Observations
Nb. Subjects
R2
Adjusted R2

0.00

0.06

0.15

0.15

0.10

0.06

0.09

0.09

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

−0.19

−1.35

1.58

1.58

0.08

0.04

0.09
5.50∗∗∗

0.10
2.44∗∗∗

0.42

0.43

2.44∗∗∗

5.45∗∗∗

0.54

0.49

−2.62∗∗−2.92∗

0.94

0.90

1062
177
0.505
0.495

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’the subjects switch to. The results are in line with our intuition: disappointed subjects choose thecomparator channel which does not include “human” intermediaries.
The results are different for subjects who first choose insurers (Model 6 of Table 2.4) here only
honesty beliefs significantly predict switching and switchers tend to change to the comparator (Model
6 bis of Table 4 bis in Appendix). Deviation by tied-agents does not affect switching. However, we
prefer to avoid any misinterpretation of these results due to a lack of power: recommendations for
only one insurer by a tied agent were only requested 10 times.
Finally, Model 7 of Table 2.4 focuses on subjects who first choose the comparator. We here include the number of deductible values uncovered. As expected, switch probability falls with the latter:revealing the deductible value is costly. We also test for linearity here, and find that when the numberof deductibles is large enough, the probability of switch increases (i.e. the coefficient on the squared
term is positive).
In Model 7 bis of Table 4 bis in Appendix, we find that these effects are significant only for those
who switch from the comparator to an insurer. Only 15 subjects switch from the comparator to brokers. This lack of observations prevents us from modelling this specific switch. However, all of theprevious results continue to hold for switchers from comparator to insurers. The positive effect of thedeductible on switches from the comparator to insurers can be explained by some subjects deciding to
adopt a “deep search strategy”, by obtaining information on at least one contract from each insurer.
As a reminder, the comparator offers information for three of the four insurers in the market. By firstusing the comparator and revealing at least 4 deductibles, subjects can then obtain complete information about at least one contract from all three of these insurers. They can then select the missing
insurer and collect information about at least one contract for all insurers on the market. This leads
us to analyze extreme search strategies.

65

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
Table 2.4 – Logistic Regression with Random EffectExplanatory Variable
Coefficients

Intraround Channel Switch
Model 5Model 6
Model 7
BrokerInsurance
Comparator

Std. error

Honesty Beliefs
Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
Round number
Deviation Degree
Deviation Deg. x Honesty Belief−0.98

0.74∗∗

0.39

0.80

0.37

0.23

−0.06

−0.06

−0.04

0.18

0.19

0.13

0.00

0.01

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00

0.00

0.00

3.96

−2.15

4.32

2.81

3.26

2.14

−0.18

−0.18

−0.20

0.17

0.23

0.15

0.59

0.53

0.30

0.45

0.58∗

−0.19

0.24

0.33

−1.38∗∗∗

Deductible no. discovered

0.40

0.17∗∗

Deductible no. discovered2
Constant (ref. level: No Switch)
Nb. Observations
Nb. Subjects
R2
Adjusted R2

0.06

−2.19

−3.73∗∗

−2.01

1.75

1.86

1.38

237
79
0.272
0.239

210
67
0.167
0.126

615
140
0.177
0.164

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’

2.4.2.1.3“Saving Search” and “Deep Search” strategies

We here focus on extreme search strategies. We split our observations up into three different
profiles (Figure 2.5). We identify two extreme strategies corresponding to extended and restrained
search, as opposed to various moderate search strategies. 6.7% of subjects obtain information about
at least one contract from all insurers: we call this the “deep search strategy”. On the contrary, 13.8%
of subjects choose “saving search”, selecting the comparator as first channel choice and underwriting
the contract with the lowest premium. Strategy choice is not affected by round number, meaning there
are no learning or boredom effects.
We carry out multinomial logistic regressions to investigate extreme search strategies. We find in Models 8, 9 and 10 of Table 2.5 that the probability of loss has a significant impact. As this probability
rises , the probability of choosing a saving search strategy decreases. In other words, subjects preferto explore the market as the probability of losing part of their wealth rises. We in addition see that
subjects who believe in others’ honesty avoid the saving search strategy (p − value < 0.001). For thesame search cost, these subjects could obtain more information by choosing a broker but could thenbe confronted with deviation.
Unlike to our other models, risk aversion is only significant in the gain domain here (p−value < 0.001).
As noted above, information search introduces uncertainty about the value of the information gain.
We can thus argue that risk-averse subjects prefer to reveal more in order to reduce this uncertainty.
However, we find no significant differences between the standard and deep extreme strategies by risk
aversion. Finally, we control in Model 10 of Table 2.5 for any learning or boredom effects by adding
the round number. This does not attract a significant coefficient and the other results are unaffected.

66

2.4. Results and Analysis
Figure 2.5 – Distribution of Search Profile by Round Number

Table 2.5 – Multinomial Logistic Regression with Random EffectExplanatory Variable
Coefficients

Rounds’ First Choice Channel
Model 8Model 9Model 10
SavingDeep
SavingDeep
SavingDeep

Std. error

Honesty Beliefs−0.46∗∗∗

0.10−0.50∗∗∗

0.16−0.49∗∗∗

0.18

0.11

0.17

0.11
−0.22∗∗

0.17

0.17

−0.11

0.11
−0.22∗∗

−0.11

0.07

0.10

0.07

0.10

Risk Aversion (gain)
Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss

−0.07

−0.06

0.07

0.10

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00
−4.50∗∗∗

0.00

0.00

3.67

0.00
−4.52∗∗∗

3.58−3.35∗∗∗

3.34

1.32

1.79

1.31

1.80

0.93

1.22

0.12

−0.18

Round number
Constant (ref. level: Others)
Nb. Observations
Nb. Subjects
R2
Adjusted R2−2.14∗∗−4.85∗∗∗

−1.30

0.66

0.94

0.66

1062
177
0.176
0.163

0.08

0.11

−4.39∗∗∗−1.73−3.02∗

0.96

0.65

0.89

1062
177
0.183
0.172

865
177
0.184
0.174

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’

2.4.2.2

Contract Choice

After determining individuals’ search strategies, we now turn to the final contract choices. We
first determine the drivers of coverage choice and highlight some focal-point and anchoring effects.
We then investigate the consequences of the quantity of information on the efficiency of choices withrespect to Expected Utility Theory.

67

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
2.4.2.2.1

Coverage choiceThis subsection focuses on subject contract choices. We first estimate the relative coverage chosen
via a linear regression. As a reminder, coverage α is defined as follows α = LossAmount−Deductible. We
LossAmounthere model relative coverage to standardize the dependent variable across subjects, as they do not face
the same choice set, defined as:
Coverage − M in. coverage available.
M ax. coverage available − M in. coverage available
Relative coverage is 0 if the subject choses the minimal coverage available and 1 if she chooses the
maximum coverage. The regressions cover subjects who have at least two choice possibilities: we saw
above that some subjects adopt a ‘saving search strategy’ so that they do not compare contracts.
As expected, coverage rises with the probability of loss (p − value < 0.1 in Models 11, 12 and
13 of Table 2.6), but falls with the number of alternatives (Model 12, Table 2.6). As the coveragechoice and the number of alternatives are both linked with the search strategy, we control for thisin Model 13. We find no significant effect of the latter, but the number of alternatives is not significant anymore. This loss of significativity is due to the Holm-Bonferroni multiple hypothesis correction.
However, this result leads us to investigate in more detail the effect of available information oncontract choice. Contrary to what might have been expected, contract choice is not mainly driven
by standard parameters (except for the probability of loss). We suspect a focal-point effect due toobfuscation here: subjects with too much information end up making their choices based only on
the simplest characteristic: the premium. As we designed the experiment to avoid strictly-dominatedalternatives, the lowest coverage here corresponds to the lowest premium.

Table 2.6 – OLS Regression with Random EffectExplanatory Variable
Coefficients

Relative Coverage Choice
Model 11

Model 12Model 13

Std. error

Honesty Beliefs
Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
Round number

0.02

0.02

0.02

0.02

0.02

0.02

0.00

0.00

0.00

0.01

0.01

0.01

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00
0.33∗∗

0.00
0.36∗∗

0.00
0.35∗

0.15

0.15

0.15

−0.02∗

−0.02

0.00
0.01

No. available alternative

0.01

First choice t: COMPARATOR

0.05

0.06

First choice t: INSURANCE
Constant
Nb. Observations
Nb. Subjects
R2
Adjusted R2

0.01

−0.01

0.36∗∗∗

0.41∗∗∗

0.04
0.43∗∗∗

0.09

0.08

0.08

915
171
0.021
0.014

915
171
0.025
0.019

915
171
0.028
0.020

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’68

2.4. Results and Analysis
2.4.2.2.2

Focal-point and Anchoring effects on contract choice

In order to control for focal-point effects regarding the premium, we run a logistic regression
on a dummy indicating individuals choosing the lowest price. With more alternatives, the probabilityof choosing the lowest price should fall. However, Models 14, 15 and 16 of Table 2.7 show that themore alternatives subjects have the more likely they are to choose the lowest price. This is consistent
with focal points.
Contrary to previous models of coverage choice, we find that the probability of loss increases theprobability of choosing the best price. This point encourages us to control for eventual anchoring effects. By construction, the higher the probability of loss the higher the premium, in a given round. We
could then argue that the probability of choosing the lowest price rises with the price, independentlyof the probability of loss.
To avoid the misinterpretation of this coefficient we add in Model 15 (2.7) the difference between
the average and minimal premium. We find a significant anchoring effect, and the probability of loss
is no longer significant. In other words, adding a coefficient for the difference between the average andthe minimal premium reduces significantly the probability of choosing the lowest price.

Table 2.7 – Logistic Regression with Random EffectExplanatory Variable
Coefficients

Lowest PriceModel 14Model 15

Model 16

Std. error

Honesty Beliefs
Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
Round number
No. available alternative

0.13

0.14

0.13

0.06

0.07

0.07

−0.02

−0.03

−0.04

0.04

0.04

0.04

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00
2.79∗∗∗

0.00

0.00

0.45

0.35

0.76

0.98

0.98

−0.01

0.00

0.00

0.04
0.31∗∗∗

0.05
0.49∗∗∗

0.05
0.49∗∗∗

0.05

Average Premium - Minimal Premium

0.06

0.06

−0.02∗∗∗

0.02∗∗∗

0.00

First choice t: COMPARATOR

0.19

−0.31

First choice t: INSURANCE
Constant (ref. level: NO)
Nb. Observations
Nb. Subjects
R2
Adjusted R2

0.00

−0.21

0.24

−0.77

−0.70

−0.48

0.09

0.08

0.08

1067
177
0.068
0.060

1067
177
0.095
0.086

1067
177
0.097
0.087

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’To sum up, subjects with “too much” information make their decisions based on the premium: thisis the focal-point effect due to obfuscation. However, their decisions are also driven by all available
contract information, and as the average premium rises relative to the minimum, the probability ofchoosing the cheapest contract falls: this is the anchoring effect.

69

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
2.4.2.2.3

Efficiency of Choices

In this section, we focus our analysis on the efficiency of subject choice. As described above,the eight available contracts are ranked according to Expected Utility Theory. On average, 44% (std.
dev. 4%) of subjects choose the optimal contract through the discovered contracts (i.e. the available
alternatives at the moment of choice). The majority of subjects hence do not make their choices
according to standard Expected Utility Theory. As previously noted, the quantity and quality of
information are significant drivers of the final contract choice.
We thus estimate models in order to determine the role of obfuscation (i.e. the quantity of information) and intermediaries’ honesty (i.e. the quality of information) on the efficiency of choice.
In Models 17, 18 and 19 of Table 2.8, the dependent variable is a dummy for the subject choosing
the optimal contract according to Expected Utility Theory. We estimate these models only on subjects
who have complete information on at least two choices and choose a contract with complete information. This leaves 892 observations from the 1062 collected (i.e. 84%). Some subjects discover morethan two contracts but finally decide to select a contract without finding out about its information. The
individual hence cannot determine the rank of this contract with respect to other contracts discovered.
We first model the probability of choosing the optimal contract with respect to the round parameters (i.e. probability of loss, loss amount and initial wealth), subject variables (honesty beliefs andrisk aversion) and the number of available alternative (Model 17, Table 2.8). We find that the roundparameters and subject variables do not significantly explain inefficient choices.
However, as the number of alternatives rises, the probability of making an inefficient choice significantly increases. While one could argue that more-informed consumers should make more efficientchoices, we show that too much choice leads to inefficient decisions. As the distribution channels do
not have the same levels of obfuscation or honesty, we add in Models 18 (Table 2.8) an estimatedcoefficient for either the interaction between the number of alternatives and the channel choice or theinteraction between the deviation and the channel choice.
Physical intermediaries, could lead to inefficiency since the quality of information depends on thehonesty of intermediaries. However, intermediaries have their own financial incentives, leading to deviations (deviation is represented here by a dummy for the intermediaries who do not advise the optimal
contract at first). We find in Model 18 of Table 8 that intermediary deviation significantly increases
the probability of inefficient choice. In other words, subjects who avoid obfuscation by delegating apart of their decision to a physical intermediary are also exposed to inefficiency in decision-making.
Interestingly, by adding variable between deviations and underwriting-channel choice we also find that
consumers who faced deviation and finally choose the comparator make more optimal choices. The
consumers who detect deviation and decide to switch channel have the highest ability of identifying
the most efficient contract.
Last, we include in Model 19 of Table 2.8 a dummy for subjects choosing the lowest price. We also
add an interaction between this dummy and the number of available alternatives since we showed inTable 7 of the previous section that obfuscation leads to focal-point effects.
While adding interaction between the number of alternatives and the channel choice in Model 18
leads to loss significativity of the number of alternative (under Holm-Bonferroni correction), Model 19
allow us to find new evidence on obfuscation effect. Indeed, we include a dummy for subjects choosing
the lowest price. We also add an interaction between this dummy and the number of available alternatives since we showed in Table 7 of the previous section that obfuscation leads to focal-point effects.
While choosing the lowest price tends to increase the probability of optimal choice, we find that when
this choice is made because of focal-point effects due to obfuscation, the efficiency of choice falls. We

70

2.4. Results and Analysis
also control for a potential learning effect in all previous models via the variable for the number ofrounds. We do not find any evidence of learning effects.

Table 2.8 – Logistic Regression with Random EffectExplanatory Variable
Coefficients

Optimal Choice (EUT)
Model 17

Model 18

Model 19

Std. error

Honesty Beliefs
Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
No. available alternative

0.07

0.06

0.08

0.06

0.07

0.07

0.06

0.08

0.07

0.04

0.04

0.04

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

−0.90

−1.09

−0.57

0.75
−0.37∗∗∗

0.76

0.78

−0.12

−0.03

0.05

Underwriting Channel t: COMPARATOR
Underwriting Channel t: INSURANCE
Deviation

0.16

0.17

−0.17

−0.08

0.68

0.70

0.06

−0.25

0.69

0.71

−0.98∗∗−0.88∗∗

0.37

Lowest Price

0.43−0.35∗

Underwriting Channel t: COMPARATOR x Nu. available alternative
Underwriting Channel t: INSURANCE x Nu. available alternative
Underwriting Channel t: COMPARATOR x Deviation
Underwriting Channel t: INSURANCE x Deviation

Constant (ref. level: NO)
Nb. Observations
Nb. Subjects
R2
Adjusted R2−0.30∗

0.18

0.18

−0.23

−0.19

0.17

0.18

1.83∗∗

1.59∗∗

0.57

0.59

0.33

0.23

0.56

0.62
−0.23∗∗

Lowest Price x No. available alternative
Round Number

0.39

2.11∗∗∗

0.11

0.03

0.03

0.02

0.04
0.89∗∗

0.05

0.05

0.83

0.08

0.44

0.75

0.78

892
171
0.065
0.055

892
171
0.081
0.064

892
171
0.129
0.111

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’

2.4.3

Intermediary strategies

We last examine the behavior of subjects playing “human” intermediaries (the B’s). On average,
each B subject is called 1.2 times per round. The B-type players have financial incentives via thebonuses: just over 60% of the time they do not propose the best policy to subjects who ask for advice.
We run a logistic regression on a dummy indicating for intermediary deviation (not first suggesting
the best contract).

71Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
The consumer variables (initial wealth, the loss amount and the probability of loss) do not affect deviation (Models 20, 21 and 22 of Table 2.9). However, risk-averse intermediaries deviate less(p − value < 0.05). As a reminder, they receive the contract bonus only if the customers choose thepolicy via their intermediation. Deviation is hence considered as a risky decision.

Table 2.9 – Logistic Regression with Random EffectExplanatory Variable
Coefficients

Deviation
Model 20

Model 21

Model 22−0.28∗∗−0.28∗∗−0.28∗∗

0.09

0.09

0.09

0.01

0.00

0.00

0.00

0.00

0.00

Std. error

Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
Optimal Contract Bonus
Status: TIED-AGENT
Dishonesty Session Level
Round Number

0.00

0.00

0.00

0.00

0.00

0.00

−2.22

−0.67

−0.53

1.66

1.68

1.71

−0.05

0.06

0.06

0.02

0.04

0.04

−1.28∗−1.47∗−1.46∗

0.68
0.59∗∗

0.64
0.63∗∗

0.62∗∗
0.26

0.26

0.26

−0.01

0.00

0.00

0.10

0.10
0.18∗∗

0.10
0.18∗∗

0.06

0.06

2.66∗∗

3.09∗∗

0.32
2.88∗

0.89

0.98

1.07

289
40
0.181
0.152

289
40
0.213
0.182

289
40
0.210
0.176

Mean Bonus - Optimal Contract Bonus

0.10

No. of demand received
Constant (ref. level: NO)
Nb. Observations
Nb. Subjects
R2
Adjusted R2

0.64

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’Moreover, the level of the bonus of the optimal contract do not affect deviation(Model 20). However, the difference between the average possible bonus and the optimal-contract bonus is significant
in explaining deviation: a smaller gap here produces less deviation (Model 21). Hence, it is not theabsolute level of remuneration which affect intermediaries deviation decisions but the difference of
remuneration between offers.
We also control for intermediary type and find that when the demand for advice comes from aparticular insurer (versus a broker) intermediaries deviate significantly less (p − value < 0.1). The
reason is simple: participants only receive two different contracts from insurers but six from brokers.
We consider honesty at the session level from the honesty game and the intermediary deviation.
After controlling for session effects (Model 20 bis of Table 9 bis in Appendix), we find a strong correlation between deviation in the honesty game (called ‘Dishonesty Session Level’ here) and intermediarydeviation. This result comforts or idea about the existence of close relationship between deontologicalethics imposed to insurance physical intermediaries and honesty concept.
Finally, we look for portfolio effects by adding the number of demands previously received during
the round (Model 22). If deviation is risky, intermediaries receiving a number of demands for advice
in the same round may diversify their risks. However, we do not find any significant evidence of this.

72

2.5. Conclusion

2.5

Conclusion

In this paper, we analyze the purchasing behavior in personal non-life insurance markets. The
personal insurance market is a useful context in which to analyze purchasing behavior, as it includes
a number of types of intermediaries. Consumers can explore the insurance market via brokers, cyberbrokers or directly visit a specific insurer and may ask tied-agents for advice. Based on the search-costtheory, we here designed an experiment to help understand distribution-channel choices.
The complexity of insurance supply, including different levels of coverage, premium and deductibles, gives rise to potential obfuscation, which appears to be a marketing device and a source of
inefficiency in decision-making.
Consumers can decide to explore the market on their own, but have to deal with many comparisons involving incomplete information. Another possibility is then to ask for advice to help choose.
Although the deontological rule implies that intermediaries should provide the best advice, they arethemselves influenced by incentives such as bonuses or lobbying. We find that this kind of delegationis conditional on honesty beliefs.
Even with considerable intermediary deviation, the complexity of choices in a risky environmentand obfuscation drive subjects to delegate part of their decisions. Even with free access to online insurance markets, our results suggest that consumers continue to value broker and tied-agent services,
although deviation leads to inefficient choices. However, deviation also explains channel choices, andconsumers who detect deviation tend to switch channel and make more efficient contract choices.
We also find that intermediary behaviors depend on their risk aversion and the general honesty
level, so that deviation is a risky decision. We present evidence of the importance of honesty andbeliefs about others’ honesty. Because both search strategies (i.e. delegation and self-exploration) aresources of inefficiency in decision-making, the development of a multi-channel distribution strategy byinsurers is required to cover a large part of market. Some consumers have sufficient belief in others’
honesty to delegate a part of their decisions and avoid obfuscation.
However, the evaluation of the economic benefits of physical intermediaries remains difficult. Whileintermediary costs are easy to evaluate (the fees and bonuses), the associated benefit is not. On the
one hand, consumers save on search costs and avoid obfuscation, but on the other these savings do
not necessarily compensate for the risk of dishonesty. As in Bergstresser et al. (2009) we find that
financial incentives for physical intermediaries produce sub-optimal contracts. Higher incentives pushintermediaries to be more aggressive and makes them less honest. There is a potential crowding out
effect of incentive towards consumers. In the context of multiple distribution channels, the design ofphysical intermediary incentives could lead to consumer mobility across channels.
In addition, we show that the quantity and quality of information have a considerable impact on
final contract choice. While standard economic theory suggests that the main drivers are consumer
risk-aversion and the nature of the risk, final choices in our experiment are based on the premium andthe probability of loss. Anchoring effects regarding both of these, consistent with obfuscation, canlead to more price competition as insurance companies compete on the premium level.
We here look at compulsory insurance; we may expect some behavioral differences under optional
insurance, in particular regarding selection. The risk-averse are more likely to purchase optional insurance (Corcos et al. (2017)). Also, as in Kuksov and Villas-Boas (2010), we can imagine that theeffect of the number of alternatives under optional insurance might reduce the focal and anchoring
effects, as consumers can decide against purchase.
In order to be able to compare the different distribution channels ceteris paribus, the proposed73

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
contracts are the same in each channel. However, real-world insurers do manage the policies thatthey propose according to the way in which they are distributed. For instance, due to the focal-point
effect we can imagine that loss-leaders will appear on the comparator and that insurers will focus theirefforts on cross-selling and upgrade options. This is particularly true when we take consumer searchcosts into consideration Diamond (1971). In addition, brokers are likely better informed about therange of contracts that are available, and may propose contracts that are more difficult to find.
Our paper has focused on first insurance purchase: we have not looked at renewal and customer
inertia due to switching costs. It would be of interest to adapt this experiment to a repeated gameincluding renewal and switching costs, as in Schram and Sonnemans (2011), as well as retaining thepossibility that consumers explore the market via the different channels.
Last, we uncover evidence of a correlation between customer risk profiles and their acquisition
strategy. Risk aversion and honesty both reflect the individual risk profile. The first could explain
adverse selection while the second could be related to fraud. The consumer’s distribution-channel
strategy could then reflect screening. We would now like to collect claims data from different insurers
who use different distribution channels in order to analyze, ceteris paribus, the differences in insurers’
portfolio loss according to the way in which the insurance contract was acquired.

Acknowledgments
We gratefully acknowledge support from the ISFA (Institut de Science Financière et d’Assurances)andPrevent’ Horizon research chair. This project was started while we were affiliated with the SAF Laboratory of the University Claude-Bernard – Lyon 1, France. We are also grateful to participants at theSecond BEE Workshop (Paris Saclay, ENS Cachan, May 2016), the 3rd EAJ conference (Lyon, September 2016), the North-American ESA Conference (Tucson, November 2016) and the CEAR/MRICBehavioral Insurance Workshop (Munich, December 2016). We are particularly grateful to Richard
Butler, Georges Dionne, Sara Fisher-Ellison, Glenn Harisson, Meglena Jeleva, Martin Kocher, Andreas
Richter,Arthur Schram, Simeon Schudy, François Pannequin and Laurent Denant Boemont for theirhelpful and pertinent comments. Finally, we thank Christophe Dutang for his time and his help asa reviewer, and Matthieu Mouminoux for his work on the development of the experimental interface.
Any errors and opinions are ours.

742.6. Appendix

2.6

Appendix
INSTRUCTIONS
You are taking part in an experiment in the context of the SAF (Sciences Actuarielle et Financières) research program of the ISFA (Institut deScience Financière et d’Assurances) of the University of Claude Bernard Lyon 1.
If you read these instructions carefully you can earn a substantial amount of money. Your final earnings will depend on your decision and thedecisions of the other participants in this session. In any case, you will receive a fixed remuneration of 3€ for your participation. Please note that
average final earnings are well above this amount. Once the experiment is over, we will give you an attestation of payment that you must sign inexchange for your earnings that will be paid directly and privately in cash.
This experimental session consists of three independent parts:The first part involves a potential exchange of coins between a wallet and a padded envelope (empty) that are already on yourtable.
The second part consists of 20 questions.
The third part includes 8 successive independent rounds. In each round, you have to choose an insurance contract in order to beprotected against a known risk.

Your final earnings in Euros is the sum of 4 amounts:

3 Euros for your attendance.
The sum of your earnings in each of the three parts.

Communication is forbidden during all the experimental session. If you do not respect this rule, we will cancel this session and you will not
be paid. For any questions, please raise your hand.
You will receive an instruction sheet and a comprehension questionnaire (if necessary) at the beginning of each part of this experimental
session.

__________________________________________

A three-letter login was placed on your table. Please enter your login in order to access the experimental interface.
__________________________________________

75

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels

First part:
During this first part, you will be assigned a type, either A or B. You will keep this type for the entire duration of this experiment. This typedefines your role in the different parts.
There is a wallet and a padded envelope (empty) on the table of each participant. Please wait for our signal to open the wallet.
The wallet can:ooEither be empty.
Or contain 10 coins of 50 cents (5 Euros in total) and a card showing the result of 10 independent draws. Each draw iscarried out using two balls: green and red.
If your wallet is NOT EMPTY you are a participant of type B:
For this part, we ask you to apply the following rule:For each green ball shown, you can take 0.50€ from the wallet and put these 0.50€ in the padded envelope.

The remaining Euros in the wallet correspond to 0.50€ times the number of red balls.
However, the experimenter and the other participants cannot know if you apply this rule. You are not monitored and all wallets are put
together in the same bag at the end of this part of the experiment.
For this part, your earnings correspond to the amount that you put in the padded envelope.
PLEASE LEAVE THE SMALL CARD IN THE WALLET

Example : If your card is the following :By applying the rule, you:

Take six 0.50€ coins, andLeave four 0.50€ coins in the wallet.

If your wallet is EMPTY you are a participant of type A:
In this part of the experiment, the different draws given to the B participants appear on your screen. You have to indicate for each of these
How much, in Euros, do you think that the B participant left in the wallet?
To calculate your earnings for this part, we randomly select one of these draws and you will receive:
5 € - |your estimation error |Example: If the draw selected is the following :B actually left 1€ (two 0.50€ coins) in the wallet and you had estimated that 2€ (four 0.50€ coins) would be left. You therefore earn:5€ - |your error of estimation| = 5€ - |2€ - 1€| = 5€ - 1€ = 4€76

2.6. Appendix

Second part:In this second part we are going to present numbers in ECU (Experimental Currency Units). This measure is converted into Euros at the endof the session at the following rate: 1 ECU = 0.02€; 1€ = 50 ECU.
In this second part, whatever your type, you have to answer two series of 10 questions. For each question, you should choose one option(A or B).
For this part, we will randomly select one question and your earnings will be calculated according to the outcome of your corresponding
chosen choice.
Example :1st possible case: in the case of a Gain

You should indicate if:
You prefer a 1-in-10 chance of winning 50 ECU and a 9-in-10 chance of winning 20 ECU (Option A)or
You prefer a 1-in-10 chance of winning 85 ECU and a 9-in-10 chance of winning 5 ECU (Option B)

2nd possible case: in the case of a Loss

You should indicate if, from 100 ECU :You prefer a 1-in-10 chance of losing 40 ECU and a 9-in-10 chance of losing 45 ECU (Option A)or
You prefer a 1-in-10 chance of losing 10 ECU and a 9-in-10 chance of losing 80 ECU (Option A)

77

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels

Third part:
This part consists of eight successive independent rounds. The first two rounds are trials and will not affect your earnings. Whatever yourtype, your earnings in this part correspond to your earnings in one of the six remaining rounds, selected randomly.
You are type A:
In each round, you have an initial wealth level, a probability of loss, and an amount of loss. You have to choose an insurance contract inorder to be protected against the loss. It is compulsory to underwrite a contract.
Each contract contains:

A fixed premium (this is the price of the contract).
A deductible that you have to pay in the case of loss.

Example: The loss amount is 1000 ECU with a probability of 17% and initial wealth of 180 ECU, If you select an insurance contract with a premium of 70 ECU and adeductible of 25 ECU:

If the loss occurs (with probability of 17%), you earn: 180 – 70 – 25 = 85 ECU
If the loss does not occur (with a probability of 83%), you earn: 180 – 70 = 110 ECU
There are eight different potential contracts. These contracts are not visible at the beginning of the round, and will partly or fully appear on
your screen as you choose different exploration actions. To do so you have an exploration credit that is used up according to the different
types of exploration you choose. Any unconsumed exploration credit is part of your earnings.
To find out about the potential contracts, you can:78



Explore one of the four menus (A, B, C and D). Each menu consists of 2 contracts. To find out about the contracts in one particularmenu you have to pay 12 ECU (debited from your exploration credit). Once you access this menu, you can ask for a recommendationfrom a type-B participant. Asking for a recommendation costs 4 ECU (debited from your exploration credit). Participants of type B
are informed of a contract-ranking according to your own attitudes towards risk. In addition, they know the bonus associated witheach contract. The B participant receives the bonus associated with the contract if you choose the contract that they recommend
for you. The A participants do not know either the contract ranking or the bonuses.



Look for an advisor (i.e. a type-B participant). This costs 12 ECU (debited from your exploration credit). A type-B participant is
randomly selected and will be informed of a contract-ranking covering six different potential contracts (including contracts fromthree of the four different menus). The ranking is based on your own attitudes towards risk. The B participant knows the bonus
associated with each contract. The A participants do not know either the contract ranking or the bonuses. The B participantsuggests a ranking of three contracts to participant A. If you choose a contract through the advisor, you will pay 14 ECU (debitedfrom your initial wealth).



Explore via a comparator. Access to a comparator costs you 12 ECU (debited from your exploration credit). Six of the eight possible
contracts will then appear on your screen (from three of the four menus). However, only the premium information will appear. To
find out about the associated deductible you have to pay 3 ECU per contract. You do not know the ranking of these contracts.

2.6. Appendix

79Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels

802.6. Appendix

You are type B:
You can be called by a type-A participant in two different ways:

If a type A seeks an advisor. There will appear on the screen of one selected type-B participant a ranking of six different contracts(from three of the four menus) and a bonus associated with each contract. This ranking depends on the premium, the deductible
and the profile of the type-A participant regarding risk.
If you are called here you should suggest a ranking of three contracts to type A. If the type-A participant chooses one of thecontracts suggested, you earn a fixed fee of 14 ECU, paid by type A, and the bonus associated with the chosen contract.



If participant A asks for a recommendation. There will appear on the screen of one selected type-B participant a ranking of twocontracts and a bonus associated with each contract.
If you are called here you should recommend only one contract. If the type-A participant chooses one of these contracts, youearn the associated bonus.

Participant A does not pay the bonus. There is a transfer from A to B only if A chooses a contract through the advisor (a fixed fee of 14 ECU).
For a request from an A participant only one B is randomly selected. It is therefore possible for each B participant to be called once, a numberof times or never during a round.
For each round, your earnings are equal to a fixed remuneration of 120 ECU and the potential additional earnings resulting frominteractions with the A participants. For this part, your final earnings are equal to those in one of the six rounds, randomly chosen.

81Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels__________________________________________

Thanks for your participation__________________________________________
Before leaving the room to receive your payment, please fill out the final questionnaire. Then, please click on “Validate”.
Before leaving your table, please:

Take your three-letter login.



Take all instruction sheets.
__________________________________________

82

2.6. Appendix

Comprehension Questionnaire for Part 1 of the Experiment:
We will mark this questionnaire in a few minutes.
If you are A:
Your wallet contains 5 Euros:

TRUE 

FALSE For each draw, you should indicate the amount in Euros that B left in the wallet:

TRUE 

FALSE For each draw, you should indicate your estimation of the amount that B left in the wallet:

TRUE 

FALSE Your earnings depend only on your estimation:

TRUE 

FALSE Your wallet only contains 5 Euros:

TRUE 

FALSE You have to leave 0.50€ in the wallet for each red ball:

TRUE 

FALSE The experimenter or the A’s know the amount that you left in the wallet:

TRUE 

FALSE No-one knows the amount that you left in the wallet:

TRUE 

FALSE If you are B:

83

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels

Comprehension Questionnaire for Part 3 of the Experiment:
We will mark this questionnaire in a few minutes.
If you are A:
It is possible to return to an exploration design if you have enough exploration credit :TRUE 

FALSE Your final earnings for this part can depend on the first two rounds:

TRUE 

FALSE The comparator displays the premium and deductible of each contract for free:

TRUE 

FALSE Your initial wealth, the amount of loss, the probability of loss and the exploration credit canchange from round to round:

TRUE 

FALSE It is possible to choose a contract without knowing its associated deductible:

TRUE 

FALSE It is possible to choose a contract suggested by the advisor without paying the fixed fee:

TRUE 

FALSE When you choose a contract suggested by the advisor, the fixed fee is debited from yourexploration credit:

TRUE 

FALSE If you do not spend your exploration credit, this amount is added to your earnings:

TRUE 

FALSE It is possible to never be called during a round:

TRUE 

FALSE You have to rank contracts in the same order as displayed:

TRUE 

FALSE The ranking displaying on your screen depends only on the bonus:

TRUE 

FALSE The bonus is paid by A:

TRUE 

FALSE If you are B:84

2.6. Appendix
Table 4 bis – Multinomial Regression with Random EffectExplanatory Variable
First choice:
Coefficients

Model 5 bisBroker

Intraround Channel Switch
Model 6 bis
Model 7 bis
Insurance
Comparator

Comparator

Insurance

Comparator

BrokerInsurance

Broker

−2.00

−0.15

0.76∗∗

0.52

0.31

0.30

1.85

0.74

0.35

2.65

0.28

0.37

−0.53

Std. error

Honesty Beliefs
Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
Round number
Deviation Degree
Deviation Deg. and Honesty Belief

0.05

−0.11

−0.04

−0.05

0.31

0.28

0.28

0.21

0.92

0.16

0.59

0.00

0.00

0.01

0.01

0.00

−0.01

0.00

0.00

0.01

0.02

0.00

0.01

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

3.28

4.47

−1.69

6.62

4.14

6.33

4.68

3.96

4.19

14.47

2.22

2.22

−0.29

−0.05

−0.10

0.74

−0.09

−0.49

0.25

0.24

0.25

2.48

0.16

0.32

1.67

0.21

−0.35

2.09
2.31

−1.83∗∗∗

−0.92

0.47
0.21∗

0.75

0.07

0.14

0.99

0.38

2.45

1.41∗

0.47

−0.41

0.06

0.84

0.29

1.68

0.94

Deductible no. discovered
Deductible no. discovered2
Constant (ref. level: No Switch)
Nb. Observations
Nb. Subjects
R2
Adjusted R2

0.13

−9.12

−4.49

−2.91

−14.8

−5.52∗

−1.14

5.74

2.13

2.04

22.78

1.80

2.48

237
79
0.293
0.221

210
67
0.224
0.132

615
140
0.281
0.256

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’85

Chapter 2. Obfuscation and Honesty: Insurance Demand with Multiple Distribution Channels
Table 9 bis – Logistic Regression with Random EffectExplanatory Variable
Coefficients

Deviation
Model 20

Model 20 bis−0.28∗∗−0.29∗∗

0.09

0.09

Std. error

Risk Aversion (loss)
Initial Wealth
Loss Amount
Probability of Loss
Optimal Contract Bonus
Status: TIED-AGENT
Dishonesty Session Level

0.01

0.01

0.00

0.00

0.00

0.00

0.00

0.00

−2.22

−2.22

1.66

1.73

−0.05

−0.04

0.02

0.02

−1.28∗−1.30∗

0.68

0.68

0.59∗∗
0.26

0.34

Session 1

0.48

−0.26

Session 2

0.66

−0.98

Session 3

1.07

−0.17

Session 4

0.77

−1.07

Session 5

0.73

−0.94

Session 6

0.55

−0.80

Session 7
Round Number
Constant (ref. level: NO)
Nb. Observations
Nb. Subjects
R2
Adjusted R2

0.53

−0.01

−0.01

0.10

0.10

2.66∗∗

2.59∗∗

0.89

0.95

289
40
0.181
0.152

289
40
0.189
0.141

Signif. codes for p-values under Holm–Bonferroni correction: 0.01 ‘*** ’ 0.05 ‘** ’ 0.1 ‘* ’Bibliography
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88

Contributions en sciences actuarielles

89Chapter 3

A game-theoretic analysis of insurers
pricing strategies—
Analyses des stratégies de prix desassureurs dans un jeu non-coopératifThis chapter is based on an article co-written with Hansjoerg Albrecher, Christophe Dutang
and Stéphane Loisel.

91

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Abstract
In this paper, we extent the repeated non-cooperative game of Dutang et al. (2013) to model a nonlife insurance market. We provide market premium equilibrium analysis and its sensitivity according
to solvency constraint as well as consumers’ price sensitivity. We also derive, using Markov chains
methodology, convergence measures of long run market shares and properties of a highly regulated
market. In a second step, we provide numerical illustrations of the repeated one period game andwe analyze market premium equilibrium path according to different price sensitivity parameters, loss
history and market size.
Keywords: game theory, non-cooperative game, consumers’ price sensitivity, solvency constraint,
stochastic behaviors, Markov chains

Résumé
Ce papier est une extension du jeu non-coopératif présenté dans Dutang et al. (2013) afin demodéliser le marché de l’assurance non-vie. Nous proposons une analyse des impacts de la sensibilité
aux prix des consommateurs et des conséquences de la présence d’une contrainte de solvabilité sur lesprimes d’équilibres. Utilisant la théorie des chaînes de Markov, nous calculons des mesures convergentes de la distribution, à long-terme, des consommateurs à travers les différents assureurs. Nous
montrons également les propriétés particulières d’un marché fortement régulé. Dans un second temps,
nous illustrons le jeu répété à l’aide de simulations numériques. Nous analysons les trajectoires deprix de marché et leurs sensibilités vis-à-vis des paramètres d’inertie des consommateurs, des sinistresobservés et de la taille du marché.
Mots-clés: théorie des jeux, jeu non-coopératif, élasticité prix, contrainte de solvabilité, comportements stochastiques, chaînes de Markov.

92

3.1. Introduction

3.1

Introduction

In the insurance world, determining an appropriate and attractive premium is always an important issue because of the strong competition among different insurers. This is particularly true fornon-life retail insurance mass markets, such auto insurance and household insurance, where an important number of insurance companies and mutuals compete. Policyholders are looking for the best
services at a lower price and thus tend to migrate to firms displaying advantageous insurance premium.
Basic economic models suggest that the equilibrium premium is the marginal cost (i.e. actuarial
premium) and in a Bertrand competition (i.e. competition in price), any upward deviation from thisequilibrium will result in losing all the policies in the next period. However, this theory suggeststhat consumers are perfectly mobile and their decisions are only based on premium level. In practice,customers do not move from an insurer to a cheaper one, and we observed inertia of the insurance
demand, preventing an insurer from the Bertrand paradox and null profit. Indeed, customer behavior
is much more complicated. In addition to customer loyalty, explain by insurers’ reputation as well
as search and switching costs (Chapter 2: Mouminoux et al. (2018), Schram and Sonnemans (2011)),
Feldblum (2001) points out that there are barriers for a new insurer to enter the non-life insurance
market, such as solvency constraint, underwriting standard,. . .
The pricing of insurance contracts is a classical research topic. In practice, insurance companies
use various approaches including general principles of premium calculation, based on the expectedclaim expenses or to its moment (average, standard deviation,...), credibility theory and generalized
linear models (GLM), see for instance Kaas et al. (2008).
However, in a highly competitive market which is dominated by a relatively small numbers of firms,
insurance pricing does not reduce to compute technical premium: each insurer attempts to predict
customers’ reactions and takes advantages against the others. The actuarial premium is thus alteredby the marketing and management departments for several reasons, such as the customers’ loyalty andthe market conditions.
Hence, game theory concepts seem to be an appropriate way to evaluate the strategic choices of
insurers in the presence of competition. The use of game theory in actuarial science is not new. The
first attempts go back to Borch (1962, 1974), Bühlmann (1980, 1984), and Lemaire (1984, 1991), whoapplied cooperative games in order to model risk transfer between insurers and reinsurers (see also
others researches and review as Aase (1993), Brockett and Xiaohua (1997), Tsanakas and Christofides(2006), Boonen (2016)). Regarding to non-cooperative games, two types of models have been considered in non-life insurance markets: a) the Cournot oligopoly where insurers’ strategies are based on the
choice of volume of business (see Powers and Shubik (1998), Powers et al. (1998)) and b) the Bertrand
oligopoly where insurers set premiums (see Polborn (1998), Rees et al. (1999) and Dutang et al. (2013)).
In this paper, we attempt to model competition in non-life insurance markets with non-cooperative
game in order to extend the insurer-vs-market reasoning of Taylor (1986, 1987), see also Kliger andLevikson (1998), Emms et al. (2007) and Moreno-Codina and Gomez-Alvado (2008) for extensions.
Extending the model of Dutang et al. (2013), we focus our analysis on consequences of customers’
price sensitivity, in others words customers inertia, as well as consequences of solvency constraints,
on price equilibrium, market share and their consequences on profits. Dutang et al. (2013) modellingseems to be the most appropriated to our analysis since they included a lapse model, an aggregate loss
model as well as a solvency constraint functions for insurers. Integration of stochastic lapse behavior
through a lapse probability function of the premiums offered by all insurers is a best way to avoid any
Bertrand paradox as well as integer consumers’ inertia in a game theoretic approach. In addition, because of strong regulation of insurance market, it is essential to consider in the insurer decision-making
strategies a solvency constraint which will affect premium level as well as long run market patterns.
Indeed, while this paper does not aim to analyze market pricing cycle but focus analysis on one-periodmarket equilibrium according to customer behavior, many researches have demonstrate the presence of93

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

market cycle in different lines of insurance (Rantala (1988), Malinovskii (2010); Emms (2012), Boonen
et al. (2018)). Hence, we also consider a repeated one-period game in order to emphasize convergenceproperties of the market using a Markov chains approach.
The rest of the paper is organized as follows. Section 3.2 remains the one-period game based on
Dutang et al. (2013). Section 3.3 present general properties of the lapse and loss models as well as
short and long run convergence measures of insurers’ market share. A subsection is also dedicated tothe analysis of a strongly regulated market under our customer behavior modelling. We also illustratein this section some results using numerical simulations of the game in a simple market. In Section
3.4, we analytically analyze premium equilibrium of a simple duopolistic market by focusing results oncustomers’ price sensitivity and impacts of comparative advantage of one insurers. We also show whensolvency and prices constraints are actives and their consequences on market equilibrium. In Section
3.5, we display some ruins theory results in order to understand the long run market pattern regarding
to the number of insurers present on the market. In Section 3.6, we provide numerical illustrationof the repeated one-period game by commenting some pattern under stochastic simulations. We also
provide a parameters sensitivity analysis. Finally, a conclusion and perspectives are given in Section
3.7.

3.2

Description of the repeated game

Consider J insurance companies competing on price in a market of N policyholders with one-year
policies, where N is considered constant across periods. In the sequel, a subscript j ∈ {1, . . . , J}
will always denote a player index (i.e. an insurer) whereas a subscript i ∈ {1, . . . , N } denotes aninsured index. Vectors will be bolded. The “game” for each insurer j ∈ {1, ..., J} consists in definepremium xj,t at the beginning of each year t, in order to maximize their profits by selling identicalpolicies to the insured market of size N . Let xt = (x1,t , . . . , xJ,t ) ∈ [x, x]J ⊂ RJ be the insurers
price vector, with xj,t representing premium of insurer j for year t. We thus denote by x−j,t =(x1,t , . . . , xj−1,t , xj+1,t , . . . , xJ,t ) the vector x without the jth component. In this section, we firstdescribe the sequence of the repeated game and then explain each component in details .

3.2.1

Game overview

Game theorie of noncooperative games deals with two main equilibirum type: the Nash equilibrium and the Stackelberg equilibrium. The Nash equilibrium assumes that 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 equilibriumas the most appropriate concept. Indeed, in majority of insurance market, in particular in retail-mass
market, it is difficult to admit that one insurer is always taking its decision at first. Pricing strategies
are generally defined at the beginning of the year, after an update of past year observations such losses
level, market shares at the end of the year, financial market features (i.e. inflation, time-value,...) andshareholders’ expectations for the future year. We thus give below the definition of a Nash equilibrium
and use this approach for the rest of our results.
Definition. For a game with J players, with payoff functions Oj and constraint function gj , a Nashequilibrium is a vector x? = (x?1 , . . . , x?J ), where xj is also constrained by price lower and upper bounds[x, x], such that for all j = 1, . . . , J, x?j solves the subproblem
max

xj,t ∈[x,x]Oj (xj , x?−j ) s.t. gj (xj ) ≥ 0.

where xj and x−j denote action of player j and the other players’ action, respectively.
A Nash equilibrium is interpreted as a point at which each players, given the strategies of others
players, have no incentive to change their strategies. Having reached Nash equilibrium a player will be

94

3.2. Description of the repeated gameworse off by changing its strategy. Note that the constraint function depends only on player action xj
and not on other players’ strategies x−j . Otherwise, we would deal with a generalized Nash equilibrium.
The elements of the objective function Oj,t corresponding to the expected profit of insurer j fortime t and the definition of the constraint function gj,t corresponding to solvency constraints requiredby regulators are detailed in the next subsections.
We define the repeated game as the iteration of expression one-period game over T years. However,since regulator constraints insurers to be solvent, we pulled out them of the market when they have
either a tiny market share (< 0.1%) or a negative capital. In the same way, we remove players fromthe 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). As a reminder, MCR can be defined
as a percentage of the SCR computed in the solvency constraint. In general in non-life insurance retailmarket MCR is between 25% and 45% of the SCR.
Repeated game. For period t = 1, . . . , T , repeat
1. The insurers among Jt maximize their objective function subject to the solvency constraint
supOj,t (xj,t , x−j,t ) s.t. gj,t (xj,t ) ≥ 0.

xj,t ∈[x,x]2. Once the premium equilibrium vector x?t is determined, customers randomly lapse or renew. We
get a realization n?j,t of the random portfolio size Nj,t (x?t ).
3. Aggregate claim amounts Sj,t are randomly drawn according to the chosen loss model and theportfolio size n?j,t .
4. The underwriting result for Insurer j is computed by U Wj,t = n?j,t × x?j,t × (1 − ej,t ) − Sj,t , where
ej,t correspond the rate of handling costs of Insurer j at time t.
5. We update the capital by the following equation Kj,t+1 = Kj,t + U Wj,t .
6. We update the set of competitors Jt+1 by removing bankrupted insurers or too small insurers.
We defined small insurers, those who have a market share lower than 0.1%. We assume that tiny
insurers will not be able to face future losses and handling costs. Indeed, they will not benefit
from mutualisation concepts, essential for insurance market and thus will decide to run-off theirbusiness.
In the current framework and for simplicity, 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 premiums are collected on January 1, (iv) every claim is (fully) paid on
December 31 and (v) there is no inflation and (vi)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 forincurred-but-not-reported claims and (v) there is inflation on both claims and premiums, (vi)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 and tractable model.
In the next subsections, we present in detail the four components of the game: a lapse model, aloss model, an objective function and a solvency constraint function.

3.2.2

Components of the game for policyholders

In the following subsections, we state the precise assumptions made for the behavior of policyholders, both in terms of lapse/renewal probabilities and in terms of loss.

95

Chapter 3.
3.2.2.1

A game-theoretic analysis of insurers pricing strategies

Assumptions for the lapse model

Being with insurer j in the previous period t − 1, the insurer choice of insured i for the current
period t follows an J-dimensional multinomial distribution MJ (1, pj→ (t)) with probability vectorpj→ (t) = (pj→1 (t), . . . , pj→J (t)) where components sum to 1.
It seems natural and it has been verified empirically (Duijmelinck et al. (2015))that the probability to choose an insurer is highly influenced by the choice of the previous period . In other words,
the probability pj→k (t) with k 6= j is generally much lower than the probability to renew pj→j (t).
To our knowledge, only the extreme UK market shows lapse rates for non-life insurance above 50%.
Those probabilities have to depend on the premiums xj , xk proposed by insurer j and k, respectively.
Therefore, we assume that pj→k (t) = pj→k (xt ).
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 a multinomial logit model, since
the probit link function does not really enhance the choice model despite its additional complexity.
Working with unordered choices, we arbitrarily set the insurer reference category for pj→k to j, thecurrent insurer. As in Dutang et al. (2013), 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

pj→k (x) =
1P 1+ efj (xj ,xl )
l6=jefj (xj ,xk ) 1+ P efj (xj ,xl )if j = k,if j 6= k,

(3.1)

l6=j

where the sum is taken over the set {1, . . . , J} and fj is a price sensitivity function. In the following,
we consider two types of price functionsx
f¯j (x, y) = µj + αj and f˜j (x, y) = µ̃j + α̃j (x − y).
y

(3.2)

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 . The two models
are denoted by premium ratio (PR) and premium difference (PD) lapse models. Parameters µj , αjrepresent a base lapse level and price sensitivity.
PJ We assume that insurance products display positive
price-elasticity of demand αj > 0. Clearly k=1 pj→k (x) = 1. The previous expression of pj→k canbe rewritten as
pj→k (x) = pj→j (x) δjk + (1 − δjk )efj (xj ,xk ) ,with δij denoting the Kronecker product. This representation helps in deriving analytic properties ofthe lapse model.
We use the following notation for year t. For customers, Ci,t is the choice of customer i at time
t take value in {1, . . . , J}, and is randomly chosen according to probabilities pj→l (t) knowing that
Ci,t−1 = j. For instance, (Ci,t )t = (1, 1, 1, 2, 2, 3, . . . ) indicates that customer i chooses Insurer 1 forperiod 1, 2, 3 and Insurer 2 for period 4,5; etc. . .
The following assumptions are made— A1: The customer choice of insurer at time t depends only on the previous choice at time t − 1.
— A2: Customers are independent, i.e. (Ci,t )i are independent variables.
— A3: Customer behaviors are identical over time, i.e. pj→k (xt ) depends on t through premiumxt .
— A4: No customer can enter or exit the market, the total market size N is constant.

96

3.2. Description of the repeated game

Hence, being with current insurer j during the t − 1 period, the insurer choice Ci,t of insured i forthe year t follows an J-dimensional multinomial distribution MJ (1, pj→ (xt )) with probability vectorpj→ (xt ). The probability mass function is given by P (Ci,t = k | Ci,t−1 = j) = pj→k (xt ).
In fact, the random choice (Ci,t )t of policyholder i is governed by a (discrete-time) Markov chain
with transition matrix
p1→1 (xt ) . . . p1→J (xt )4 ..
P→ (xt ) = ,.
pJ→1 (xt ) . . .

pJ→J (xt )see Proposition 3.3.1 for details. In general, the Markov chain is time-inhomogeneous as P→ depends
on the price vector xt which evolves over time. Note that using Markov chains to study customer
behavior was also done in Marker (1998), where a special duopolistic situation is considered: one
insurer versus the market.
From an insurer point of view, we are interested in the number of policyholders at time t. Let
nj,t−1 be the portfolio size at the previous period t − 1. We define C j,t = (C j,1,t , . . . , C j,J,t ) as therandom assignment of customers of Insurer j at time t, where C j,k,t denotes the (random) number ofpolicyholders moving from Insurer j to Insurer k.
Therefore, policyholders of Insurer j will choose insurers according to a multinomial
 distribution
C j,t ∼ MI (nj,t−1 , pj→ (xt )) given the portfolio size nj,t−1 . Let N t = N1,t , . . . , NJ,t be the vectorof (random) portfolio sizes at time t. They are obtained by summing the choices of each Insurers’
customers




 
N1,t
C 1,1,t
C J,1,t
J
X




 
C j,t =  ...  + · · · +  ...  =  ...  .
Nt =
j=1
NJ,t
C 1,J,tC J,J,tIn other words, the portfolio sizes are obtained a sum of J independent multinomial vectors N t =C 1,t + · · · + C J,t . We denote by nt = n1,t , . . . , nJ,t the realizations of the random vector.
PJ
For Insurer j, his portfolio size Nj,t is a sum of independent variables: Nj,t = k=1 C k,j,t , where
C j,j,t corresponds to renewed policies and the rest of the sum corresponds to new business comingfrom other insurers. It is important to note that the insurers’ portfolio sizes are not independent, since
the total market size remains constant (A4).
For instance, consider a three-insurer market with n0 = (100, 50, 60). C 1,1 = (70, 20, 10) denotesthat 70 customers choose to renew, 20 go to Insurer 2, 10 to Insurer 3. If in addition C 2,1 = (18, 30, 12)
and C 3,1 = (2, 8, 50), then n1 = (90, 58, 74) is the realized portfolio size at time 1, i.e. Insurer 1 gets
90 customers, Insurer 2 gets 58 and Insurer 3 gets 74.
Finally, the process (N t )t is a J-dimensional discrete-time Markov process, see Proposition 3.3.4
for details. The process (N t )t takes values in the set of portfolio sizes


J


X
SN = n ∈ NJ ,nj = N ,
j=1

which has

N +J−1Nelements, see e.g. Breuer and Baum (2005).

The lapse/renewal process of policyholders at each time period considered can be seen as a closed
Markovian network of discrete-time queues with batch services (see e.g. (Boucherie and van Dijk, 2011,

97

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Chap. 6)), for which the service time corresponds to the number of years a policyholder stays withthe same insurer. In that context, pj→k are called routing probabilities, an insurer is a server and apolicyholder is a customer.

3.2.2.2

Assumptions for the loss modelNow we turn our attention to the loss model faced by policyholders. Let Yi,t be the aggregate lossof policy i during the coverage period t. The following assumptions are made— A5: There is no adverse selection, i.e. Yi,t are independent and identically distributed (i.i.d.)random variables, ∀i = 1, . . . , N .
— A6: Catastrophic events are excluded and Yi follows a frequency – average severity loss model.
— A7: The insurance business is short-tailed, i.e. the loss Yi is paid in total on December 31 ofeach year.
Assumption A5 allows us to simplify the simulation process because the i.i.d. assumption implies
that individual losses Yi,t do not need to be simulated. The aggregate claim amount per insurer Sj,tcould be simulated directly. Assumption A6 means that we assume a simple frequency – average
severity loss model
Mi,tX
Yi,t =
Zi,l,t ,l=1

where the claim number Mi,t is independent of the claim severity Zi,l,t . Therefore, the aggregate claim
amount for insurer j is
Nj,t (xt )X
Sj,t (xt ) =Yi,t ,
i=1

where Nj,t (xt ) is the portfolio size of insurer j given the price vector xt . Assumption A5 means thatboth claim numbers Mi,t and claim amounts Zi,l,t are i.i.d. random variables. As already mentioned,
we focus on short-tail business and so Assumption A7 is in line with our objective of modelling personal business lines.
For the numerical illustrations, we assume two different distributions for both claim frequencies andclaim severities. Claim numbers follow either a Poisson or a negative binomial distribution, whereas
claim amounts follow either a lognormal or a gamma distributions. Hence, we consider four different
cases for the loss model. Taking into account the time t, we obtain the following cases
i.i.d.
— PG: a Poisson-gamma with Mj,t ∼ P(λ) and (Zj,l,t )l ∼ G(µ1 , σ1 ),i.i.d.

— NBG: a negative binomial-gamma with Mj,t ∼ N B(r, p) and (Zj,l,t )l ∼ G(µ1 , σ1 ).
i.i.d.

— PLN: a Poisson-lognormal with Mj,t ∼ P(λ) and (Zj,l,t )l ∼ LN (µ2 , σ22 ),i.i.d.

— NBLN: a negative binomial-lognormal with Mj,t ∼ N B(r, p) and (Zj,l,t )l ∼ LN (µ2 , σ22 ).
fj,t = PNj,t (xt ) Mi,t be the total number of claims for Insurer j in period t. We assume that
Let M
i=1
f1,t , . . . , M
fI,t ) is a vector with independent margins.
(M
We could have considered correlated loss models by introducing dependence either among claim
fj,t )j at portfolio level or claim frequency (M1,t , . . . , MN,t ) or on the claim severity (Zj,i,1 , . . . , Zj,i,m )numbers (Miat individual level. This procedure is not considered here.

3.2.3

Components of the game for insurers

3.2.3.1

Objective function

In Section 3.2.2.1, we assumed that the price elasticity of demand for the insurance product ispositive. Thus, if the entire market underwrites below costs, any action of a particular insurer to get

983.2. Description of the repeated gameback to profitability will result in a reduction of his business volume. This has two consequences forthe choice of the objective function: (i) it should involve a decreasing demand function of price xj,tgiven the competitors’ price vector x−j,t and (ii) it should depend on an assessment of the insurer
break-even premium πj,t per unit of exposure.
Insurers have a history of past premium levels x?j,t , gross written premium GWPj,t , portfolio size
nj,t and capital Kj,t at the beginning of year t. Let d be the past number of years considered for which
economic variables (e.g. market premium) will be computed.
The parameter πj,t corresponds to the estimated mean of overall costs including handling costs
and claims and depends on the assessment of the loss expectation by insurer j. We thus define πj,t as
πj,t = ωj āj,t−1 + (1 − ωj )mt−1 ,(3.3)

where, at the beginning of each time period, the average market premium is determined asd1X
mt−u , with mt−u =mt−1 =
d u=1

PJ?
j=1 GWPj,t−u × xj,t−u

GWP.,t−u,PIwhich is the mean of last d market premiums and GWP.,t−u = j=1 GWPj,t−u . With current portfolio
size nj,t−1 and initial capital Kj,t−1 , each insurer computes its actuarially based premium asāj,t−1 =d
1 X Sj,t−u
1,
1 − ej,t d u=1 nj,t−u| {z }
avg ind loss

where Sj,t denotes the observed aggregate loss of insurer j during year t and ej,t denotes the expense
rate as a percentage of gross written premium. Note that the claim amount is not adjusted against
large claims (i.e. yi,t are not capped).
aj,t represents the actuarial premium based on the past loss experience of Insurer j, mt is the market premium, available for instance, via rating agencies or through insurer associations and ωj ∈ [0, 1]
is the credibility factor of insurer j. ωj reflects the confidence of insurer j in its own loss experience:the closer to 1, the more confident insurer j is. Note that πj takes into account expenses implicitly
via the actuarial and the market premiums.
As in Dutang et al. (2013) ,we choose the demand function asnj,t−1xj,t
Dj,t (xt ) =1 − βj−1,N
mj (xt )(3.4)

where βj > 0 is the elasticity parameter and mj (x) is a market premium proxy. Indeed, we assume
that the insurance product is a normal product where price elasticity of consumers is negative. In thisform, Dj (xt ) approximates the expected market share E(Nj,t (xt ))/N presented in Section 3.2.2.1.
As the elasticity parameter βj is positive, a premium increase (of insurer j) will result in a decreaseof the demand for insurance. The market proxy used in Equation (3.4) is the mean price of the othercompetitors
1 X
mj (x) =
xk .
(3.5)
J −1
k6=j

The market proxy aims to assess other insurers’ premiums without specifically targeting one competitor. It can be interpreted as the premium of an ideal medium competitor. Consequently, Insurer j
typically does not target the cheapest, the most expensive or the leader insurers. From a mathematical99

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

point of view, we will lose the continuity of the demand function if we choose the cheapest premiumxj− 1 in the demand function is closed related to the average
mink6=j xk . Furthermore, the term mj (x)of the relative premium differences since
1 X xk1 1 X
J −1
mj (x)(− 1) =
xk −
=
− 1.
J −1
xj
xj J − 1
J −1
xj
k6=j

k6=j

Consider now an alternative market proxy value depending on the previous period market share of
insurers. Hence, firms do not attribute the same weight to each competitor. We thus consider thatfirms put more importance on the biggest competitors’ price. In the following, we analyze the following
market proxy being the weighted mean of other competitors’ prices
mj (x, n) =X1
nk xk .
N − nj

(3.6)

k6=j

Now we state the objective function defined as



nj,t
xj,t
Oj,t (xt ) =1 − βj−1(xj,t − πj,t ) ,N
mj (xt , nt )(3.7)

where πj,t is the break-even premium j in (3.3) and mj (xt , nt ) is either (3.5) or (3.6).
The objective function Oj,t defined as the product of a demand function and an expected profit per
policy represents a company-wide expected profit. Thus, maximising the objective function Oj,t leads
to a trade-off between increasing premium to favour higher projected profit margins and decreasing
premium to defend the current market share. Note that Oj,t has the nice property to be infinitely
differentiable.

3.2.3.2

Constraint function

In our game context, we want to avoid the simplistic Solvency I framework and follow the Solvency
II framework. However, we still want to keep the tractability for the SCR computation rule. We recall
that the aggregate claim amount is assumed to be a frequency - average severity model and catastrophic
events are excluded. 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,
(3.8)
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 the second term is related
to the standard deviation.
Numerical experiments show that the normal approximation is less conservative for high quantiles(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 corresponding respectively to the computation of the MCR andthe SCR. Thus, using the approximation (3.8), the minimum and solvency capital requirement (MCR
and SCR) is deduced as√
SCRq ≈ kq σ(Y ) n.
Therefore, we decide to choose the adapted solvency constraint function
1
gj,t(xj,t ) =100

Kj,t−1 + nj,t−1 (xj,t − πj,t )− 1,√
k995 σ(Y ) nj,t−1(3.9)

3.3. Properties of policyholder behaviors and consequences on insurer portfolio sizes and losses
where k995 is the solvency coefficient. Indeed, insurers are constrained to have capital above the SCR.
When insurers get capital between the MCR and SCR, regulator completely monitor them while whencapital drops down the MCR, regulators constraint them to run-off their business. 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,t − πj ) ≥ k995 σ(Y ) nj , but gj1 is normalized with respect to capital, providing a better
numerical stability.

3.3

Properties of policyholder behaviors and consequences on
insurer portfolio sizes and losses

3.3.1

Properties of the lapse model

In this first subsection, we study the theoretical properties of the lapse model at both policyholder
level and insurer level. We derive three propositions for the process (Ci,t )t of Policyholder i as well as
two propositions for the process (Nj,t )t of Insurer j. All proofs are postponed to Appendix 3.8.1.

3.3.1.1

Properties of the lapse model at policy level

For a single policyholder i, the following results show that the insurer choices (Ci,t )t of that
customer follow a time-inhomogeneous Markov chain.
Proposition 3.3.1. The choice (Ci,t )t of customer i at time t is a time-inhomogeneous Markov chain.
(t)= P→ (x1 ) × · · · × P→ (xt ) is the transition matrix of Ci,t knowing Ci,0 . The Markov
Therefore, P→
chain (Ci,t )t has an invariant measure.
This result can be extended to all policyholders simultaneously:
Proposition 3.3.2. The choice vector (C1,t , . . . , CN,t )t of all customers at time t is a time-inhomogeneous
Markov chain with transition matrix P→ (xt )⊗n (the n-times Kronecker product of the matrix P→ (xt )).
From this, the distribution of policyholders for each insurers can be defined.
Proposition 3.3.3. For all t ∈ N? , C j,t ∼ MI (nj,0 , p̃j ) given Nj,0 = nj,0 where p̃j is the jth row of(t)the matrix P→. In particular, C j,t ∼ MI (nj,t−1 , pj→ (xt )) given Nj,t−1 = nj,t−1 .

3.3.1.2

Properties of the lapse model at portfolio size level

Following Prop. 3.3.3 and Prop. 3.3.1, we show that insurer portfolio sizes (N t )t is a (multidimensional) Markov chain in the set of all possible portfolio sizes SN . Then we also characterize theconditional distribution N t | N t−1 = n which in the limit is a multinomial distribution, see Prop. 3.3.4.
In Prop. 3.3.5, we derive the conditional distribution of a particular portfolio size Nj,t = mj |N t−1 = n
of Insurer j. It is difficult to derive other general properties of the distribution of a sum of multinomial or binomial variables with different probabilities pl→j , except when the size parameters nj arereasonably large, in which case the normal approximation is appropriate.
Proposition 3.3.4. The insurer portfolio size vector (N t )t is a time-inhomogeneous Markov chain
with state space SN . Let µ be the invariant measure of (Ci,t )t (policy level). N t | N = n tends to a
multinomial distribution MJ (N, µ); and the invariant measure of (N t )t (insurer level) is the vector
with all probabilities of that multinomial distribution MJ (N, µ).
The probability generating function of N t | N t−1 = n is tractable and is given bynGP
N t |N t−1 =n (z) = (P→ (xt ) × z) ,101

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

where z ∈ RJ and the exponentiation n is a component-wise operation on the vector P→ (xt ) × z. Notethat N t | N t−1 = n does not follow a multinomial distribution unless the rows of P→ (xt ) become
identical. This is easy to show at least asymptotically.
Similarly, the probability generating function
n(t)of N t | N 0 = n is GP
(z)
=
P
×
z
.
→
N t |N 0 =n
Finally, the probability mass function is
P (N t = m|N t−1 = n)X=0≤c11 ,...,c1J ≤N,
Ps.t.

lwhere N =c1l =n1

J
Y

X

...

0≤cJ1 P,...,cJJ ≤N, P j=1,
s.t.

lcJl =nJ

k

nj !
c
c
(pj→1 (xt )) j1 . . . (pj→J (xt )) jJ ,cj1 ! . . . cjJ !

ckj =mjP

i ni .

Proposition 3.3.5. The distribution of Nj,t = mj |N t−1 = n has probability mass functionX
Y nl n −c
P (Nj,t = mj |N t−1 = n) =(pl→j (xt ))cj (1 − pl→j (xt )) l j ,cj(3.10)

0≤c1 ,...,cJ ≤n l∈JPs.t.

lcl =mjand probability generating functionn1
GP× · · · × (1 − pJ→j (xt ) + pJ→j (xt )z)nJ .
Nj,t |N t−1 =n (z) = (1 − p1→j (xt ) + p1→j (xt )z)P
In particular, E(Nj,t |N t−1 = n) = nj × pj→j (xt ) + l6=j nl × pl→j (xt ).

3.3.1.3(3.11)

Additional properties for a constant regulated price vector

Now we turn our attention to the special case of a strongly regulated market. In a regulated
market, premiums are either set by regulators to a particular level or subject to an upper and lower
bound. Insurance price regulation has important consequences on competition, capital and thus on
insurers’ solvability (Klein et al. (2002)). Using convergence measure, we investigate long-run market
share distribution among insurers benefiting from different consumers demand patterns. In our setting,
we focus on the case where regulators set the premium at a single level (for a given line of business).
Since premiums are the same for all insurers, premium level defined by the regulators does not haveconsequences on our lapse model, i.e. ∀t > 0, xt = (x, . . . , x), and that the regulated price evolves fromyear to another, i.e. x1 = (x, . . . , x) x2 = (y, . . . , y) x3 = (z, . . . , z),. . . . Indeed the price sensitivity
functions remains the same: both f j (xj , xl ) = µj + αj and fej (xj , xl ) = µ̃j from (3.2).
In this subsection, we assume that the price vector is a constant vector xt = (x, . . . , x) and timedependence on t is thus omitted. Thus, pj→k simplifies to1
P
if j = k, (1
 1+ efj
if j = k, p
if j = k,l6=j1+(J−1)efj
j→j
fj
fj
pj→k (x) ===(3.12)eepif
j 6= k,

P
if
j
=
6
k,
if
j
=
6
k,
j6
=
f
f
 1+ e j1+(J−1)e j
l6=j

Note that this expression is only function of j (and not of k). Since(1 − pj→j )/(J − 1).

P

k pj→k= 1, we get pj6= =The two following results are directly derived from Propositions 3.3.4 and 3.3.5 in the general case.
Proposition 3.3.6. N t |N t−1 = n has the following transition probabilities
X

P (N t = m|N t−1 = n) =0≤c11 ,...,c1J ≤N,
P
s.t.

l

102

c1l =n1

J
Y

X

...

0≤cJ1 P,...,cJJ ≤N, P j=1,
s.t.

lcJl =nJkckj =mjnj !
n −cpcjj p j jj ,cj1 ! . . . cjJ ! = j6=(3.13)

3.3. Properties of policyholder behaviors and consequences on insurer portfolio sizes and losses=
where p= = pj→j , pj6= = 1−p
J−1 .

Proposition 3.3.7. Nj,t |N t−1 = n is a sum of two binomially distributed random variables B(nj , p= )=and B(n − nj , 1−pJ−1 ). In particular, given N t−1 = n, the probability generating function isn−

P (Nj,t = mj |Nj,t = nj ) =

 
j
X
nj
k=0+

k

pk= (1 − p= )nj −k
n − nj mj −kn−nj −mj +kp(1 − pj6= ), (3.14)mj − k j6=−+=
where p= = pj→j , pj6= = 1−p
J−1 , 0 = max(0, (mj − (n − nj )) and nj = min(nj , mj ).

The two following propositions at policy level are derived from the corresponding propositions in
the general case.
Proposition 3.3.8. The choice (Ci,t )t of customer i at time t is a time-homogeneous Markov chain(t)when xt = x. In particular, P→= (P→ (x))t . There exists a unique invariant measure µ for (Ci,t )tgiven by
J


Y
cΠ
cΠ
J
1
with cΠpj6= .
(3.15)µ = cΠ +···+c
Π , . . . , cΠ +···+cΠ
i =
1

J

1

J

j=1,j6=i

If in addition, the choice probabilities pj→k are identical for all Insurer j, then µ = (1/J, . . . , 1/J).
Proposition 3.3.9. The choice vector (C1,t , . . . , CN,t )t of all customers at time t is a time-homogeneous
Markov chain when xt = x. The invariant measure is obtained by applying N times the Kronecker
product µ⊗N of µ defined in Equation (3.15). If, in addition, the choice probability pj→k are identicalacross insurers, then µ⊗N = (1/J N , . . . , 1/J N ).
Therefore, we can deduce two propositions at portfolio level from the two previous propositions.
Proposition 3.3.10. For all t ∈ N? , C j,t ∼ MJ (nj,0 , p̃j ) given Nj,0 = nj,0 where p̃j is the jth rowof the matrix (P→ )t . In particular, C j,t ∼ MJ (nj,t−1 , pj→ (x)) given Nj,t−1 = nj,t−1 .
Proposition 3.3.11. The portfolio size vector (N t )t at time t is a time-homogeneous Markov chain
with state space SN . The probability generating function of N t | N t−1 = n is GP
N t |N t−1 =n (z)nJn= (P→ × z) , where z ∈ R and the exponentiation is a component-wise operation.
Let µ be the invariant measure at individual level in (3.15). N t | N 0 = n converges in distribution
to a multinomial distribution with parameters MJ (N, µ). Therefore, the invariant measure of (N t )t(insurer level) is the vector with all probabilities of that multinomial distribution MJ (N, µ).
In other words for a constant price vector x and a large t, the distribution of N t is independentof N 0 and for n ∈ SN , the probability mass function is
P (N t = n) =N!
µn1 . . . µnJ J with µ defined in (3.15).
n 1 ! . . . nJ ! 1(3.16)

Furthermore, E(Ni,t ) = µi N and V ar(Ni,t ) = µi (1 − µi )N . The last equation is in line with theso-called product form solution of a closed Markovian network of queues with batch services. In fact,
Theorem 2 of Henderson et al. (1990) show a similar form to (3.16), where the invariance conditionµT = µT P→ is called the balance equation.
In the special case of identical insurers leading to µ = (1/J, . . . , 1/J), we further obtain
N!
P (N t = n) =n1 ! . . . nJ !

 N
1
N
N (J − 1), E(Ni,t ) = , V ar(Ni,t ) =
.
I
J
J2

103

Chapter 3.

A game-theoretic analysis of insurers pricing strategies(a) Transition matrix for a policyholder(b) Invariant measure

Figure 3.1 – The Markov chain of a single policyholder (Ci,t )t3.3.1.4

Illustration purposes of a simple market

For illustration, let us consider a small three-insurer market with 6 policyholders. For a single
client, the Markov chain (Ci,t )t is plotted in Figure 3.1a.
The lapse/renewal function (3.1) is computed using a price sensitivity function f˜j (x, y) in (3.2).
The lapse probabilities have been chosen such that the central lapse rates are 10%, 14% and 18%
respectively. If all insurers propose the same premium (1.25 when 1 is the pure premium), and lapse
rates increase by 5 points when premium increase by 5%. These lapse parameters correspond to
those used in Dutang et al. (2013). In this setting, we assume that Insurer 1 has a better reputation(solvency, brand value) than Insurer 2 and Insurer 3 so that Insurer 1 retains more customers thancompetitors (when premium are identical). As plotted in Figure 3.1a, policyholders of Insurer 1 havea renewal probability of 90% and lapse towards Insurer 2 or Insurer 3 with a probability of p16= = 5%.
From Equation (3.15), the invariant measure for a single policyholder rewrites as
p16= p36=
p16= p26=
p26= p36=µ = p26= p36= +p16= p36= +p16= p26= , p26= p36= +p16= p36= +p16= p26= , p26= p36= +p16= p36= +p16= p26= .
Considering the parametrization p16= = 0.05, p26= = 0.07, p36= = 0.09 leads to µ= (0.441, 0.314, 0.245).
Figure 3.1b, depicts a barplot with the invariant measure of a (single) policyholder. For instance, inthe long run, the probability of a policyholder to be with Insurer 1 is 44.1%. This convergence isrelatively quick. Below, we compute the transition probability (at policy level) after 10, 20 and 30
periods. We observe the geometric convergence towards µ.






0.520 0.263 0.218
0.452 0.307 0.241
0.442 0.314 0.244
(10)
(20)
(30)
P→
= 0.368 0.390 0.242 , P→
= 0.429 0.324 0.246 , P→
= 0.439 0.316 0.245 .
0.392 0.312 0.296
0.434 0.317 0.249
0.440 0.315 0.245
The set of portfolio sizes isSN = (n1 , n2 , n3 ) ∈ N3 , n1 + n2 + n3 = 6 ,104

3.3. Properties of policyholder behaviors and consequences on insurer portfolio sizes and losseswhich has 6+3−1= 28 elements. In other words, all possible portfolio sizes are integer-valued 36
component vectors that sum up to 6. All possible portfolio sizes are given in Table 3.1. The Markov
chain of insurers portfolio sizes (N t )t is plotted in Figure 3.2.
As already mentionned, all states of the Markov chain of insurers portfolio sizes (N t )t are communicating and recurrent. Therefore, any probability to go from n = (n1 , n2 , n3 ) to m = (m1 , m2 , m3 )is strictly positive for n, m ∈ SN . For readability, we do not plot all transition probabilities of the
28 × 28 transition matrix of (N t )t in Figure 3.2. The full transition matrix is given in Appendix
3.8.1.5. In Figure 3.2, we set a threshold of 5% and plot only arrows with the probability value abovethat threshold. In Appendix 3.8.1.5, we plot the transition matrix with a threshold of 1%, see Figure
3.12. To emphasize the highest probabilities, the thickness and the darkness of an arrow are increasing
with its probability value, see the legend of Figure 3.2.
Irrespective of the state, we observe that the highest transition probabilities are the probabilities
to remain in the same state, that is policyholders’ moves are neutral for Insurers. However, for thestate (0, 0, 6), the probability to remain in that state is only 39% whereas for the state (6, 0, 0), it is59%. This is a direct consequence of pi6= since Insurer 1 is the preferred insurer.

6,0,0
5,1,0
4,2,0
3,3,0
2,4,0
1,5,0
0,6,0

5,0,1
4,1,1
3,2,1
2,3,1
1,4,1
0,5,1
4,0,2

3,1,2
2,2,2
1,3,2
0,4,2
3,0,3
2,1,3
1,2,3

0,3,3
2,0,4
1,1,4
0,2,4
1,0,5
0,1,5
0,0,6

Table 3.1 – All possible portfolio sizes with 3 insurers and 6 policyholders
Using Equation (3.13), we compute the invariant measure of insurers portfolio sizes (N t )t . The
invariant measure is plotted in Figure 3.3. Colors for the state in Figure 3.2 are chosen such that thedarkest (from light blue to slate blue) colors correspond to the most probable states according to theinvariant measure of (N t )t . According to the invariant measure, the top 5 most probable states are(4, 1, 1) (2, 3, 1) (3, 1, 2) (2, 2, 2) (3, 2, 1). In these states, Insurer 1 is either an absolute leader (portfolio
size greater than 2) or a co-leader (all insurer of size 2) except the fourth most probable state (2, 3, 1)
see Table 3.2.

105

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.2 – The Markov chain of insurers portfolio sizes (N t )t (transition probabilities lower
than 5% are not plotted and line thickness increases the probability value)1063.3. Properties of policyholder behaviors and consequences on insurer portfolio sizes and losses

Ins. 1
Ins. 2Ins. 3

Leadershipstatesnb. states(6,0,0);(5,1,0);(4,2,0);(5,0,1)
8(4,1,1);(3,2,1);(4,0,2);(3,1,2)(2,4,0);(1,5,0);(0,6,0);(2,3,1)
8(1,4,1);(0,5,1);(1,3,2);(0,4,2)(2,1,3);(1,2,3);(2,0,4);(1,1,4)
8(0,2,4);(1,0,5);(0,1,5);(0,0,6)

cum. prob.
0.46
0.22
0.13

states(3,0,3);(3,3,0)(2,2,2)(0,3,3);(3,3,0)(2,2,2)(0,3,3);(3,0,3)(2,2,2)co-leadershipnb. statescum. prob.

3

0.18

3

0.17

3

0.14

Table 3.2 – Long-term leadership and co-leadership probabilities by insurer

Figure 3.3 – The invariant measure of insurer portfolio sizes (N t )t3.3.1.5

Additional properties for a time-independent price vector

Finally, we consider the case of a deviation from a regulated price or a market-accepted level by
one insurer, yet the other competitors remain at the same level. Therefore, we restrict our analysis toxt = (x, ρx, . . . , ρx) with ρ > 0 a fixed parameter. Note that the case where x = (x1 , . . . , xJ ) without
assuming that some insurers propose the same price is very complex and should be solved numerically.
We cannot use the simplification of transition probability pj→k in this case, yet the transition matrix
P→ will be a circulant matrix.
The dependence on time t is omitted. In that case using (3.2), we have two cases to consider either

α1
x1

 f 1 (x1 , xl ) = µ1 + α1 xl = µ1 + ρxjl 6= 1, f j (xj , xl ) = µj + αj xl = µj + αj
 l = 1, f (x , x ) = µ + α xj = µ + α ρ
j
1
j
j x1
j
j
j



 fe1 (x1 , xl ) = µ1 + α1 (x1 − xl ) = µ1 + α1 (1 − ρ)xorl 6= 1, fej (xj , xl ) = µj + αj (xj − xl ) = µj l = 1, fe (x , x ) = µ + α (x − x ) = µ + α (ρ − 1)x.
j
j
1
j
j
j
1
j
j
(3.17)
So there are three possible exponents denoted respectively f1 , fj , fj,ρ . Therefore, pj→k is only afunction of j and not of k and simplifies to

p1→j (x) =
1P 1+ ef1
l6=j
ef1


 1+ P ef1

if j = 1,
if j 6= 1,(=

1
1+(I−1)ef1
ef1
1+(I−1)ef1

if j = 1,
if j 6= 1,


=

p1→1
p16=

if j = 1,
if j 6= 1,(3.18)

l6=j107

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

pj→l (x) =


1

f
f

 1+(I−2)e j +e j,ρefj
fj

1+(I−2)e


efj,ρ

f

+efj,ρ

1+(I−2)e j +efj,ρ

if j = l,pj→j
if j 6= l 6= 1, = pj6=

pj→1
if j 6= 1, l = 1,

if j = k,if j 6= l 6= 1,
if j 6= 1, l = 1,

(3.19)

Now we are able to derive the invariant measure in the following proposition for the case xt =(x, ρx, . . . , ρx).
Proposition 3.3.12. The choice (Ci,t )t of customer i at time t is a time-homogeneous Markov chain(t)when xt = x. In particular, P→= (P→ (x))t . There exists a unique invariant measure µ for (Ci,t )tgiven by
dΠ
−1 −

J
P
j=2

µ1 =
dΠ
−1 +

J
P
j=2

dΠ
−1,−j pj6=
, µj =dΠ−1,−j (p16= − pj6= )dΠ
−1,−j p16=
, j = 2, . . . , J.
J
P Π
Π
d−1 +
d−1,−j (p16= − pj6= )(3.20)

j=2

QJ
QJ
Π
with dl = (J − 1)pl6= + pl→1 and dΠ−1,−j =
l=2,l6=j dl , d−1 =l=2 dl .
p2→1
In the special case of identical insurers, the invariant measure becomes µ1 = p2→1 +(J−1)p
,
16=
p16=
µj = p2→1 +(J−1)p16= for j = 2, . . . , I.
Let us analyze the case ρ > 1, i.e. Insurer 1 is cheaper than others. From (3.17), we deduce thatfj < fj,ρ . Therefore, we can order the transition probabilities
efj < efj,ρ ⇒ ∀l, j 6= 1, pj6= = pj→l < pj→1 .
In order to easily compare transition probabilities, we further assume that insurers are identical, withlapse parameters µj = µ1 and αj = α1 . So f1 < f2 < f2,ρ yields
 fe 2,ρ > ef1⇒ ef2,ρ + (I − 2)ef1 +f2,ρ > ef1 + (I − 2)ef1 +f2
ef1 +f2,ρ > ef1 +f2⇔ ef2,ρ (1 + (I − 1)ef1 ) > ef1 (1 + (I − 2)ef2 + ef2,ρ ) ⇔

ef1
ef2,ρ
>
.
f
f
2,ρ
2
1 + (I − 2)e + e
1 + (I − 1)ef1
p

p2→1
16=
Since the invariant measure simplifies in that case to µ1 = p2→1 +(I−1)p, µj = p2→1 +(I−1)p, we have16=
16=µ1 > µj for j = 2, . . . , I.

Proposition 3.3.13. The portfolio sizes (vector) (N t )t at time t is a time-homogeneous Markov
chain with state space SN . Let µ be the invariant measure at individual level in (3.20). N t | N 0 = n
converges in distribution to a multinomial distribution with parameters MJ (N, µ). Therefore, theinvariant measure of (N t )t (insurer level) is the vector with all probabilities of that multinomial distribution MJ (N, µ).
Using the same setting as in Section 3.3.1.4, we illustrate this result. Consider a premium vectorx = (1.25, 1.25, 1.1), that Insurer 3 (the least preferred insurer in the previous setting) proposes thelowest premium, ρ = 1.136364. We can then apply Propostion 3.3.12 to derive the invariant measure.
Note that it is not Insurer 1 who deviates from ρx but Insurer 3.
From Equation (3.20), taking into account the index change (i.e. d−3,−2 = d1 = 2p16= + p1→3 ,d−3,−1 = d2 = 2p26= + p2→3 , d−3 = d1 d2 ) the invariant measure for a single policyholder rewrites as
µ3 =

108

(2p16= + p1→3 )(2p26= + p2→3 ) − p16= (2p26= + p2→3 ) − p26= (2p16= + p1→3 )
P
,
i µi

3.3. Properties of policyholder behaviors and consequences on insurer portfolio sizes and lossesµ2 =p36= (2p16= + p1→3 )p36= (2p26= + p2→3 )PP, µ1 =.
i µi
i µi

Numerically, we get
0.82 0.05 0.14
0.295
P→ = 0.06 0.78 0.15 , µ = 0.228 .
0.05 0.05 0.91
0.477
Therefore, Insurer 3 is largely preferred by policyholders both in short-term and long-term horizons.
Using P→ , we can also compute the transition matrix of insurers’ portfolio size (N t )t . And usingµ, we can again compute the invariant measure of (N t )t . In Figure 3.4, we plot the transition matrix
with the same graphic procedure. At one-period level, Insurer 3 is the most preferred insurer: moststates on the left-hand of Figure 3.4 have darkest colors. This contrasts with the situation in Figure 3.2.
The invariant measure is plotted in Figure 3.5 and the leadership probabilities are given in Table
3.3. In Figure 3.5, we observe that the most probable state is (1, 1, 4) when Insurer 3 gets two thirdsof the market. In the top 5 most probable states, Insurer 3 is always a leader (strictly). In Table 3.3,
the leadership of Insurer 3 is even more striking with 76% of leadership compared to competitors.
So, Insurer 3 is largely the big winner of the price setting. Asking a premium 1.1 when competitors
set 1.25 changes completely the previous situation of equalled premium of 1.25. Obviously, competitors
will not remain inactive or blind of that situation and promptly react to an insurer stealing the market.

Ins. 1
Ins. 2Ins. 3

Leadershipstatesnb. states(6,0,0);(5,1,0);(4,2,0);(5,0,1)
8(4,1,1);(3,2,1);(4,0,2);(3,1,2)(2,4,0);(1,5,0);(0,6,0);(2,3,1)
8(1,4,1);(0,5,1);(1,3,2);(0,4,2)(2,1,3);(1,2,3);(2,0,4);(1,1,4)
8(0,2,4);(1,0,5);(0,1,5);(0,0,6)

cum. prob.
0.08
0.05
0.76

states(3,0,3);(3,3,0)(2,2,2)(0,3,3);(3,3,0)(2,2,2)(0,3,3);(3,0,3)(2,2,2)co-leadershipnb. statescum. prob.

3

0.10

3

0.08

3

0.12

Table 3.3 – Long-term leadership and co-leadership probabilities by insurer

3.3.2

Properties of the loss model

In this subsection, we focus on the properties of the loss model.
Proposition 3.3.14. The aggregate claim amount per insurer Sj,t at period t is a compound distribution of the same kind as the individual loss amount Yi,t .
PM
fj,t (xt )Hence, the insurer aggregate claim amount Sj,t (xt ) is a compound distribution l=1
Zl suchfj,t : (i) a Poisson-lognormal with M
fj,t (xt ) ∼ P(Nj,t (xt )λ); (ii) a negathat the total claim number M
fj,t (xt ) ∼ N B(Nj,t (xt )r, p).
tive binomial-lognormal with M
Taking into account the time t, the insurer aggregate claim amount t Sj,t for year is the compoundPMfj,tl=1 Zj,l,t given that Nj,t = nj,t . We assume that the claim number Mj,t and claim severity Zj,l,t
follow
fj,t ∼ P(nj,t λ) and (Zj,l,t )l i.i.d.
— PG: a Poisson-gamma with M∼ G(µ1 , σ1 ),fj,t ∼ N B(nj,t r, p) and (Zj,l,t )l i.i.d.
— NBG: a negative binomial-gamma with M∼ G(µ1 , σ1 ).
i.i.d.

fj,t ∼ P(nj,t λ) and (Zj,l,t )l ∼ LN (µ2 , σ 2 ),— PLN: a Poisson-lognormal with M
2fj,t ∼ N B(nj,t r, p) and (Zj,l,t )l i.i.d.
— NBLN: a negative binomial-lognormal with M∼ LN (µ2 , σ 2 ).
2109

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.4 – The Markov chain of insurers portfolio sizes (N t )t (transition probabilities lower
than 5% are not plotted and line thickness increases the probability value)i.i.d.

fj follows a Poisson distribution P(nj λ), while if we
Therefore, if we assume Mi ∼ P(λ), then M
i.i.d.
fj follows a negtive binomial distribution N B(rnj , p). In addition
assume Mi ∼ N B(r, p), then M
to considering two frequency distributions (Poisson and negative binomial), we also consider two loss
severity distributions: (i) gamma and (ii) lognormal.
The expectation and the variance of the aggregate claim amount can be easily derived
fj,t )E(Z), V ar(Sj,t ) = V ar(Mfj,t )E 2 (Z) + E(Mfj,t )V ar(Z).
E(Sj,t ) = E(M110

3.3. Properties of policyholder behaviors and consequences on insurer portfolio sizes and losses

Figure 3.5 – The invariant measure of insurers’ portfolio sizes (N t )tThe survival function of the aggregate claim amount is given by
P (Sj,t > s) =X

P (Nj,t = m)

0≤m≤n

X



fj,t = k|Nj,t = m P
P M

0≤k

k
X

!
Zl > s .

(3.21)l=1

Pk
The expression is explicit in the case of gamma variables Zl , where the term l=1 Zl still follows a
gamma distribution G(kµ1 , σ12 ). When it exists, the moment generating function is given by


P
P
GM
GM
,
Sj,t (z) = GNj,t GM
Z (z)
f
j,t

σ1
where GM (.) denotes the moment generating function with GM
Z (z) = (µ1 /(µ1 − z)) . If we consider
lognormal
P losses, the moment generating function is not explicit. Furthermore, the distribution of thesum l Zl is not a lognormal distribution, but a large number of approximations exist, see Asmussen
et al. (2012) and the references therein.

In the special case where x = (x, . . . , x) is a constant vector and fj is independent of j, using(3.14) we have for all sn−

P (Sj > s)

=

X

 
j
X
nj

0≤m≤n l=0+

X
0≤k

l

l

nj −l

(p= ) (1 − p= )
fj = k|Nj = m P
P M

k
X

m−l


n − nj1 − p=
n−nj −m+l(pj6= )
J −1
m−l!

Zl > s ,l=1

where 0+ = 0 ∨ (mj − (n − nj )) and n−
j = nj ∧ mj . As explained above, the random variable
fj = k | Nj = m follows a Poisson distribution P(mλ) or a negative binomial distribution N B(mr, p).
M
Therefore, the probability can be computed explicitly when Zl follows a gamma distribution or approximatively when Zl follows a lognormal distribution.
When x = (x, ρx, . . . , ρx), we can also have an explicit expression of the distribution of Nj (x)|N t−1 =n which will lead to a similar expression of the survival function of the aggregate loss.

111

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Proposition 3.3.15. If Insurer j is the cheapest insurer (i.e. for all k 6= j, xj ≤ xk ), then the loss
average by policy of Insurer j is stochastically ordered as
Nj (x)∀k 6= j,Nk (x)

XX
1
1
Yi ≤cxYi .
Nj (x) i=1
Nk (x) i=1

This proposition remains us results from Wang et al. (2010), where they find in a dynamic modelthat larger firms experience less premium variation than smaller firms. Indeed, since premium equilibrium is highly correlated to loss history, we can reasonably expect that, in the long run, firm proposing
lowest premium benefits from a largest market share. The loss average is therefore less volatile, allowing insurer to be less constrained by solvency regulation and potential loss shocks and leading, in a
dynamic pattern, to a more stable premium.

3.4

Properties of insurer underwriting strategies

In this section, we derive theoretical properties of the insurance game with respect to insurers.
Firstly, we focus on the one-period game in a general setting, then we study the special case of two
insurers. As for the previous section, all proofs are postponed to Appendix 3.8.2.

3.4.1

Properties of the premium equilibrium in the one-period game (thegeneral case)

The first proposition shows that the premium equilibrium of the one-period game admits a uniquepremium equilibrium.
Proposition 3.4.1. Consider the J-player insurance game with objective function (3.7) and solvency
constraint (3.9) where the market proxy is either the arithmetic mean (3.5) or the weighted mean (3.6)
of the competitors’ price. The game admits a unique (Nash) premium equilibrium when βj > 0 for
j = 1, . . . , J.
Furthermore, in a very special case of no active constraint, a simple linear system can be solved to
get the premium equilibrium, see Appendix 3.8.2.1 for details. In this case, the premium equilibrium
does not depend (directly) on capital levels and the loss model. It may depend on portfolio sizes only
when the market proxy is the weighted mean (3.6) of the competitors’ price, not for the arithmetic
mean (3.5).
Proposition 3.4.2. Let x? be the premium equilibrium of the J-player insurance game with objectivefunction and solvency constraint function defined in Equations (3.7) and (3.9), respectively, where
the market proxy is either the arithmetic mean (3.5) or the weighted mean (3.6) of the competitors’
price. When no constraint functions are active, the premium equilibrium x? solves a linear system of
equations M x? = v.
It is important to note that the linear system deduced in the previous proposition is not equivalent to the original Nash equilibrium since it relies on the strong assumptions of non-active constraintfunctions. Nevertheless, it helps to understand the effect of some parameters on the premium equilibrium in that particular case.
Getting a linear system for the premium equilibrium, we are looking for a necessary and sufficient condition. The linear system for the premium equilibrium has a solution when the determinantof M is non null. Using M = M1 M2 from Appendix 3.8.2.1,
Q we have det (M ) = det (M1 ) det (M2 ).
M2 being diagonal yields a positive determinant det (M2 ) = i wi > 0 for the weights considered. So
det (M0 ) = 0 is equivalent to det (M1 ) = 0. As shown in Appendix 3.8.2.4, we have
Σdet (M1 ) = 0 ⇔ 2β1 w−1β̃1 = w1

J
X
j=2

112

(1 + βj )β̃j .

3.4. Properties of insurer underwriting strategies
Many βj solve this equation. But there is a unique solution when wj = w is constant and βj ’s are allidentical. Indeed, βj = 1 is a solution for det (M1 ) = 0 both when the market proxy is the arithmetic
mean (3.5) or the weighted mean (3.6).
Again, when the determinant of M cancels, it does not mean that premium equilibrium does notexist, but that the premium equilibrium does not solve that linear system.
A sufficient condition for the linear system to have a solution is given now. A sufficient condition
for the linear system to have a (unique) solution is M to be diagonally dominant. That is ∀j = 1, . . . , J,XXΣ|2bj | >| − aj wk | ⇔ 2bj > aj
wk ⇔ 2bj > aj w−j
k6=j

k6=j

Σ
In the case of objective function (3.7), we choose aj = 1 + βj , bj = βj w−j. So the sufficient conditionisΣΣ
2βj w−j> (1 + βj )w−j
⇔ βj > 1.

In that case, one can check that the determinant det (M1 ) is strictly positive. This fact was also seenin Dutang et al. (2013).
Proposition 3.4.3. Let x? be the premium equilibrium of the J-player insurance game. For each
Insurer j, the premium equilibrium x?j depends on the parameters as given in Table 3.4, where eachcolumn analyzes the different situations of active constraints. As the break-even premium πj is increasing in āj,t−1 and mt−1 , the effect of āj,t−1 and mt−1 on the premium equilibrium x?j is identicalto the effect of πj .

z
πj
βj
nj
Kj
σ(Y )

x?j = x, x
z 7→ x?j (z)x?j solv. constr.
z 7→ x?j (z)%unknown&%no act. constraint
z 7→ x?j (z) if (3.5) z 7→ x?j (z) if (3.6)%%&&% if πj > 2, & oth.

Table 3.4 – Sensitivity analysis of premium x?j

3.4.2

Properties of the premium equilibrium in the one-period game : J = 2

The objective of this section is to better understand price equilibrium, in a simple case whereI = 2. We first look at parameters conditions leading to constraints activation. As sum up in Table
3.5, we determine bounds of costs i according to price constraint level as well as consumers sensitivity
parameters βj . We thus determine the value of Nash Equilibrium (table6) , which is under active
constraints, for at least one competitor.
In a second part, we investigate the case where constraints are not active (i.e. price equilibriumof each firm depends on their consumer sensitivity and competitor consumer sensitivity). We thus
analyze effect of competitor price elasticity on price equilibrium. Interestingly, explicit formula allow
us to show that marginal effect is not monotone and highly depend on comparative competitive advantage in term of consumers’ inertia.

113

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Considering a two-insurer market x = (x1 , x2 ), n = (n1 , n2 ) leads to the same market proxy.
Indeed using Equations (3.5) and (3.6), we obtainmj (x, n) =1
n−j x−j = x−j = mj (x),n1 + n2 − nj

where the index −j denotes the other player when considering player j. The objective function (3.7)
simplifies to



nj
xj
Oj (xj , x−j ) =1 − βj−1(xj − πj ) .
n1 + n2x−j
The computation of the premium equilibrium involves the use of Lagrange multipliers to take into
account constraints in the KKT system. Table 3.21 of Appendix 3.8.2.5 sums up the 9 different cases
for the KKT system (3.26): (x?1 , x?2 ). As shown in Appendix 3.8.2.5, we get the following Proposition
3.4.4 for β1 , β2 > 1. We refer to Appendix 3.8.2.5 for other cases.
Proposition 3.4.4. The premium equilibrium of the two-player insurance game when solvency constraints are not active is given in Table 3.5 with premium value given in Table 3.6.

π2

. < l1 (β2 )l1 (β2 ) < . < l2 (β2 )l2 (β2 ) < . < l3 (β2 )l3 (β2 ) < . < l4 (β2 )l4 (β2 ) < .

. < l1 (β1 )(x, x)(x, x)(x, x̃b )(x, x̃b )(x, x)l1 (β1 ) < . < l2 (β1 )(x, x)(x, x)(x, x̃b )(x, x̃b ) or (x̃d , x)(x̃d , x)π1l2 (β1 ) < . < l3 (β1 )(x̃a , x)(x̃a , x)
C , xN C )(xN
1
2(x̃d , x)(x̃d , x)l3 (β1 ) < . < l4 (β1 )(x̃a , x)(x̃a , x) or (x, x̃c )(x, x̃c )(x, x)(x, x)l4 (β1 ) < .
(x, x)(x, x̃c )(x, x̃c )(x, x)(x, x)

Table 3.5 – Summary of all cases for a unique equilibrium (x?1 , x?2 ) (only valid for βj > 1),
β−1
β−1
1+β
with l1 (β) = 2x − 1+ββ x, l2 (β) = β x, l3 (β) = β x, l4 (β) = 2x − β x.
Value of x?j
Player 1

no constraint
1 β2 π1 +(1+β1 )β2 π2NCx1 = 2β
4β1 β2 −(1+β1 )(1+β2 )Player 2

C = 2β1 β2 π2 +(1+β2 )β1 π1xN
24β1 β2 −(1+β1 )(1+β2 )x?−j = xx?−j = x1 )xx̃a = β1 π1 +(1+β2β12 )xx̃b = β2 π2 +(1+β
2β2

1 )x
x̃d = β1 π1 +(1+β2β12 )xx̃c = β2 π2 +(1+β2β2Table 3.6 – Unconstrained premium equilibrium value
We see in Table 3.5 that Nash Equilibrium of our game highly depends on price bounds (x, x) but
also on consumers costs sensitivity βj , j = {1, 2}.First and as expected an increase of x or a decreaseof x lead to constraint NE (Nash Equilibrium).
More interestingly, competitive advantage resulting from low costs πj is limited by regulatory conC
straint. Indeed, since xNis an increasing function of πj , low πj could lead firm j to have first bestjNC
price xj below x, in particular when their consumers are sensitive (i.e. βj large enough).
In response, competitor −j of insurer j, which is unconstrained by bounds will not propose theCunconstrained prices xN−j since it could benefit from this limitation by proposing highest price. Inversely, an important costs πj lead firm j to be limited by upper bounds in particular when consumers
are not sensitive.
From Table 3.6, we find that when firm j is contrained, the best response of firm −j does notdepend on j consumers’ sentivitity. We thus focus our analysis on the special case where there areCNC
no active constraints. In that case, x? = (xN
1 , x2 ) by Table 3.5 or as shown directly from thelinear system in Appendix 3.8.2.6. From Table 3.4, we already know that πj 7→ x?j is an increasing114

3.5. Theoretical properties of the repeated game
function. From the explicit value, it is easy to show that π−j 7→ x?j is an increasing affine function
when det (M ) > 0 or a decreasing affine function otherwise. From Table 3.4, we already know thatan increase of consumers’ price sensitivity with respect to firm j decrease the price equilibrium of firm j.
However, an increase of consumers’ price sensitivity with respect to firm −j could have positiveor negative impact depending on β2 based level. Indeed from Appendix 3.8.2.6, we haveβ 2 π2 (3β1 − 1) − 2β2 π2 (1 + β1 ) − 2β1 π1 (1 + β1 ) − π2 (1 + β1 )
∂x?1
.
= 2
∂β2
(3β1 β2 − 1 − β1 − β2 )2The numerator admits two real roots β˜21 , β˜22 such that β˜21 < 0 and β˜22 > (1 + β1 )/(3β1 − 1). Thus,
2
there always exists a threshold of β˜2 for which the derivative of firm j price equilibrium changes signwith respect to the price sensitivity of competitor −j. This means that when (1 + β1 )/(3β1 − 1) <2
β2 < β˜2 , an increase of price sensitivity of consumers with respect to competitor −j lead to a decreaseof price equilibrium for both insurers. Intuitively, the decrease of price competitor −j over pass the2
price elasticity advantage of j. However, when β2 > β˜2 , an increase of β2 leads to a decrease of x?2
and an increase of x?1 . Hence, when β2 is high enough, a marginal increase of this parameter leads toan important comparative advantage for firm 1.
The simple case consider here (i.e J = 2), allows us to analytically demonstrate the importanceof consumers’ price sensitivity, in particular when one firm benefit from highest inertia. It conducts
either to determine importance of regulatory constraints on market equilibrium but also on premium
level, particularly when firms get important difference in their consumers sensitivity.
First, firms displaying important (resp. weak) consumers’ price sensitivity will be more constraint
by lower (resp. upper) bounds. In addition, when they are not constrained, consumers’ price sensitivity
has also an important impact on market equilibrium. While we consider here a Bertrand competition(competition in price), taking into account stochastic demand of consumers by including sensitivity
parameter β allow us to avoid the "Bertrand Paradox". This is particularly evident when firms haveimportant difference in consumers’ price sensitivity.

3.5

Theoretical properties of the repeated gameThe two following propositions give us some insights of the complexity of the repeated game. The
first proposition ensures that the game will end in a long term horizon, whereas the second propositionshows that the cheapest insurer at one period has an advantage on its underwriting result by policy
from a stochastic order point of view.
Proposition 3.5.1. For the repeated J − player insurance game defined in the previous subsection,
the probability that there are at least two non-bankrupt insurers at time t decreases geometrically as tincreases.
Proposition 3.5.2. For the repeated J − player insurance game defined in the previous subsection, if
for all k 6= j among non-bankrupt insurers, 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) i=1

115

Chapter 3.

3.6

A game-theoretic analysis of insurers pricing strategies

Empirical evidences

All numerical applications are carried out with the R software, R Core Team (2018) : we refer toAppendix 3.8.4.1 for computation details. This section is divided into two parts: one part for a twoinsurer market and the other for a three-insurer market. In each subsection, we begin by presenting
the base parameters for which simulates the repeated game; and then we follow by analyzing single
random paths and over a large number of runs. Finally we investigate the effect of parameters on therepeated game.

3.6.1

A duopoly

3.6.1.1

Description of the setting

We consider a duopoly where two insurers fight for a market of N = 500 policyholders. In thecentral setting, Insurer 1 has 50% more policies than Insurer 2. Thus Insurer 1 will be called theleader, whereas Insurer 2 the challenger. Objective and constraint functions are given in Equations(3.7) and (3.9) with parameters given in Table 3.9. Policyholders which are the source of randomness(both for lapse and loss) have parameters given in Tables 3.7 and 3.8 respectively loss and lapse. Lapse
rates and price elasticity are similar to those of Dutang et al. (2013). Price are normalized to 1 as itis the pure premium without any expense.

Loss frequency
dist.
λPoisson
0.1(E(M ) = V ar(M ) = 0.1)

Policyholder(Expec. & Var.)

Loss severity
dist.
µ1
σ1
lognormale 1.931616 0.8613578(E(Z) = 10, V ar(Z) = 110)

Table 3.7 – Parameters of the loss model PLN

fj
f˜j
f˜j

Policy of Insurer 1
Policy of Insurer 2µj-2.1973-1.8153

Lapse MLN PD modelαj
1 − pj→j ((1, 1)) 1 − pj→j ((1, 1.05))
7.402010%15%5.844514%
19%

Table 3.8 – Parameters of the lapse model

Insurer 1
Insurer 2
Market

βj
3
4
x
1

ωj
1
1
x
3

Obj. and constr. param.
ej
πj,0
nj,0
30% 1.28494
300
30% 1.28612
200
N
E(Y )
V ar(Y )500
1
4.5826Kj,0317.49259.23k95
3solv. ratio
133%.
133%
d
3

Table 3.9 – Parameters of insurers

3.6.1.2

Analysis of some random paths

Let us consider a single random path (that can be obtained by setting the seed). In Figure 3.6,
we observe that the challenger’s premium is lower for the first four periods than the leader’s premium.

116

3.6. Empirical evidences
This leads to a change of leadership both in terms of portfolio sizes and gross written premium. However in period 4, Insurer 2 experiences a large loss depicted by a loss ratio around 120% damaging
the solvency ratio. In period 5, Insurer 2 is forced to readjust the premium to a high level: this is asituation favorable to Insurer 1 which gets back some market share. After period 6, Insurer 1 retrieves
the leadership and holds it until period 10.
In this simple setting, we notice that the competition is driven by many effects that are temporarily
prevalent or temporarily insignificant. The first effect playing a major role is the solvency constraint,
see e.g. periods 4 and 5. The second effect is the portfolio size which mitigates the loss volatility.

117Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.6 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2

3.6.1.3

Analysis of an aggregate number of random paths

Now we analyze 100 runs of the repeated game. For each plot in Figure 3.7, we compute the firstquartile, the median and the third quartile over 100 runs at each period. From Figure 3.7, we can first118

3.6. Empirical evidencesnote that most of the time premium of Insurer 1 are always higher than Insurer 2’s premium, since
all quartiles of Insurer 1 are higher than those of Insurer 2. This leads to a huge difference in termsof portfolio sizes, for which at period 10 the first quartile of Insurer 2 is even higher than the third
quartile of Insurer 1. This situation is a little tempered when looking gross written premium.
Secondly, the level of premium equilibria are higher than initial premium. Typically the mediansof premium equilibria are around 2 when initial premiums are 1.25. This leads to a drop in terms ofloss ratio and a gain for solvency ratios.
Thirdly, Insurer 2 having a higher number of policies, in particular for the last five periods, reduces
the interquartile range of claim frequencies by policies. The third quartiles of Insurer 2 claim numbers
tends towards the median, while the opposite effect is observed for Insurer 1.

3.6.1.4

Sensitivity analysis with respect to parameters

In this subsection, we make a sensitivity analysis with respect parameters in the following order:
the loss model, the lapse model, the portfolio size, the total market size and the objective parameters.
Most figures and tables are put in Appendix 3.8.7.1.
Let us start with the loss distribution. Recalling that the base model is a Poisson - lognormalcompound distribution. We consider two other models : a negative binomial - lognormal compound
distribution labelled NBLN and a Poisson - gamma compound distribution labelled PG. The new setof parameters of the frequency distribution or the severity distribution are given in Table 3.10. These
two concurrent loss models analyze either an increase of the frequency variance (Poisson vs. negative
binomial) or a decrease of the severity tail (lognormal vs. gamma). In the first case, the resulting
variance of the aggregate claim distribution is increased, while in the second case the resulting varianceis reduced.

Policyholder(Expec. & Var.)

Loss frequency for NBLN
dist.
r
p
neg. binom. 0.014286 0.125(E(M ) = 0.1, V ar(M ) = 0.81)

Loss severity for PG
dist.
µ2
σ2
gamma 1.51515 0.151515(E(Z) = 10, V ar(Z) = 66)

Table 3.10 – New parameters of loss models NBLN and PG
On a single run, the NBLN loss model produces a similar pattern for the volume indicators (portfolio size, gross written premium) but the premium equilibrium and the loss ratio are more erratic forboth insurers, see Figure 3.14 vs. Figure 3.6. The range of the premium and the loss ratio are higher
than for the base model. For the PG loss model, portfolio sizes and gross written premium of insurers
have also similar pattern but the premium equilibrium and the loss ratio are far less erratic for both
insurers, see Figure 3.15 vs. Figure 3.6. Therefore the SCR ratio is higher and more stable for the PG
model.
Over 100 runs, the NBLN loss model has a major impact of the quartiles of premium equilibrium.
Indeed the higher volatility of the frequency distribution increases the claim number by policy (theinterquartile range triples from 4 to 12 percentage points) implying a more volatile break-even premium πj,t which leads to more erratic premium equilibrium, see Figure 3.23 vs. Figure 3.7. This effectis logical from the sensitivity Table 3.4. By a domino effect, the portfolio size and hence the grosswritten premium are more unstable for both insurers (the interquartile range nearly doubles).
The opposite effect is observed for the PG loss model. For that case, the lower volatility of theseverity distribution slightly decreases the volatility of the loss ratio. (the interquartile range triples

119

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.7 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles

from 4 to 12 percentage points). However, the PG loss model has a relative small impact on otherquantities, see Figure 3.24 vs. Figure 3.7.
Let us continue with the lapse model. Recalling that the base model is a multinomial logit model
based on the differences of premium values called MLN PD. We test two lapse models : a multinomial
logit model based on the differences of ratio premium values called MLN PR and a multinomial logit
model based on the differences of premium values with a new set of parameters leading to different
central lapse rates called MLN PD 2.

120

3.6. Empirical evidences

Policy of Insurer 1
Policy of Insurer 2

fj
f¯j
f¯j

µj
-11.4497-9.1209

Policy of Insurer 1
Policy of Insurer 2

fj
f˜j
f˜j

µj
-2.1973-2.1973

Lapse MLN PR modelαj
1 − pj→j ((1, 1)) 1 − pj→j ((1, 1.05))
9.252510%15%7.305614%
19%
Lapse MLN PD 2 modelαj
1 − pj→j ((1, 1)) 1 − pj→j ((1, 1.05))
7.402010%15%7.402010%
15%

Table 3.11 – New parameters of lapse models MLN PR and MLN PD 2
On a single run, the MLN PR lapse model produces a similar pattern for the portfolio size but
the premium equilibrium and the loss ratio seems more cyclical for both insurers, see Figure 3.16
vs. Figure 3.6. Furthermore, the premium values are more closed to the initial premium leading to amore stable gross written premium. Customers are less elastic to price changes for the MLN PR lapse
model. For the MLN PD 2 lapse model exhibits very different patterns of premium and portfolio sizevalues. In this particular path, Insurer 2 takes and keeps the leadership on the market, see Figure 3.17
vs. Figure 3.6.
Over 100 runs, we note that the quartiles of the premium equilibrium for the MLN PR are closer
than the base situation (MLN PD). Thus the quartiles of the portfolio sizes tends to be identicalbetween the two insurers, Figure 3.25 vs. Figure 3.7. After 6 periods, the gross written premium, theloss ratio and the claim numbers have very similar quartiles between both insurers.
For the MLN PD 2 lapse model, the leadership of Insurer 2 after 5 periods is more overwhelming than in the base situation: the first quartile of Insurer 2’s portfolio size is higher than the third
quartile of Insurer 3’s portfolio size after 5 periods. The MLN PD 2 lapse model seems to accelerate
the leadership of Insurer 2 which yields to a more stable claim numbers by policy and loss ratio, see
Figure 3.26 vs. Figure 3.7.
Let us pursue with the initial parameters. Recalling that the base model assumes initial portfolio
sizes n0 = (300, 200) and initial solvency ratios of 133%. We study first another value of initial portfolio sizes n0 = (250, 250) and then another values of solvency ratios 100%.
On a single run, we observe that insurers have both a five-period leadership, however the domination of Insurer 1 is striking on this scenario. For instance, both the portfolio size and the gross written
premium of Insurer 1 are dominant which leads to better solvency ratio, see Figure 3.18 vs. Figure
3.6.
Over 100 runs, Insurer 2 takes most of the time the leadership in terms of gross written premiumor portfolio sizes. But the tendency towards of a full leadership of Insurer 2 is observed as in the basesetting, see Figure 3.27 vs. Figure 3.7.
When considering new solvency ratios (of 100%), the effect is negligible compared to the basescenario for a single path, see Figure 3.19 vs. Figure 3.6. Over 100 runs, the pattern for all indicators
are identical between the two situations, see Figure 3.28 vs. Figure 3.7. The novelty appears by thefact that insurers have been bankrupted in some random path: 4% for Insurer 2 of ruin and 1% for
Insurer 1 in the first two periods, no ruin after.
Let us continue with the total market size. Originally the total market size is N = 500. We test
the effect of portfolio sizes by doubling it. On a single run, the range of the premium equilibrium as

121

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

well as the premium differences between the two insurers are much lower than in the base situation,
see Figure 3.20 vs. Figure 3.6. This implies that the market shares make more periods to change sothat Insurer 1 almost remains the leader.
Over 100 runs, we note that the interquartile range decreases for many indicators, in particularloss ratios, claim number by policies. The direct consequence is an increase of the solvency ratios for
all quartiles, see Figure 3.29 vs. Figure 3.7.
Let us finally consider the lapse sensitivity parameters βj and the credibility factor ωj . In thebase situation, β1 = 3 and β2 = 4. We test the opposite situation. This has a major influence on
the premium equilibrium value both on a single run or over 100 runs. For instance, on Figure 3.21,
we observe that Insurer 1 is more competitive and remains the leader for 8 out of 10 periods. But
the drawback is that Insurer 1’s solvency ratio is worse for this alternative situation than in the basescenario, see Figure 3.21 vs. Figure 3.6.
Over 100 runs, Insurer 1 remains most of the time the leader, see the quartiles of the gross written
premium on Figure 3.30. Consequently both the loss ratio and the claim number by policy are morestable: the interquartile range falls for Insurer 1 and widens for Insurer 2.
Finally we study another set of values for credibility factors ωj . In the base scenario, they are set
to zero, while in the alternative scenario we assume that ωj = 1/2. That is the break even premiumπj is the mean between past market premium and past actuarial premium.
On a single run, we observe that the break even premium increases a lot leading to large premium
equilbrium. The premium upper bound x is reached after 6 periods, see Figure 3.22. As soon as both
insurers ask the same premium, only the lapse model plays a role : we see a convergence towards anequilibrium depicted in Proposition 3.3.11. The loss model has no longer any effect: the loss ratios
shrinks and the solvency ratios explodes.
Over 100 runs, a similar effect is observed on Figure 3.31. The first quartiles of premium equilibrium reach almost the upper bound so that the portfolio sizes quartiles towards the invariant measure.
However, we see that Insurer 2 takes the leadership very soon. Again the solvency ratios and the loss
ratios appear abnormal. This is due to the fact that the total market size is constant and cannot exit
the market as it would happen in such situation.

3.6.2

A three-insurer market

3.6.2.1

Description of the setting

We consider a game where three insurers fight for a market of N = 500 policyholders. In thecentral setting, Insurer 1 is the leader with more than one half of the market, whereas Insurer 2 is thechallenger with 30% of policyholders and Insurer 3 the outsider with the last 10% of policyholders.
Objective and constraint functions are still given in Equations (3.7) and (3.9) with parameters given
in Table 3.13. In this three-player setting, we also study both market proxies m(x, n) used in theobjective function given in Equations (3.5), (3.6). Policyholders face the same of the loss model as theprevious subsection, see Table 3.7, but follow a new multinomial logit model with parameters given in
Table 3.12.

3.6.2.2

Analysis of some random paths

In Figure 3.8 for a simple market proxy (see (3.5)), we observe that Insurer 1 remains the leader
for the first six periods, until competitors have a better loss experience with very low loss ratios around1223.6. Empirical evidences

Policy of Insurer 1
Policy of Insurer 2
Policy of Insurer 3

fj
f˜j
f˜j
f˜j

µj
-2.890372-2.508437-2.209495

Lapse MLN PD modelαj
1 − pj→j ((1, 1))
7.401976
10%5.844477
14%4.928581
18%

1 − pj→j ((1, 1.05))
15%
19%
23%

Table 3.12 – Parameters of the lapse model

Insurer 1
Insurer 2
Insurer 3
Market

βj
3
3
4
x
1

ωj
1
1
1
x
3

Obj. and constr. param.
ej
πj,0
nj,0
30% 1.28494
300
30% 1.28612
150
40% 1.48396
50
N
E(Y )
V ar(Y )500
1
4.5826Kj,0317.49224.4997.21
k95
3

solv. ratio
133%.
133%
100%
d
3

Table 3.13 – Parameters of insurers

20% compared to 60% in Periods 5 and 6. Hence inevitably, Insurer 1 sets a very high premium in
Period 1 due to a high break-even premium and Insurer 3 capture half of the market. Globally on
this run, Insurer 2 keeps a very stable market share. When we compare to the weighted market proxy(see (3.6)), the pattern mainly different. For the whole period, Insurer 1 remains the leader both interms of portfolio size and gross written premium by setting premium slightly lower than competitors,
see Figure 3.9. The weighted market proxy seems to induce higher values of premium equilibrium: wealmost reach the upper bound x in Figure 3.9.

3.6.2.3

Analysis of an aggregate number of random paths

Now we turn our attention on the analysis of 100 runs of the repeated game. Again, for each plot
in Figures 3.10 and 3.11, we compute the first quartile, the median and the third quartile over 100
runs at each period. From Figure 3.10, we observe that premium equilibrium are very similar withalmost the same quartiles for every insurers.
However quartiles of portfolio sizes, gross written premium and solvency ratios are ordered: for
instance third quartiles of Insurer 1 is greater than third quartiles of Insurer 2 which is in turn greaterthan third quartiles of Insurer 3. Hence Insurer 1 seem to stay leader in most scenarios. In addition,
we notice that that the interquartile range of loss ratio and claim numbers per policy is lower thanthose of Insurers 2 and 3.
In Table 3.14, we observe that the leadership of Insurer 1 is greater than those of competitors, yetthere is a decreasing trend over time. The longer the repeated game is player, the lower the probability
that Insurer 1 is the leader is. In Table 3.14, the ruin probability is also computed: having a lower
initial solvency ratio, Insurer 3 is more exposed to the insolvency risk (13% against 1% for Insurer 2
and 0% for Insurer 1).
Now we analyze 100 runs of the repeated game when the market proxy is the weighted mean (3.6).
In Figure 3.11, the pattern observed are very similar to those of Figure 3.10. The quartiles of premium
equilibrium are similar, yet for some periods an order seems to appear: Insurer 1 has the highest firstquartiles for the first five period but the lowest for the last four periods.

123Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.8 – Some indicators of the repeated game for a single run, black solid line for Insurer
1, red dashed line for Insurer 2, green dotted line for Insurer 3

In terms of volume indicators (portfolio sizes and gross written premium), Insurer 1 has the highest
values for all quartiles. In particular for portfolio sizes, Insurer 3’s third quartiles are most of time124

3.6. Empirical evidences

Figure 3.9 – Some indicators of the repeated game for a single run, black solid line for Insurer
1, red dashed line for Insurer 2, green dotted line for Insurer 3

lower than Insurer 1’s first quartiles. The domination of Insurer 1 makes that the interquartile range
of Insurer 1’s loss ratio and claim numbers by policy is alway lower than those of competitors.

125

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.10 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1, red dashed line for Insurer 2, green dotted line for Insurer 3; bottom-up triangles
for 1st quartiles, numbers for medians and top-down triangles for 3rd quartiles

Looking at Table 3.15, we observe similarly to Table 3.14 that Insurer 1 is the leader and is neverruined. At the opposite, Insurer 3 gets bankrupted more often as time evolves (from 2% to 14%) andhas few chances to become leader (around 10%). Finally Insurer 2 remains the best competitor of
Insurer 1 with a significant of probability of leadership.

126

3.6. Empirical evidences

Period
1
2
3
4
5
6
7
8
9
10

Ruin probabilities
Insurer 1 Insurer 2 Insurer 3
0.00
0.01
0.05
0.00
0.01
0.09
0.00
0.01
0.10
0.00
0.01
0.12
0.00
0.01
0.12
0.00
0.01
0.13
0.00
0.01
0.13
0.00
0.01
0.13
0.00
0.01
0.13
0.00
0.01
0.13

Leadership probabilities
Insurer 1 Insurer 2 Insurer 3
1.00
0.00
0.00
0.94
0.00
0.00
0.86
0.02
0.03
0.73
0.12
0.06
0.59
0.16
0.14
0.52
0.18
0.16
0.45
0.23
0.18
0.42
0.25
0.18
0.38
0.25
0.21
0.44
0.24
0.15

Table 3.14 – Empirical probabilities of ruin and leadership over 100 runs, 3-player game withsimple market proxy

Period
1
2
3
4
5
6
7
8
9
10

Ruin probabilities
Insurer 1 Insurer 2 Insurer 3
0.00
0.01
0.02
0.00
0.01
0.04
0.00
0.01
0.05
0.00
0.01
0.04
0.00
0.01
0.05
0.00
0.01
0.07
0.00
0.01
0.09
0.00
0.01
0.10
0.00
0.01
0.11
0.00
0.01
0.14

Leadership probabilities
Insurer 1 Insurer 2 Insurer 3
1.00
0.00
0.00
0.98
0.00
0.00
0.88
0.04
0.03
0.65
0.15
0.09
0.62
0.17
0.10
0.60
0.18
0.11
0.57
0.21
0.10
0.59
0.20
0.09
0.55
0.20
0.13
0.56
0.23
0.08

Table 3.15 – Empirical probabilities of ruin and leadership over 100 runs, 3-player game withweighted market proxy
3.6.2.4

Sensitivity analysis with respect to parameters

Every figures and tables of the sensitivity analysis are put in Appendix 3.8.7.2. Let us start thesensitivity analysis with respect to the loss model. For the 3-insurer game, we only test the NBLN
loss model where the claim frequency follows a negative binomial compared to a Poisson distribution,
see Tables 3.10 and 3.7 respectively.
Comparing Figures 3.10 and 3.32 for a simple market proxy, we note that the interquartile rangesof all indicators largely increase. In particular, portfolio sizes and claim numbers by policy are more
erratic for all insurers, even for Insurer 1. In this more randomized setting, the leadership is harder toget. Typically quartiles of Insurer 1’s gross written premium are more closed to those of competitors,
see Table 3.23 for leadership probabilities. This increase of loss randomness translates to a lower solvency ratios. For instance, the medians of Insurer 2’s solvency ratios is increasing from 133% to 300%
for NBLN loss model while for PLN loss model it rockets to 500%. In many situations, Insurer 3 goesto bankrupt. In Table 3.23, we compute the ruin probabilities and observe very large probabilities ofruin for Insurer 3.

127Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.11 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1, red dashed line for Insurer 2, green dotted line for Insurer 3; bottom-up triangles
for 1st quartiles, numbers for medians and top-down triangles for 3rd quartiles

Studying the weighted market proxy, we observe similar patterns between Figures 3.11 and 3.33.
For examples, quartiles of the premium equilibrium are similar across insurers, yet Insurer 3 seem
below most of the time. Premium equilibrium have also higher values. The increase in loss randomness also deteriorates the solvency ratios, where first quartiles are much lower on Figure 3.33 than on
Figure 3.11. In Table 3.24, the empirical ruin probabilities are higher even for Insurers 1 and 2.
Regarding the lapse model, we compare the MLN PD lapse model given in Table 3.12 with the
MLN PR lapse model given in Table 3.16. Reference lapse rates are identical, only the sensitivity
function fj changes from premium differences to premium ratios.

1283.7. Conclusion and perspectives

The lapse model seems to produce more favorable situations for Insurer 1, see Figure 3.34. Yet
the quartiles of premium equilibrium for all insurers are very close, the quartiles of portfolio sizes arealmost separated. Indeed first quartiles of Insurer 1’s portfolio sizes are above third quartiles of Insurer 2’s portfolio sizes. Therefore, Insurer 1 is almost always the leader, see quartiles of gross written
premium in Figure 3.34 and Table 3.25. As a consequence the interquartile ranges fall for loss ratio
and claim numbers by policy for Insurer 1.

Policy of Insurer 1
Policy of Insurer 2
Policy of Insurer 3

fj
f¯j
f¯j
f¯j

µj
-12.14284-9.814033-8.370220

Lapse MLN PR modelαj
1 − pj→j ((1, 1))
9.2524710%7.305596
14%6.160726
18%

1 − pj→j ((1, 1.05))
15%
19%
23%

Table 3.16 – New parameters of lapse models MLN PR and MLN PD 2
We do not comment the sensitivity with respect to initial portfolio sizes or the total market sizeas the effects are similar to the situation decreasing the volatily of the loss model. Finally we focus on
the objective parameters βj and ωj . We consider a test with ωj = 1/2 and a test with β3 = 3 rather
than β3 = 4.
Changing the credibility factors have the effect of smoothing quartiles of all indicators. As thebreak-even premium is more closed among insurers, the value of the premium equilibrium are very
stable, see Figures 3.36 and 3.37. This leads to smooth quartiles of portfolio sizes and gross written
premium. In this situation, Insurer 1 remains the leader in most situations. The values of premium
equilibrium are also much higher and touch the upper limit x. As soon as all insurers set the upper
limit x, then we get the convergence towards equilibrium described in Proposition 3.3.11. This effect
was previously observed for the two-insurer game.
Comparing Figure 3.38 against Figure 3.10, we see that premium values are higher for Insurer
3 (quartiles of Insurer 3’s premium are always higher than those of competitors). Hence Insurer 3
obtains a worse situation with fewer policies and the competition between Insurers 1 and 2 is fiercer.
Similar patterns are also observed when studying Figure 3.39 against Figure 3.11 where a weighted
market proxy is considered.

3.7

Conclusion and perspectives

We address in this paper an analysis of consumers’ price-sensitivity impacts as well as solvency
constraint effects on market premiums equilibrium in a competitive non-life insurance market. We
extent the repeated non-cooperative game of Dutang et al. (2013), where consumers behaviors andloss are both stochastic. Thus, we consider that each insurer maximizes its own objective function(i.e. the profit under solvency constraint) while taking into account that its strategy will affect others
insurers’ strategies.
We show that price-sensitivity of consumers has a major effect on prices equilibrium. An insurer
has a higher probability to be a leader in the long-run if it benefits from a comparative advantage on
consumers inertia. In addition, we demonstrate that a largest portfolio size implies a lowest volatilityof losses. Therefore, it reduces the level solvency constraint. Numerical simulations allow us to support its results by evincing how losses affect premiums equilibrium under a regulated market. Insurers
suffering from losses shock on the previous period could be obliged to increase its market premium to129

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

the detriment of portfolio size. Therefore, insurer benefiting from important inertia of its consumers
is less impacted by a losses shock and is be able to keep a relatively important size, which guarantee
long-run profit.
Indeed, while important losses shocks are generally the main sources of insurers’ ruin, it could also
result from the fact that insurers are not able to reach a sufficient level of business to guarantee mutualization principle. Therefore, further researches about price-sensitivity of consumers in a competitivemarket could bring a new dimension to the ruin theory. For instance it could explain why, even undersolvency constraint, insurer bankruptcy could not be avoid.
Of course, we could have considered different modellings to approach our research question. For
instance, we could improve the objective function by considering correlation of consumers’ lapse behaviors across time (Barsotti et al. (2016)).It could also be interesting to examine the game under adividend rule such as a solvency coverage ratio upper bounded, with for instance
Kj,t+1≤ 300%.
√
k995 σ(Y ) nj,tIn reality, insurers have to guarantee a minimum level of dividend in order to keep investments fromthe financial market. In addition, they are not allow to transfer are all benefits into their capital.
Indeed, by doing so, they would avoid to pay income taxes.
Another interesting extension of our model could also consists on take into account information
asymmetry. For now, we consider that all players observe passed claims of others and adjust theirstrategies with respect to these information. We can imagine to add information asymmetry where
each player j has beliefs about others’ claims such ŝ−j = sj and ê−j = ej .
Finally, we assume that consumers are independent and identically distributed. However, we knowthat insurance market is characterized by an important asymmetry of information between insureds andinsurers, leading to adverse selection effect. Considering the extending model of Albrecher and DailyAmir (2017), it could be interesting to analyze how price-sensitivity impacts on premiums equilibrium
are different when considering different type of consumers.

1303.8. Appendix

3.8

Appendix

3.8.1

Proofs of Section 3.3

3.8.1.1

Properties of lapse model at customer level

Proof of Prop. 3.3.1. By A3, (Ci,t )t is a Markov chain with transition matrix P→ (xt ) defined as

p1→1 (xt )...
...
P→ (xt ) = 
pJ→1 (xt ) . . .

p1→J (xt ).

pJ→J (xt )In fact, P→ is a matrix function. By Proposition 3.8.2, the transition matrix has no null terms. It
is immediate that the transition from Ci,0 to Ci,t is the multiplication of the t matrices P→ (x1 ),. . . ,P→ (xt ). On the finite state space {1, . . . , J}, the Markov chain is both irreducible and aperiodic using
Proposition 3.8.2 in Appendix 3.8.3.2.
Proof of Prop. 3.3.2. By (Norris, 1997, p. 41), the process (C1,t , C2,t )t is still a Markov chain on thespace E 2 = {1, . . . , J}2 with transition matrix P→ (xt ) ⊗ P→ (xt ). Iterating N − 1 more times leads to
the result.
Proof of Prop. 3.3.3. Let Nj be the set of customers of Insurer j at time 0. That is ∀i ∈ Nj , Ci,0 = j.
As (Ci,t )t is a Markov chain, the transition from Insurer j to Insurer k is governed by the jth row p̃jof the matrix


(t)
(t)
(t)
P→
= P→ (x1 ) × · · · × P→ (xt ), p̃j = P→,j,1 , . . . , P→,j,J .
Thus ∀i ∈ Nj , Ci,t | Ci,0 = j ∼ MJ (1, p̃j ). By A1 and A2, those policyholders of Insurer j will choose
insurers according to a multinomial distribution C j,t ∼ MJ (nj,t−1 , p̃j ) given Nj,0 = nj,0 . From periodt − 1 to period t, the transition matrix simplifies to P→ (xt ) and p̃j = pj→ (xt ).

3.8.1.2

Properties of lapse model at portfolio level

Proof of Prop. 3.3.4. The portfolio sizes vector is the sum of choice vectors N t = C 1,t + · · · + C J,t .
By A2, Proposition 3.3.3 and given N t−1 = n, (C j,t )j are independent multinomial vectors withparameters MJ (nj , pj→ (xt )) for j = 1, . . . , J. Therefore, N t (obtained by summing over j) has aknown distribution given N t−1 = n. Since (N t )t is a discrete-time process taking values in N , (N t )tis a Markov chain. By recurrence, the number of elements of Sms is Card(Sms ) = N n+J−1. The
mstransition matrix of size Card(Sms ) × Card(Sms ) has a complex expression
P t = (P (N t = m|N t−1 = n))n,m
where n, m ∈ Sms and
P (N t = m|N t−1 = n)=

P (N1,t = m1 , . . . , NJ,t = mJ |N1,t−1 = n1 , . . . , NJ,t−1 = nJ ).
X

=

...

0≤c11 ,...,c1J ≤N,
Ps.t.

lc1l =n1X=l

c1l =n1

J
Y

P (C j,t = cj |Nj,t−1 = nj )11P

0≤cJ1 P,...,cJJ ≤N, j=1
s.t.

l

J
Y

0≤cJ1 P,...,cJJ ≤N, P j=1,
s.t.

l


ckj =mjkcJl =nJ

X...

0≤c11 ,...,c1J ≤N,P
s.t.

X

cJl =nJ

k

nj !
c
c
(pj→1 (xt )) j1 . . . (pj→J (xt )) jJ .
cj1 ! . . . cjJ !

ckj =mj131

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

The probability P (C j,t = cj |Nj,t−1 = nj ) depends on the price vector xt , and therefore is timedependent. By A2, the probability generating function of N t |N t−1 = n is in constrast simpler
GP
N t |N t−1 =n (z)= GP(z) × · · · × GP
(z)
C 1,t
C J,t
 nJ

n
1
n
= (P→ (xt ) × z) ,× · · · × z T pJ→ (xt )= z T p1→ (xt )where z ∈ RJ , the exponentiation n is a component-wise operation and GP (.) denotes the probability
generating function. Using Proposition 3.3.3, we have C j,t | N j,0 = n follows a multinomial distri(t)nbution with parameters MJ (nj , p̃j ). By similar arguments, GP
N t |N 0 =n (z) = (P→ × z) . If µ is theinvariant measure of (Ci,t )t , then
 
 T 
µ
µ z
(t)
(t)
P→
−→ . . . ⇒ P→
× z =  ... 
t→+∞
µ
µT z
T
n1
⇒ GP
× · · · × (µT z)nJ = (µT z)
N t |N t−1 =n (z) = (µ z)

P

i ni

.

In other words, the probability generating function of N t | N 0 = n is the p.g.f. of a multinomial
distribution. Since we obtain a limiting distribution for the Markov chain (N t )t is also its invariant
measure, see e.g. (Norris, 1997, p. 33).
Proof of Prop. 3.3.5. As a sum of independent binomially distributed variables, the mass probability
function of the portfolio size Nj,t is given byX
Y nl n −c
P (Nj,t = mj |N t−1 = n) =(pl→j (xt ))cj (1 − pl→j (xt )) l j .
cj
0≤c1 ,...,cJ ≤n l∈JPs.t.

lcl =mjBy A4, the probability generating function of the sum constituting Nj,t is the product of generating
function of each binomially distributed random variablesn1
GP
× · · · × (1 − pJ→j (xt ) + pJ→j (xt )z)nJ .
Nj,t |N t−1 =n (z) = (1 − p1→j (xt ) + p1→j (xt )z)

Differentiating with respect to z, we get
0GP
Nj,t |N t−1 =n (z) =JX

nk pk→j (xt )(1 − pk→j (xt ) + pk→j (xt )z)nk −1

k=1

Y(1 − pl→j (xt ) + pl→j (xt )z)nl .

l6=kTaking z = 1 leads to the result.

3.8.1.3

Properties of a constant regulated price vector

Proof of Prop. 3.3.8. By construction, (Ci,t )t is a Markov chain with transition matrix P→ (xt ). Whenxt = x, P→ (xt ) is fixed across time that is
p1→1 (x) . . . p1→J (x) = P→ .
...
P→ (xt ) = 
pJ→1 (x) . . . pJ→J (x)When x = (x, . . . , x) using (3.12), we have
pj→k (x) =

132

pj→j
pj6=

if j = k,if j 6= k,

3.8. Appendix
Since

P

k pj→k = 1, we deduce pj6= = (1 − pj→j )/(J − 1). Therefore,
p1→1
P→ = 
...

1−p1→1
J−1

...

pj→j
1−pJ→J
J−1

pJ→J


.

Since the number of state is finite (J) and the Markov chain is irreducible by Proposition 3.8.2, there
exists a unique invariant measure µ, see e.g. Norris (1997).
Let us consider the general matrix M ∈ RJ×J with general term Mi,j = ai (1 − δij ) + (1 − (J −1)ai )δij . Note that the rows of M equal 1. In other words, M has only two different terms by rows
1 − (J − 1)a1



...
M =




a1
..
.

...



ai

1 − (J − 1)ai
..
.

ai




.




...

aJ

1 − (J − 1)aJ

The reversibility conditions for a measure µ are µ1 M1,2 = µ2 M2,1
 µ1 a1 = µ2 a2
...
...
⇔


µ1 M1,J = µJ MJ,1
µ1 a1 = µJ aJ
Let aΠ
−i =JQ µ1 a1 /a2 = µ2...
⇔
µ1 a1 /aJ = µJ

aj . Using µ1 + · · · + µJ = 1, we get by multiplying both sides by aΠ−ij=1,j6=i

µ1 +

X
i>2

µ1

X
aΠ
a1
−1
Π
.
= 1 ⇔ µ1 aΠaΠ−1 + µ1−i = a−1 ⇔ µ1 = J
P Π
ai
i>2
a−i
i=1




Thus, the following measure µ is in detailed balance with M µ = 

aΠ
−1
J
P
aΠ
−i
i=1

,...,

aΠ
−J
J
P
aΠ
−i
i=1

 . µ is also aninvariant measure for M . Setting aj = pj6= , we obtain the following invariant measure for P→
Q
pj6=
µi =

j6=i
J Q
P

.
pj6=

l=1 j6=l

In the special case where pj→j are identical across insurers, pj6= = p6= is constant. Hence for all
j = 1, . . . , JY(p6= )J−1(p6= )J−1
1=
pj6= = (p6= )J−1 ⇒ µi = J
= .
J−1
P
J × (p6= )J
j6=i(p6= )J−1
l=1

Proof of Prop. 3.3.9. As the transition matrix of (C1,t , . . . , CN,t )t is Pt ⊗N = P ⊗N , we get a time
homogeneous Markov chain. By (Norris, 1997, p. 41), if two independent Markov chains have the sameinvariant measure, then the Markov chain product is the Kronecker product of the invariant measure.
So the process (C1,t , . . . , C2,t )t has the invariant measure µ ⊗ µ = (1/J 2 , . . . , 1/J 2 ). Iterating N − 1
more times leads to the result.

133Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Proof of Prop. 3.3.10. Direct application of Proposition 3.3.3.
Proof of Prop. 3.3.11. Direct application of Proposition 3.3.4.
Proof of Prop. 3.3.6. Using Proposition 3.3.4 and the particular form of the transition matrix

1→1
p1→1 1−p
...
J−1
.
pj→j
P→ = 
1−pJ→J...
pJ→J
J−1
we deduce that the probabilities appearing in Proposition 3.3.4 simplify to
nj !
cc(pj→j ) jj (pj6= )cj1 . . . (pj6= ) jJ
cj1 ! . . . cjJ !



1 − pj→j
J −1

 P cjl=

nj !
c(pj→j ) jj
cj1 ! . . . cjJ !
nj !
c(pj→j ) jj
cj1 ! . . . cjJ !



1 − pj→jJ −1

nj −cjj

=

l6=j

.

Therefore we get the desired by summing over appropriate indexes.
Proof of Prop. 3.3.7. For a constant price vector, identical players (pj→j = p= and pj6= = p6= ) and
Proposition 3.3.5, the probability generating function isGP
Nj,t |N t−1 =n (z)=(1 − p1→j + p1→j z)n1 × · · · × (1 − pJ→j + pJ→j z)nJY
(1 − p= + p= z)nj
(1 − p6= + p6= z)nl=

(1 − p= + p= z) (1 − p6= + p6= z)N −nj .

=

l6=j
nj

In other words, Nj,t |N t−1 = n is a sum of two binomially distributed random variables B(nj , p= ) and=B(n − nj , 1−pJ−1 ).

3.8.1.4

Properties of a time-independent price vectorP
1→1
Proof of Prop. 3.3.12. Since k pj→k = 1, we get pj6= = (1−pj→j −pj→1 )/(J −2), and p16= = 1−p
J−1 .
We use the following notation 
a1 b1 . . .
p1→1 p16=...
 p2→1 p2→2 p26=
  b2 a2 c2 . . .
...

 

 p3→1 p36= p3→3 p36=
  b3 c3 a3 c3 . . ..
.
.
P→ = 
(3.22)
.
=

 

..
..




.
.
pJ→1

pJ6=

...

pJ6=

pJ→J

bJ

cJ

...

cJ

aJ

A first series of equation for the invariant measure is obtained from µ0 = µ0 P→ . That is µ1 = a1 µ1 + b2 µ2 + b3 µ3 + b4 µ4 + · · · + bJ µJ µ2 = b1 µ1 + a2 µ2 + c3 µ3 + c4 µ4 + · · · + cJ µJµ3 = b1 µ1 + c2 µ2 + a3 µ3 + c4 µ4 + · · · + cJ µJ

..


.



µJ = b1 µ1 + c2 µ2 + · · · + cJ−1 µJ−1 + aJ µJ
Ignoring the first equation and subtracting the second equation to all others, we getµ2 = b1 µ1 + a2 µ2 + c3 µ3 + c4 µ4 + · · · + cJ µJ µ3 − µ2 = (c2 − a2 )µ2 + (a3 − c3 )µ3
..

.



µJ − µ2 = (c2 − a2 )µ2 + (aJ − cJ )µJ

134

3.8. Appendix −b1 µ1 = −µ2 + a2 µ2 + c3 µ3 + c4 µ4 + · · · + cJ µJ
 µ3 (1 − a3 + c3 ) = (c2 − a2 + 1)µ2
⇔
..

.



µJ (1 − aJ + cJ ) = (c2 − a2 + 1)µ2
The J − 2 equations giveµj = µ2c2 − a2 + 1,j > 2cj − aj + 1

Recalling that any row of M sums up to 1, ai + bi + ci (J − 2) = 1 for i 6= 1, we have∀j = 3, . . . , J, aj = 1 − cj (J − 2) − bj ⇒ cj − aj + 1 = 1 − (1 − cj (J − 2) − bj ) + cj = (J − 1)cj + bj = dj
Note that in the special case bj = cj , dj = Jbj leads back to µj = b2 µ2 /bj . So in general, µj = µ2 dd2j
for j = 3, . . . , J. The first equation for µ1 becomesb1 µ1 = µ2 − a2 µ2 −

J
X

cj µ2

j=3

J
X
d2
d2
⇔ b1 µ1 = µ2 (1 − a2 ) − µ2
cj
dj
dj
j=3

J
J
X
cj
(b2 + (J − 2)c2 )d2
b2
d 2 X c2( − ).
− µ2
cjµ1 = µ2 + µ2
b1
b
d
b
b
d
d
1
j
1
1
2
j
j=3
j=3
P
The probability condition i µi = 1 gives⇔ µ1 = µ2J
X d2
d 2 X c2cjb2( − ) + µ2 + µ2
=1µ2 + µ2b1
b1 j=3 d2
dj
d
j>2 j


−1
J
J
X
c
cj
b1
b
d
c
b
b2
d2 X c2
2
2
2
j
1
( −
( −
⇔ µ2 + µ2 + µ2+ ) = 1 ⇔ µ2 = 1 +++ ) .
b1
b1 j=3 d2
dj
djb1
b1 j=3 d2
dj
dj
Reintroducing the product notation dΠ−1,−j =
µ2

=

=

Q

l6=1,j dl yields the following reformulation

dΠ
−1,−2
b2
Π 1
Π
dΠ
−1,−2 + d−1,−2 b1 + d−1 b1

cj
b1
c2
j=3 ( d2 − dj + dj )PJdΠ
−1,−2 b1ΠdΠ−1,−2 (b1 + b2 ) + (J − 2)c2 d−1,−2 +PJΠ
j=3 d−1,−j (b1 − cj )=dΠ
−1,−2 b1PJΠdΠ
−1,−2 (b1 + d2 − c2 ) +j=3 d−1,−j (b1 − cj )=dΠ
−1,−2 b1
J
P
dΠ
dΠ
−1 +
−1,−j (b1 − cj )j=2

Using µj = dd2j µ2 , we obtain (also valid for j = 2)
J
P
dΠ
dΠ
−1 +
−1,−j (−cj )Π
d−1,−j b1
j=2, j > 2, µ1 =
.
µj =
J
J
P
P
Π
Π
Π
Π
d−1 +
d−1,−j (b1 − cj )d−1 +d−1,−j (b1 − cj )
j=2
j=2

135

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Let us go back to the original transition matrix (3.22)
a1 = p1→1 , b1 = p16= =1 − p1→1
1 − pj→j − pj→1
, ∀j > 1, aj = pj→j , bj = pj→1 , cj = pj6= =
.
J −1
J −2

So dj = (J − 1)cj + bj = (J − 1)pj6= + pj→1
dΠ
−1 −

J
P
j=2

µ1 =
dΠ
−1 +

J
P
j=2

dΠ
−1,−j pj6=dΠ
−1,−j (p16= − pj6= ), µj =dΠ−1,−j p16=
, j = 2, . . . , J.
JPΠΠd−1,−j (p16= − pj6= )
d−1 +
j=2

In the special case where of identical insurers, then ∀j 6= 1, pj6= = p26= and pj→1 = p2→1 . So dj = d2
J−2
J−1
⇒ dΠ
⇒ dΠ
. Hence−1 = d2−1,−j = d2
∀j ≥ 2, µj =

dJ−2
p16=
p16=
p2→1
2
=
, µ1 =.
JP
p2→1 + (J − 1)p16=
p2→1 + (J − 1)p16=
J−2
dJ−1
+
d
(p
−
p
)
16=
26=
2
2
j=2

Proof of Prop. 3.3.11. Direct application of Proposition 3.3.4.

3.8.1.5

136

Transition matrix of insurers portfolio sizes

3.8. Appendix

Figure 3.12 – The markov chain of insurers portfolio sizes (N t )t (transition probabilities lower
than 1% are not plotted and line thickness increases the probability value)137

Chapter 3.
6,0,0
5,1,0
4,2,0
3,3,0
2,4,0
1,5,0
0,6,0
5,0,1
4,1,1
3,2,1
2,3,1
1,4,1
0,5,1
4,0,2
3,1,2
2,2,2
1,3,2
0,4,2
3,0,3
2,1,3
1,2,3
0,3,3
2,0,4
1,1,4
0,2,4
1,0,5
0,1,5
0,0,6
6,0,0
5,1,0
4,2,0
3,3,0
2,4,0
1,5,0
0,6,0
5,0,1
4,1,1
3,2,1
2,3,1
1,4,1
0,5,1
4,0,2
3,1,2
2,2,2
1,3,2
0,4,2
3,0,3
2,1,3
1,2,3
0,3,3
2,0,4
1,1,4
0,2,4
1,0,5
0,1,5
0,0,6

A game-theoretic analysis of insurers pricing strategies

6,0,0
5.31e-01
4.13e-02
3.21e-03
2.50e-04
1.94e-05
1.51e-06
1.18e-07
5.31e-02
4.13e-03
3.21e-04
2.50e-05
1.94e-06
1.51e-07
5.31e-03
4.13e-04
3.21e-05
2.50e-06
1.94e-07
5.31e-04
4.13e-05
3.21e-06
2.50e-07
5.31e-05
4.13e-06
3.21e-07
5.31e-06
4.13e-07
5.31e-07
2,3,1
3.04e-04
2.69e-03
1.96e-02
9.84e-02
1.64e-01
5.80e-02
1.31e-02
1.11e-03
1.07e-02
8.49e-02
4.53e-01
1.51e-01
3.19e-02
2.08e-03
1.75e-02
1.02e-01
1.09e-01
3.17e-02
2.81e-03
1.74e-02
3.00e-02
1.95e-02
2.63e-03
6.07e-03
6.23e-03
1.05e-03
1.47e-03
2.91e-04
6,0,0
5,1,0
4,2,0
3,3,0
2,4,0
1,5,0
0,6,0
5,0,1
4,1,1
3,2,1
2,3,1
1,4,1
0,5,1
4,0,2
3,1,2
2,2,2
1,3,2
0,4,2
3,0,3
2,1,3
1,2,3
0,3,3
2,0,4
1,1,4
0,2,4
1,0,5
0,1,5
0,0,6

5,1,0
1.77e-01
5.19e-01
7.97e-02
9.26e-03
9.58e-04
9.30e-05
8.67e-06
6.79e-02
5.58e-02
8.27e-03
9.49e-04
9.76e-05
9.44e-06
1.18e-02
5.97e-03
8.58e-04
9.73e-05
9.95e-06
1.68e-03
6.36e-04
8.88e-05
9.97e-06
2.18e-04
6.75e-05
9.19e-06
2.69e-05
7.14e-06
3.19e-06
1,4,1
8.44e-06
9.88e-05
1.06e-03
9.70e-03
6.63e-02
1.85e-01
8.04e-02
3.33e-05
4.27e-04
5.11e-03
5.41e-02
4.30e-01
1.80e-01
7.87e-05
9.77e-04
1.08e-02
9.22e-02
1.10e-01
1.50e-04
1.70e-03
1.50e-02
2.88e-02
2.40e-04
2.18e-03
5.63e-03
2.97e-04
9.46e-04
1.45e-04
1,2,3
1.69e-05
1.09e-04
5.05e-04
1.25e-03
2.19e-03
2.68e-03
1.07e-03
1.49e-04
1.08e-03
5.59e-03
1.30e-02
1.82e-02
6.51e-03
1.12e-03
9.20e-03
5.46e-02
1.03e-01
3.30e-02
6.39e-03
5.91e-02
3.99e-01
1.19e-01
2.16e-02
1.68e-01
1.79e-01
4.54e-02
8.10e-02
2.41e-02

4,2,0
2.46e-02
1.42e-01
5.03e-01
1.15e-01
1.77e-02
2.29e-03
2.66e-04
1.64e-02
6.31e-02
5.78e-02
1.23e-02
1.86e-03
2.38e-04
7.77e-03
1.16e-02
6.56e-03
1.32e-03
1.95e-04
1.86e-03
1.72e-03
7.37e-04
1.42e-04
3.43e-04
2.32e-04
8.21e-05
5.46e-05
2.95e-05
7.97e-06
0,5,1
9.38e-08
1.37e-06
1.92e-05
2.58e-04
3.18e-03
3.31e-02
1.98e-01
3.97e-07
6.38e-06
1.02e-04
1.63e-03
2.57e-02
4.03e-01
1.13e-06
1.86e-05
3.05e-04
5.01e-03
8.22e-02
2.76e-06
4.61e-05
7.68e-04
1.28e-02
6.31e-06
1.06e-04
1.78e-03
1.37e-05
2.32e-04
2.91e-05
0,3,3
3.13e-07
2.91e-06
2.31e-05
1.43e-04
5.57e-04
1.69e-03
4.36e-03
2.84e-06
3.01e-05
2.78e-04
2.01e-03
8.30e-03
2.58e-02
2.25e-05
2.75e-04
3.06e-03
2.79e-02
1.23e-01
1.45e-04
2.05e-03
2.83e-02
3.79e-01
6.66e-04
1.01e-02
1.51e-01
2.48e-03
3.88e-02
8.04e-03

3,3,0
1.82e-03
1.57e-02
1.09e-01
4.83e-01
1.46e-01
2.82e-02
4.36e-03
1.73e-03
1.23e-02
5.80e-02
5.91e-02
1.63e-02
3.03e-03
1.38e-03
6.89e-03
1.12e-02
7.07e-03
1.80e-03
8.12e-04
1.75e-03
1.74e-03
8.31e-04
2.49e-04
3.34e-04
2.43e-04
5.61e-05
5.49e-05
1.06e-05
4,0,2
2.46e-02
1.28e-02
4.70e-03
8.77e-04
1.25e-04
1.55e-05
1.76e-06
1.36e-01
4.70e-02
6.78e-03
7.92e-04
8.40e-05
8.40e-06
4.63e-01
4.32e-02
3.94e-03
3.55e-04
3.15e-05
1.35e-01
1.16e-02
9.85e-04
8.35e-05
2.67e-02
2.22e-03
1.84e-04
4.43e-03
3.62e-04
6.62e-04
2,0,4
7.59e-05
7.28e-05
6.40e-05
4.87e-05
2.85e-05
8.40e-06
1.76e-06
8.33e-04
7.26e-04
5.46e-04
3.09e-04
7.92e-05
1.45e-05
8.24e-03
6.11e-03
3.34e-03
7.11e-04
1.11e-04
6.82e-02
3.59e-02
5.89e-03
7.61e-04
3.84e-01
4.18e-02
4.32e-03
1.85e-01
1.74e-02
5.49e-02

2,4,0
7.59e-05
8.73e-04
9.04e-03
7.83e-02
4.59e-01
1.74e-01
4.02e-02
9.37e-05
9.78e-04
8.61e-03
5.28e-02
5.96e-02
2.00e-02
1.06e-04
9.45e-04
6.04e-03
1.08e-02
7.49e-03
1.04e-04
6.87e-04
1.63e-03
1.74e-03
7.77e-05
2.25e-04
3.23e-04
2.95e-05
5.28e-05
7.97e-06
3,1,2
5.47e-03
1.84e-02
2.00e-02
1.25e-02
3.20e-03
5.79e-04
8.67e-05
3.18e-02
1.15e-01
8.69e-02
1.90e-02
2.98e-03
3.96e-04
1.23e-01
4.53e-01
8.34e-02
1.14e-02
1.36e-03
1.60e-01
1.42e-01
2.32e-02
2.91e-03
5.65e-02
3.01e-02
4.61e-03
1.35e-02
5.32e-03
2.65e-03
1,1,4
8.44e-06
3.22e-05
6.18e-05
9.35e-05
1.18e-04
1.14e-04
4.34e-05
9.49e-05
4.00e-04
7.41e-04
1.02e-03
1.03e-03
3.49e-04
9.78e-04
4.65e-03
7.93e-03
8.68e-03
2.57e-03
8.74e-03
4.80e-02
6.54e-02
1.64e-02
5.97e-02
3.74e-01
8.06e-02
1.98e-01
1.95e-01
1.10e-01

1,5,0
1.69e-06
2.42e-05
3.34e-04
4.31e-03
4.97e-02
4.33e-01
1.98e-01
2.56e-06
3.54e-05
4.59e-04
5.33e-03
4.76e-02
5.96e-02
3.75e-06
4.88e-05
5.72e-04
5.21e-03
1.03e-02
5.19e-06
6.13e-05
5.71e-04
1.50e-03
6.56e-06
6.23e-05
2.02e-04
6.79e-06
2.58e-05
3.19e-06
2,2,2
4.56e-04
2.84e-03
1.21e-02
2.18e-02
2.19e-02
7.26e-03
1.60e-03
2.78e-03
1.91e-02
8.91e-02
1.19e-01
3.54e-02
6.98e-03
1.21e-02
8.94e-02
4.38e-01
1.20e-01
2.18e-02
2.63e-02
1.46e-01
1.47e-01
3.46e-02
3.31e-02
5.51e-02
3.31e-02
1.41e-02
1.38e-02
3.97e-03
0,2,4
2.34e-07
1.56e-06
7.82e-06
2.43e-05
6.03e-05
1.32e-04
2.66e-04
2.70e-06
2.04e-05
1.16e-04
3.74e-04
9.45e-04
2.09e-03
2.90e-05
2.54e-04
1.70e-03
5.74e-03
1.48e-02
2.77e-04
2.92e-03
2.49e-02
8.79e-02
2.20e-03
2.88e-02
3.58e-01
1.24e-02
1.78e-01
5.49e-02

0,6,0
1.56e-08
2.69e-07
4.62e-06
7.95e-05
1.37e-03
2.35e-02
4.05e-01
2.81e-08
4.84e-07
8.32e-06
1.43e-04
2.46e-03
4.23e-02
5.06e-08
8.71e-07
1.50e-05
2.58e-04
4.43e-03
9.11e-08
1.57e-06
2.70e-05
4.64e-04
1.64e-07
2.82e-06
4.85e-05
2.95e-07
5.08e-06
5.31e-07
1,3,2
1.69e-05
1.53e-04
1.17e-03
6.54e-03
1.87e-02
3.15e-02
1.31e-02
1.08e-04
1.10e-03
9.52e-03
6.07e-02
1.43e-01
5.46e-02
5.17e-04
5.81e-03
5.72e-02
4.18e-01
1.53e-01
1.54e-03
1.63e-02
1.33e-01
1.50e-01
3.36e-03
2.87e-02
5.32e-02
5.20e-03
1.31e-02
2.65e-03
1,0,5
1.69e-06
1.99e-06
2.27e-06
2.44e-06
2.40e-06
1.93e-06
7.06e-07
2.31e-05
2.62e-05
2.82e-05
2.75e-05
2.18e-05
7.04e-06
3.04e-04
3.25e-04
3.15e-04
2.45e-04
6.81e-05
3.74e-03
3.60e-03
2.75e-03
6.30e-04
4.12e-02
3.08e-02
5.40e-03
3.44e-01
4.02e-02
2.00e-01

5,0,1
1.77e-01
5.28e-02
7.14e-03
7.92e-04
8.00e-05
7.65e-06
7.06e-07
4.99e-01
4.27e-02
3.63e-03
3.06e-04
2.56e-05
2.13e-06
9.80e-02
8.01e-03
6.54e-04
5.32e-05
4.32e-06
1.46e-02
1.18e-03
9.45e-05
7.58e-06
1.94e-03
1.55e-04
1.24e-05
2.42e-04
1.92e-05
2.91e-05
0,4,2
2.34e-07
2.80e-06
3.08e-05
2.98e-04
2.31e-03
1.11e-02
4.02e-02
1.56e-06
2.10e-05
2.69e-04
3.14e-03
3.06e-02
1.60e-01
8.20e-06
1.24e-04
1.85e-03
2.73e-02
3.94e-01
3.04e-05
4.81e-04
7.58e-03
1.19e-01
9.38e-05
1.52e-03
2.45e-02
2.59e-04
4.23e-03
6.62e-04
0,1,5
9.38e-08
3.78e-07
8.75e-07
1.71e-06
3.07e-06
5.25e-06
8.67e-06
1.31e-06
5.88e-06
1.39e-05
2.74e-05
4.95e-05
8.48e-05
1.77e-05
9.12e-05
2.21e-04
4.39e-04
7.97e-04
2.29e-04
1.41e-03
3.50e-03
7.03e-03
2.76e-03
2.17e-02
5.54e-02
2.87e-02
3.33e-01
2.00e-01

Table 3.17 – Values of the transition matrix
138

4,1,1
4.92e-02
1.55e-01
9.81e-02
2.01e-02
2.98e-03
3.77e-04
4.34e-05
1.53e-01
4.88e-01
8.25e-02
1.05e-02
1.17e-03
1.23e-04
1.21e-01
1.03e-01
1.61e-02
1.93e-03
2.08e-04
3.17e-02
1.65e-02
2.44e-03
2.86e-04
6.05e-03
2.33e-03
3.34e-04
9.83e-04
3.09e-04
1.45e-04
3,0,3
1.82e-03
1.35e-03
8.36e-04
3.82e-04
9.10e-05
1.60e-05
2.35e-06
1.50e-02
9.15e-03
3.99e-03
7.94e-04
1.19e-04
1.53e-05
9.98e-02
4.14e-02
6.36e-03
7.81e-04
8.66e-05
4.24e-01
4.28e-02
4.18e-03
3.97e-04
1.64e-01
1.47e-02
1.30e-03
4.04e-02
3.47e-03
8.04e-03
0,0,6
1.56e-08
2.19e-08
3.06e-08
4.29e-08
6.00e-08
8.40e-08
1.18e-07
2.56e-07
3.59e-07
5.02e-07
7.03e-07
9.84e-07
1.38e-06
4.20e-06
5.88e-06
8.24e-06
1.15e-05
1.61e-05
6.89e-05
9.65e-05
1.35e-04
1.89e-04
1.13e-03
1.58e-03
2.22e-03
1.85e-02
2.60e-02
3.04e-01

3,2,1
5.47e-03
3.28e-02
1.28e-01
1.35e-01
3.75e-02
7.00e-03
1.07e-03
1.85e-02
1.18e-01
4.73e-01
1.19e-01
2.00e-02
2.80e-03
2.49e-02
1.12e-01
1.07e-01
2.40e-02
3.79e-03
1.98e-02
3.10e-02
1.81e-02
3.78e-03
6.50e-03
6.17e-03
2.70e-03
1.51e-03
1.04e-03
2.91e-04
2,1,3
3.04e-04
1.09e-03
1.65e-03
1.76e-03
1.29e-03
4.03e-04
8.67e-05
2.59e-03
1.02e-02
1.34e-02
1.05e-02
2.88e-03
5.47e-04
1.83e-02
7.89e-02
7.59e-02
1.78e-02
2.93e-03
9.18e-02
4.15e-01
8.27e-02
1.20e-02
1.85e-01
1.73e-01
2.95e-02
8.34e-02
4.57e-02
2.41e-02

3.8. Appendix
3.8.1.6

Properties of the loss model

Proof of Prop. 3.3.14. Using assumptions A5, A6, A7, we can show that the moment generating function of Sj,t given that Nj,t = nj isM
nj∀u, GM
Sj,t (u) = (GYi,t (u)) .

Since Yi,t is a compound distribution, we getP
MGM
Yi,t (u) = GMi,t (GZ (u)).

As we use a classic frequency distribution (either Poisson or negative binomial), we obtain

r
p
GP
(u)
=
exp(λ(u
−
1))
or
Mi,t
1 − (1 − p)uleading toM
GMSj,t (u) = exp(λnj (GZ (u) − 1)) orp
1 − (1 − p)GM
Z (u)

rnj
.

That is the scale parameter of a Poisson distribution is λ and the scale parameter of a negative binomial
is r, see also Appendix 3.8.5.3 for details.
Proof of Prop. 3.3.15. When a price vector x is such that xj < xk for all k 6= j, pk→j (x) > pk→l (x)for l 6= j given the initial portfolio sizes nj ’s are constant, since the change probability pk→j (fork 6= j) is a decreasing function (see Appendix 3.8.3.1). Using the stochastic orders (≤st , ≤cx ) andthe majorization order (≤m ) whose definitions and main properties are recalled in the Appendices3.8.1.7 and 3.8.1.8 respectively, we can show a stochastic order of the portfolio size by applying theconvolution property of the stochastic order J times: Nk (x) ≤st Nj (x), ∀k 6= j. Let us consider theempirical loss average of an insurer with portfolio size nnA(n) =

1X
Yi ,
n i=1

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

ñ
X
1
i=1

ñ

Yi ≤cx

n
X
1
i=1

n

Yi ⇔ A(ñ) ≤cx A(n).

Using Theorem 3.A.23 of Shaked and Shanthikumar (2007), except that for all φ convex, E(φ(A(n)))
is a decreasing function of n and Nk (x) ≤st Nj (x), we can show A(Nj (x)) ≤cx A(Nk (x)).

3.8.1.7

Notation 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 are various otherequivalent characterizations, including expectectation order for all increasing function E(φ(X)) ≤−1
E(φ(Y )), quantile order FX(u) ≤ FY−1 (u), distribution function order FX (x) ≥ FY (x).

139

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

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 thismean, we can show that an ordering of binomial distributions. If X ∼ B(n, p) and Y ∼ B(n, q), suchthat p ≤ q, then X ≤st Y . Theorem 1.A.3 of Shaked and Shanthikumar (2007) also shows that thestochastic 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 as E(φ(X)) ≤ E(φ(Y )) for all convex
functions φ. If we restrict to differentiable functions, it means ∀x, φ(2) (x) ≥ 0. This explains therelation 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 ) andE((X −a)+ ) ≤ E((Y −a)+ ). By theorem 3.A.12 of Shaked and Shanthikumar (2007), the convex orderis 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.

3.8.1.8

Notation and definition of majorization orders

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

k
X

ai ≤

i=1

k
X

ãi and

i=1

n
X
i=1

ai =

n
X

ãi .

i=1

A direct consequence of property B.2.c ofPMarshall andP Olkin (1979) is that if X1 , . . . , Xn are exchangeable and a, ã ∈ Rn , a ≤m ã implies i ai Xi ≤cx i ãi Xi .

3.8.2

Proofs of Section 3.4

3.8.2.1

Existence and uniqueness properties

Proof of Prop. 3.4.1. Since the strategy set is [x, x]J , it guarantees the market proxy mj = mj (x) or
mj = mj (x, n) to be positive. Therefore, given x−j , the function xj 7→ Oj (x) is a quadratic function
with second-degree term −βj x2j /mj , hence concave. For all insurer, the constraint functions (3.9) arelinear. By Theorem 1 of Rosen (1965), the existence of a premium equilibrium is guaranteed. The
proof of uniqueness is exactly the same as Dutang et al. (2013).
Proof of Prop. 3.4.2. Consider a generic objective functionXxj
Oj (x) = aj − bj
(xj − cj ), with mj (x) =w i xi ,mj (x)
i6=jwith known weights wi > 0 and positive constant aj , bj , cj > 0. For simplicity, we remove themultiplicative term nj /N . Note that the weights wi do not sum up to 1. By assumption, no constraintfunctions are active, i.e. λj? = 0. Thus if x? is a Nash equilibrium, x? must verify∀j, ∇xj Oj (x? ) = 0 ⇔ −bj

1
xj(xj − cj ) + aj − bj
= 0 ⇔ −bj (2xj − cj ) + aj mj (x) = 0
mj (x)
mj (x)
X⇔ 2bj xj − ajwi xi = bj cj ⇔ M x = v,
i6=j

with



2b1
−a2 w2

M =


140

−a1 w12b2−a1 w2−a2 w3..
.

...

−aJ wJ−1





b1 c1

...
 . 
 , v =  ..  .

bJ cJ
2bJ

3.8. Appendix
The matrix M can be rewritten as M = M1 M2 with


2b1 /w1 −a2
M1 = 

−a12b2 /w2

...
−a2
..
.

...

...

−aJ

2bJ /wJ


w1



,
M
=


2

...

0
..
.

...

0

wJ



.

P
In the case of objective function (3.7), we choose aj = 1 + βj , bj = βj i6=j wi and cj = πj . When themarket proxy is the arithmetic mean (3.5), we set wj = 1 and bj = βj (J − 1) leading to
2β1 (J − 1)π1 β1 (J − 1)



.
..
v=
 , M1 = πJ βJ (J − 1)...

−1 − β1..
.
−1 − βJ





...

1



 , M2 = ...
2βJ (J − 1)

0
..
.

...

0

1



.

When the market proxy is the weighted mean (3.6), we set wj = nj and bj = βj (N − nj ) leading to
2β1 (N − n1 )π1 β1 (N − n1 )



..
,
M
=
v=


1
.


πJ βJ (N − nJ )3.8.2.2...

−1 − β1..
.
−1 − βJ



...



n1



 , M2 = 
2βJ (N − nJ )
...

0
..
.

...

0

nJ



.

Recursive formula for the determinant of a multidiagonal matrix

Proposition 3.8.1. Consider the following multi-diagonal matrix for n ≥ 2



u1
 v2


Mn = 





v1
u2

...
v2
..
.

...

...

vn−1
...

un−1
vn




.

vn−1 
un

Let w1 = u1 and wj = vj , ∀j = 2, . . . , n. The determinant is given bydet (Mn ) = (−1)n+1

n
X
j=1

wj

Y

(vk − uk ).

k6=j

141

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Proof. Let us find a recursive formulau1
v2
det (Mn )

v1
u2

...
v2
..

=
...

.
un−1
vn

v1 − u10..
.
0
un − vnv2==(un − vn )det (Mn−1 ) + (−1)n+1 (v1 − u1 )

u2
..
.

v2 − u200..
(un − vn )det (Mn−1 ) + (−1)n+1 (v1 − u1 )
.
..
.
0

v2

...

un−1
vn

u2
v3
v4

v2
u3
v4

=

(un − vn )det (Mn−1 ) + (−1)

...
v4
..
.

...

un−1
vn

...
v3
v4

n+1

v3
u4

u3
v4

(v1 − u1 )(v2 − u2 )

v3
u4

...
v4
..
.

...
un−1
vn

...
=

(un − vn )det (Mn−1 ) + (−1)n+1 (v1 − u1 )(v2 − u2 ) . . . (vn−2 − un−2 )

vn−1
vn

un−1
vn

(un − vn )det (Mn−1 ) + (−1)n+1 (v1 − u1 )(v2 − u2 ) . . . (vn−1 − un−1 )vn
Pn
Q
Let us guess that det (Mn ) = (−1)n+1 j=1 wj k6=j (vk − uk ). We can check=det (Mn+1 )==(un+1 − vn+1 )det (Mn ) + (−1)n+1 (v1 − u1 )(v2 − u2 ) . . . (vn − un )vn+1
n
X
Y
n+1
(un+1 − vn+1 )(−1)
wj(vk − uk ) + (−1)n+1 (v1 − u1 )(v2 − u2 ) . . . (vn − un )vn+1
j=1=(−1)n+2

n+1
X
j=1

wj

Y

k6=j

(vk − uk ).

k6=jLet us check the recurrence for n = 2: det (M2 ) = u1 u2 − v1 v2 = (u2 − v2 )u1 + (−1)2+1 (v1 − u1 )v2 .

3.8.2.3

Sensitivity analysis

Proposition 2.2 of Dutang et al. (2013) state the sensitivity of premium equilibrium with respect
to parameters when the objective function is (3.7) and the market proxy is the arithmetic mean (3.5).
For sake of completeness, we unify the proof for both market proxies the arithmetic mean (3.5) or theweighted mean (3.6).
Proof of Prop. 3.4.3. The premium equilibrium x? solves the necessary Karush-Kuhn-Tucker conditions: for all Insurer j,
X j?
∇xj Oj (x? ) +λl ∇xj gjl (x?j ) = 0,1≤l≤3
(3.23)
0 ≤ λj? , gj (x?j ) ≥ 0, gj (x?j )T λj? = 0,142

3.8. Appendix
where λj? ∈ R3 are Lagrange multipliers, see e.g. Facchinei and Kanzow (2009). In the last part ofequation (3.23), gj (x?j )T λj? = 0 is the complementarity equation implying that the lth constraint gjl
is either active (gjl (x?j ) = 0) or inactive (gjl (x?j ) > 0), but λj?
l = 0.
j?
?
If λj?
>
0
or
λ
>
0,
then
x
=
x
or
x
respectively,
thenx?j is independent of any parameter.
j2
3
j?
If λ1 > 0, then the solvency constraint (3.9) is active, i.e. the premium equilibrium x?j verifies
1 ?
gj (xj ) = 0. Hence we have√k995 σ(Y ) nj − Kj?
xj = πj +.
(3.24)
nj
Here, the implicit function theorem is not necessary since x?j does not depend on x?−j . We deduce thatx?j is an increasing function of πj , k995 , σ(Y ) and a decreasing function Kj . The function nj 7→ x?j (nj )is not necessarily monotone. Let z = nj . Differentiating Equation (3.24) with respect to z, we getkσ(Y ) Kj
10+√,
ϕ (z) = 3/2 −2zz
whose sign depends on the value of the other parameters.
j?
j?
?
Otherwise λj?
1 = λ2 = λ3 = 0 Insurer j’s premium equilibrium verifies ∇xj Oj (x ) = 0. If themarket proxy is the arithmetic mean (3.5), then


x?j
nj
πj
1 + βj − 2βj
+
β
= 0.
j
N
mj (x? )mj (x? )Let xjy be the premium vector with the jth component equal to y, i.e. xjy = (x1 , . . . , xj−1 , y, xj+1 , . . . , xJ ).
Given a parameter z on which we want to investigate the sensitivity, we define the bivariate function
Fxj as


x?j
∂Oj j
nj
πj
Fxj (z, y) =(xy , z) =+β
1 + βj − 2βj
.
j
∂xj
N
mj (x? )mj (x? )Hence, the KKT condition can be simply 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∂F j
can invoke the implicit function theorem, see Appendix 3.8.6.1 for details. So, if ∂yx (z0 , y0 ) 6= 0, thederivative of ϕ is given by0

ϕ (z) = −∂Fxj∂z (z, y).
∂Fxj∂y (z, y) y=ϕ(z)In our case, the sign of the denominator can be easily deduced
 j

∂Fxj
∂ 2 Oj jnj
0∂Fx(z, y) =
(x
,
z)
=
−2β
<
0
⇒
sign(ϕ
(z))
=
sign
.
(z,
ϕ(z))
j
y
∂z
∂y
∂x2j
N mj (x)The sign of the derivative w.r.t. πj , βj , nj is given in Table 3.18.

z
πj
βj
nj

j
∂ ∂Oj
∂z ∂xj (xy , z)(z, y)n j βjN mj (x)


nj
πj
y
1
−
2β
+
j mj (x)Nmj (x)x?jπj
1 + βj − 2βj mj (x? ) + βj mj (x
?)

j
∂ ∂Oj
∂z ∂xj (xϕ(z) , z)nj βj
N mj (x? )−njzNsign ϕ0 (z)

0

0

>0
<0

Table 3.18 – Sensitivity analysis of premium x?j when market proxy is (3.5).
143

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

If the market proxy is the weighted mean (3.6), then


x?j
nj
πj
1 + βj − 2βj
+ βj
= 0.
N
mj (x? , n)mj (x? , n)Here, we define the bivariate function Gjx as∂Oj jnj
Gjx (z, y) =(x , z) =
∂xj y
N


1 + βj − 2βj

x?j
πj
+ βj
mj (x? )mj (x? ).

Hence, the KKT condition can be simply rewritten as Gjx? (z, x?j ) = 0. Again, we apply the implicit
function theorem. In our case, the sign of the denominator can be easily deduced

 j
∂Gjx
nj
∂ 2 Oj j
0
∂Gx
(x
,
z)
=
−2β
.
(z, y) =<0⇒sign(ϕ
(z))
=
sign
(z,
ϕ(z))
j
y
∂z
∂y
∂x2j
N mj (x, n)
The sign of the derivative w.r.t. πj , βj , nj is given in Table 3.19.

z
πj
βj
nj

j
∂ ∂Oj
∂z ∂xj (xy , z)(z, y)n j βjN mj (x,n)


nj
πj
y
1
−
2β
+
j
N
mj (x,n)mj (x,n)2βj yβj π j(πj −2)zβj
1 + βj − mj (x,n) + mj (x,n) + N (N −z)mj (x,n)j∂ ∂Oj∂z ∂xj (xϕ(z) , z)nj βj
N mj (x? ,n)−njzN(πj −2)zβj
N (N −z)mj (x? ,n)sign ϕ0 (z)>0<0sign(πj − 2)

Table 3.19 – Sensitivity analysis of premium x?j when market proxy is (3.6).
3.8.2.4

A necessary and sufficient condition

Using the proposition of Appendix 3.8.2.2, we get the determinant of M1
det (M1 )

J
X
Y
2b1 Y
(−ak − 2bk /wk ) + (−1)J+1(−aj )(−ak − 2bk /wk )
w1
j=2

=

(−1)J+1

=

J
X
Y
2b1 Y
(ak + 2bk /wk ) −aj(ak + 2bk /wk ).
w1
j=2

k6=1

k6=1

k6=j

k6=j

Σ
Let aj = 1 + βj and bj = βj . Thus, det (M ) = 0 is equivalent to det (M1 ) = 0. Using w−j=that isJΣ YXY
2β1 w−1ΣΣ(1 + βk + 2βk w−k/wk ) =
(1 + βj )(1 + βk + 2βk w−k
/wk )
w1
j=2
k6=1
k6=j
Q
Σ
Π
Using β̃j = 1 + βj + 2βj w−j/wj and β̃−j= k6=j β̃k ,Σ
det (M1 ) = 0 ⇔ 2β1 w−1β̃1 = w1

J
X

P

k6=j wk ,(1 + βj )β̃j .

j=2There are many solutions so that βj ’s solve this equation. But there is a unique solution when wj = w
is constant and βj ’s are all identical. Say βj = β leading to β̃j = 1 + β + 2β(J − 1). Hence det (M1 ) = 0
yields to β = 1⇔ 2βw(J − 1)(1 + β + 2β(J − 1)) = w

JX(1 + β)(1 + β + 2β(J − 1))j=2⇔ 2β(J − 1)(1 + β + 2β(J − 1)) = (1 + β)(J − 1)(1 + β + 2β(J − 1)) ⇔ 2β = 1 + β ⇔ β = 1144

3.8. Appendix
3.8.2.5

The two-insurer case

The objection function is the following generic for j = 1, 2

xj
Oj (x) = aj − bj
(xj − cj ), x ≥ xj ≥ x,x−jNote that ai , bi , ci are always positive. We recall that g1 (xj ) is the solvency constraint, g2 (xj ) = xj − x
and g3 (xj ) = x − xj .
In the following, we use in red the (strict) positivity Lagrange multipliers (enforcing x or x) andin orange the (strict) premium bound inequalities.
The premium equilibrium x? for each Insurer j solves the necessary Karush-Kuhn-Tucker conditions:
X j?
∇xj Oj (x? ) +λl ∇xj gjl (x?j ) = 0,1≤l≤3
(3.25)j?
?
? T j?
0 ≤ λ , gj (xj ) ≥ 0, gj (xj ) λ = 0,
where λj? ∈ R3 are Lagrange multipliers, see e.g. Facchinei and Kanzow (2009). In the last part ofequation (3.23), gj (x?j )T λj? = 0 is the complementarity equation implying that the lth constraint gjl
is either active (gjl (x?j ) = 0) or inactive (gjl (x?j ) > 0), but λj?
l = 0.
j?
We suppose that no solvency constraint. Hence, λ1 = 0. (3.23) becomes for j = 1, 2 (only 2
insurers)
xj
1
 bj cj x−j+ aj − 2bj x−j+ λj2 − λj3 = 0 j
λ2 (xj − x) = 0(3.26)

λj3 (x − xj ) = 0xj ∈ [x, x], λji ≥ 0
We consider 9 cases for the premium equilibrium (x?1 , x?2 ) depending on Lagrange multipliers λlj , see
Table 3.20.

Player 2(x?1 , x?2 )2
λ2 > 0, λ23 = 0
λ22 = 0, λ23 = 0
λ22 = 0, λ23 > 0

λ12 > 0, λ13 = 0(x, x)(x, x̃b )(x, x)

Player 1
λ12 = 0, λ13 = 0(x̃a , x)NC(x1 C , xN
2 )(x̃d , x)

λ12 = 0, λ13 > 0(x, x)(x, x̃c )(x, x)

Table 3.20 – 9 cases for Lagrange multipliers (x?1 , x?2 ) with x < x̃. < x3.8.2.5.1

C
NC
Case (xN
1 , x2 ) (3.26) becomes


x1
1
(
 b1 c1 x2 + a1 − 2b1 x2 = 0
1 b2 c2 +a2 b1 c1
x2 = 2b4bx2
1
1 b2 −a1 a2
b2 c2 x1 + a2 − 2b2 x1 = 0 ⇔2b1 b2 c1 +a1 b2 c2
x1 = 4b1 b2 −a1 a2
 jj
λ2 = 0, λ3 = 0
This case is possible iff 4b1 b2 − a1 a2 6= 0 and prem. bound constraints areCx < xN=12b1 b2 c1 + a1 b2 c2
2b1 b2 c2 + a2 b1 c1C< x, x < xN=
< x.
2
4b1 b2 − a1 a2
4b1 b2 − a1 a2

As the numerator is positive, 4b1 b2 − a1 a2 < 0 makes impossible these inequalities.

145

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

In the case of objective function (3.7) and the market proxy is the arithmetic mean (3.5), we choose
aj = 1 + βj , bj = βj (J − 1) = βj and cj = πj . The constraints becomeCx < xN=12β1 β2 π1 + (1 + β1 )β2 π22β1 β2 π2 + (1 + β2 )β1 π1C=< x, x < xN<x2
4β1 β2 − (1 + β1 )(1 + β2 )4β1 β2 − (1 + β1 )(1 + β2 )In the case of positive determinant 4β1 β2 − (1 + β1 )(1 + β2 ) > 0, we getx(4β1 β2 − (1 + β1 )(1 + β2 )) < 2β1 β2 π1 + (1 + β1 )β2 π2 < x(4β1 β2 − (1 + β1 )(1 + β2 ))x(4β1 β2 − (1 + β1 )(1 + β2 )) < 2β1 β2 π2 + (1 + β2 )β1 π1 < x(4β1 β2 − (1 + β1 )(1 + β2 ))((1+β1 )(1+β2 )2)1)x(4 − (1+ββ11)(1+β) < 2π1 + (1+β)β2
β1 π2 < x(4 −β1 β2⇔(1+β2 )
(1+β1 )(1+β2 )(1+β1 )(1+β2 )x(4 −) < 2π2 + β2 π1 < x(4 −
)
β1 β2
β1 β2
(
(1+β1 )(1+β2 )1)2)x(4 − (1+ββ11)(1+β) < 2π1 + (1+β)β2
β1 π2 < x(4 −β1 β2⇔(1+β1 )
(1+β1 )(1+β2 )(1+β1 )(1+β2 )(1+β1 )
(1+β1 ) (1+β2 )(1+β1 )− β1 x(2 −) > − β1 π2 − β1)2β1 β22β2 π1 > − β1 x(2 −2β1 β2
(1 + β1 )(1 + β1 )(1 + β2 )(1 + β1 )(1 + β2 )⇒ 2x −x (2 −) < π1 2 −β12β1 β22β1 β2(1 + β1 )(1 + β1 )(1 + β2 )< 2x −x (2 −)β1
2β1 β2⇒ 2x −(1 + β1 )(1 + β1 )x < π1 < 2x −x
β1
β1

In the case of negative determinant, there are two times a sign change in inequalities leading to thesame result. Similarly for π2 , we get
2x −3.8.2.5.2(1 + β2 )(1 + β2 )x < π2 < 2x −
x.
β2
β2

Case (x, x) (3.26) becomes



xj
j
1
1
(
1
 bj cj x−j + aj − 2bj x−j + λ2 = 0
 b1 c1 x + a1 − 2b1 + λ2 = 0
λ12 = 2b1 − a1 − b1xc1
12
b2 c2 x + a2 − 2b2 + λ2 = 0 ⇔⇔xj − x = 0
λ22 = 2b2 − a2 − b2xc2
 j
x1 = x2 = x, λ13 = λ23 = 0
λ3 = 0This case is possible iff (positivity of Lagrange multipliers) 2b1 − a1 − b1xc1 > 0 and 2b2 − a2 − b2xc2 > 0.
The objective value at (x, x) is Oj (x, x) = (aj − bj )(x − cj ).
In the case of objective function (3.7) and the market proxy is the arithmetic mean (3.5), we choose
aj = 1 + βj , bj = βj and cj = πj . The positivity of Lagrange multipliers leads to∀j, xβj − 1> πj .
βj

If 0 < βj ≤ 1 then this condition is impossible.

3.8.2.5.3

Case (x, x) (3.26) becomes

 j
xj
j
1
 bj cj x−j + aj − 2bj x−j − λ3 = 0
 λ3 = −2bj + bj cj x1 + aj
xj = x⇔x−x =0 j j
 j
λ2 = 0λ2 = 0This case is possible iff (positivity of Lagrange multipliers) −2bj + bj cj x1 + aj > 0. The objective value
at (x, x) is Oj (x, x) = (aj − bj )(x − cj ) > Oj (x, x).

146

3.8. Appendix
In the case of objective function (3.7) and the market proxy is the arithmetic mean (3.5), we choose
aj = 1 + βj , bj = βj (J − 1) = βj and cj = πj . The positivity of Lagrange multipliers leads to∀j, πj >βj − 1x.
βj

If 0 < βj ≤ 1 then this condition is impossible.

3.8.2.5.4

Case (x̃a , x) (3.26) becomes
b1 c1 +a1 xx1
 b1 c1 x + a1 − 2b1 x1 = 0
 x1 =
2b1
x+4b1 b2 x
b2 c2 x11 + a2 − 2b2 x1 + λ22 = 0 ⇔λ22 = −2b1b1b2cc12+a
− a2
1x



1
1
2
1
x2 = x, λ2 = λ3 = λ3 = 0
x2 = x, λ3 = λ23 = 0

This case is possible iff b1 c1 + a1 x 6= 0 (always true because positive) and the prem. bound inequality
and the lagr. pos. constraint are resp.
x < x̃a =b1 c1 + a1 x
−2b1 b2 c2 + 4b1 b2 x
> a2 .
< x,
2b1
b1 c1 + a1 x

In the case of objective function (3.7) and the market proxy is the arithmetic mean (3.5), we choose
aj = 1 + βj , bj = βj and cj = πj . The premium isx̃a =β1 π1 + (1 + β1 )x.
2β1

We know the sign of the denominator β1 π1 + (1 + β1 )x > 0. The prem. bound constraint isx

1 + β1β1 − 1< π1 < 2x −
x.
β1
β1

The Lagr. pos. constraint becomes⇔−2β1 β2 π2 + 4β1 β2 x
> 1 + β2 .
β1 π1 + (1 + β1 )x−2β1 β2 π2 + 4β1 β2 x > (1 + β2 )(β1 π1 + (1 + β1 )x)⇔ (4β1 β2 − (1 + β1 )(1 + β2 ))x > (1 + β2 )β1 π1 + 2β1 β2 π2(1 + β2 )β1 π1 + 2β1 β2 π2
C
⇔ x>
= xN
2
4β1 β2 − (1 + β1 )(1 + β2 )Using the prem. bound ineq., this implies
4x −(1 + β2 )(β1 − 1)
(1 + β2 )(1 + β1 )(1 + β2 )β2 − 1
x > 2π2 +x ⇔ 4x − 2
x > 2π2 ⇔x > π2
β1 β2
β2 β1
β2
β2

3.8.2.5.5

Case (x, x) (3.26) becomes 1
b1 c1 x1 + a1 − 2b1 xx − λ13 = 0
λ = b1 c1 x1 + a1 − 2b1 xx





 23
x
1
2
b2 c2 x + a2 − 2b2 x + λ2 = 0
λ2 = −b2 c2 x1 − a2 + 2b2 xx⇔
1
λ = 0, x2 = x
λ1 = 0, x2 = x


 22
 22
λ3 = 0, x1 = x
λ3 = 0, x1 = x

This case is possible iff the Lagr. multipliers are positiveb1 c1 + a1 x − 2b1 x > 0, −b2 c2 − a2 x + 2b2 x > 0147

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

In the case of objective function (3.7) and the market proxy is the arithmetic mean (3.5), we choose
aj = 1 + βj , bj = βj and cj = πj . The constraints become
π1 > 2x −1 + β1
1 + β2x > π2x, 2x −β1
β2

Equivalently the positivity of multipliers areβ1 π1 + (1 + β1 )x > 2β1 x⇔
2β2 x > β2 π2 + (1 + β2 )x

(

π1
1
x
2x + 2β1 + 1/2 > xxπ2
1x > 2x + 2β2 + 1/2

Or xx > 1 > xx So λ22 > 0 is impossible when β2 ≤ 1 but λ13 > 0 is always true when β1 ≤ 1.

3.8.2.5.6

Case (x, x̃c ) (3.26) becomes

 1
b1 c1
1 b2 x+ a1 − b24b
 b1 c1 x12 + a1 − 2b1 xx2 − λ13 = 0
 λ3 = b22bc22+a
c2 +a2 x
2xx⇔
b c 1 + a2 − 2b2 x2 = 0
x = c2 /2 + a2 x/(2b2 ) 2 2x 2
x1 = x, λ12 = λ22 = λ23 = 0
x1 = x, λ12 = λ22 = λ23 = 0

This case is possible iff b2 c2 + a2 x 6= 0 (always true because positive) and the lagr. pos. and the prem.
bound constraints are resp.
2b2 b1 c1 − 4b1 b2 x
+ a1 > 0, x < c2 /2 + a2 x/(2b2 ) < x
b2 c2 + a2 x
In the case of objective function (3.7) and the market proxy is the arithmetic mean (3.5), we choose
aj = 1 + βj , bj = βj (J − 1) = βj and cj = πj . We havex̃c =β2 π2 + (1 + β2 )x.
2β2The Lagr. pos. constraint becomes
2β2 β1 π1 − 4β1 β2 x
+ 1 + β1 > 0
β2 π2 + (1 + β2 )x⇔ 2β2 β1 π1 − 4β1 β2 x + (1 + β1 )β2 π2 + (1 + β1 )(1 + β2 )x > 0⇔ 2β2 β1 π1 + (1 + β1 )β2 π2 > 4β1 β2 x − (1 + β1 )(1 + β2 )x2β2 β1 π1 + (1 + β1 )β2 π2⇔>x4β1 β2 − (1 + β1 )(1 + β2 )
C
⇔ xN
> x.
1

The prem. bound inequalities rewritex < x̃c < x ⇔ 2x −1 + β2β2 − 1x < π2 < x
.
β2
β2

C
Using the last inequality and x̃N> x we arrive to1π1 > 2x − x

3.8.2.5.7

1 + β1β1 − 1=xβ1
β1

Summary with respect to Lagrange multipliers We plot the 4 bounds forthe break-even premium in Figure 3.13. We summarize the constraints either lagr. pos. or prem.
bound ineq. in Table 3.21 for the value of KKT system solution (3.26).
148

3.8. Appendix
bounds for, π, β = 2, x = 3, x = 1

bounds for, π, x = 3, x = 1

bounds for, π, β = 0.5, x = 3, x = 1β+1β2x − x

β

-50

3

4

π2
0

β
-2

β
β+1

2

β−1

x

β+1

2x − x1xx

2x − x

0

β
x

-2

β

β−1

0

β
β−1

x

β

2

2

β−1

π2

-20
-30

β−1

-40

bounds for pi

-10

4

4

0

β+1

2x − xβ−1β+1

2x − x

β

β

x

β

5

-2

0

2

β

4

-2

π1

0

2

4

π1

β−1
Figure 3.13 – Bound examples for π with solid line x β−1
β , dashed line x β , dotted line
1+β
2x − 1+ββ x, dash-dotted line 2x − β x

λ12 > 0, λ13 = 0
π1 < x β1β−1prem. eq.
λ22 > 0λ23 = 0P2

λ22 = 0λ23 = 0λ22 = 0λ23 > 01

π2 < x β2β−12no pos. lagr. if βj ≤ 1?
x = (x, x)
C < x ⇒ π < β1 −1 x
xN
1
1
β
1

2
x β2β−1 < π2 < 2x − 1+βxβ22
no pos. lagr. if β1 ≤ 1
x? = (x, x̃b )1
π1 < 2x − 1+βxβ122x − 1+βx < π2β2
no pos. lagr. if β1 ≤ 1
x? = (x, x)

Player 1
λ12 = 0, λ13 = 01x β1β−1 < π1 < 2x − 1+β
x
β
1

1

C < x ⇒ π < β2 −1 x
xN
2
2
β2
no pos. lagr. if β2 ≤ 1?
x = (x̃a , x)112x − 1+βx < π1 < 2x − 1+β
x
β
β
1

2x −

1

(1+β2 )
(1+β )
x < π2 < 2x − β 2 x
β22need positive det.
C , xN C )x? = (xN121+β1
2x − β x < π1 < x β1β−1
1

1

C > x ⇒ π > x β2 −1
xN
2
2
β2
no prem. bound if β1 ≤ 1
x? = (x̃d , x)

λ12 = 0, λ13 > 0
12x − 1+βx < π1β12π2 < 2x − 1+βxβ2
no pos. lagr. if β2 ≤ 1
x? = (x, x)
NC
x1 > x ⇒ π1 > x β1β−1122x − 1+βx < π2 < x β2β−1
β22no prem. bound if β2 ≤ 1?
x = (x, x̃c )β1 −1x < π1β1β2 −1x < π2β2x? = (x, x)

Table 3.21 – Summary of 9 cases for the KKT system (3.26) (x?1 , x?2 ): in red the pos. Lagr.
constraints (enforcing x or x) and in orange the prem. bound inequalities.
Computation of the Unique Equilibrium for each π conditions According
to 3.4.1, there exists a unique equilibrium for each π condition. Table 3.22 summarizes the different
solutions of the KKT system, but we should determine the unique equilibrium for each condition onπ.
Let us denote the 4 bounds for π by

3.8.2.5.8

l1 (β) = 2x −

1+β
β−1
β−1
1+β
x, l3 (β) =x.
x, l2 (β) =x, l4 (β) = 2x −ββββThere are all plotted in Figure 3.13. Note that l1 (β) < l2 (β), l3 (β) < l4 (β) for all β and l2 (β) < l3 (β)
for β > 1, l2 (β) > l3 (β) for β >= 1.
As a reminder
C
xN
=
1

2β1 β2 π1 + (1 + β1 )β2 π2 N C
2β1 β2 π2 + (1 + β2 )β1 π1
β1 π1 + (1 + β1 )x,x=, x̃a =,
4β1 β2 − (1 + β1 )(1 + β2 ) 2
4β1 β2 − (1 + β1 )(1 + β2 )
2β1
x̃b =

β2 π2 + (1 + β2 )x
β2 π2 + (1 + β2 )x
β1 π1 + (1 + β1 )x, x̃c =, x̃d =
.
2β2
2β2
2β1

When solving KKT system (3.26) we previously use some implications instead of equivalences inorder to surround π. Therefore, and because of unicity of equilibrium, we have to clarify which is theunique equilibrium for each π condition.

149

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Firstly, for l1 (β1 ) < π1 < l2 (β1 ) and l1 (β2 ) < π2 < l2 (β2 ), we get two possible conditions
CNC(x, x), (xN
1 , x2 ). Using condition of solution (x, x) under equivalence, we get
(
β1 −1
1
2x − 1+ββ1 x < π1 < β1 x
β2 −12)2x − (1+ββ2 x < π2 < β2 x
2xβ1 β2 − (1 + β1 )β2 x < π1 β1 β2 < (β1 − 1)β2 x⇔2xβ2 (1 + β1 ) − (1 + β2 )(1 + β1 )x < π2 β2 (1 + β1 ) < (β2 − 1)(1 + β1 )x⇒ π1 β1 β2 + π2 β2 (1 + β1 ) < (β1 − 1)β2 x + (β2 − 1)(1 + β1 )x = (3β1 β2 − 1 − β1 − β2 )xC
And when 4β1 β2 − (1 + β1 )(1 + β2 ) > 0 (i.e. det(M ) > 0), then xN< x. We thus eliminate
1NCNC(x1 , x2 ) when l1 (β2 ) < π2 < l2 (β2 ) and l1 (β2 ) < π1 < l2 (β2 ).
Secondly, using same reasoning for the case l3 (β2 ) < π2 < l4 (β2 ) and l3 (β2 ) < π1 < l4 (β2 ), we canCNC
also eliminate (xN
1 , x2 ) solution.
Thirdly, for l2 (β2 ) < π2 < l3 (β2 ) and l1 (β2 ) < π1 < l2 (β2 ), we get two possible conditions
C
NC
NC
C
NC
< x . Thus (xN(x, x̃b ), (xN
1 , x2 ). However, (x, x̃b ) is a solution and imply x1
1 , x2 ) is not anNCCequilibrium. Using same reasoning, we can also eliminate the unconstrained solution (x1 , xN2 ) casesfor— l2 (β1 ) < π1 < l3 (β1 ) and l1 (β2 ) < π2 < l2 (β2 ),— l3 (β1 ) < π1 < l4 (β1 ) and l1 (β2 ) < π2 < l2 (β2 ),— l1 (β1 ) < π1 < l2 (β1 ) and l2 (β2 ) < π2 < l3 (β2 ),— l1 (β1 ) < π1 < l2 (β1 ) and l3 (β2 ) < π2 < l4 (β2 ),— l2 (β1 ) < π1 < l3 (β1 ) and l3 (β2 ) < π2 < l4 (β2 ),— l3 (β1 ) < π1 < l4 (β1 ) and l2 (β2 ) < π2 < l3 (β2 ).
Furthermore

O1 (x̃a , x) = −β1 π1
1 + β1(−π1 /2 +x); O1 (x, x̃c ) = (1 − β1 (x/x̃c − 1))(x − π1 )2x
β1How to order?
This leads to Table 3.22.
π2 |π1. < l1 (β2 )l1 (β2 ) < . < l2 (β2 )l2 (β2 ) < . < l3 (β2 )l3 (β2 ) < . < l4 (β2 )l4 (β2 ) < .

. < l1 (β1 )(x, x)(x, x)(x, x̃b )(x, x̃b )(x, x)l1 (β1 ) < . < l2 (β1 )(x, x)(x, x),(x, x̃b ),(x, x̃b ), (x̃d , x)(x̃d , x)π1l2 (β1 ) < . < l3 (β1 )(x̃a , x)(x̃a , x),
C , xN C )(xN
12, (x̃d , x)(x̃d , x)l3 (β1 ) < . < l4 (β1 )(x̃a , x)(x̃a , x), , (x, x̃c ), (x, x̃c )(x, x),(x, x)l4 (β1 ) < .
(x, x)(x, x̃c )(x, x̃c )(x, x)(x, x)

Table 3.22 – Summary of all cases for Unique Equilibrium (x?1 , x?2 ) (only valid for βj > 1)

3.8.2.6

The two-insurer case without constraint

Let use Proposition 3.4.2.

M=

2β1
−1 − β2−1 − β1
2β2

⇒ det (M ) = 4β1 β2 − (1 + β1 )(1 + β2 ) = 3β1 β2 − 1 − β1 − β2 .
The value of the determinant cancels when β2 = (1 + β1 )/(3β1 − 1) for β1 6= 1/3. The determinant is
positive when
β2 > (1 + β1 )/(3β1 − 1)
β2 = (1 + β1 )/(3β1 − 1)
β2 < (1 + β1 )/(3β1 − 1)

150

β1 > 1/3
det (M ) > 0
det (M ) = 0
det (M ) < 0

β1 < 1/3
β1 = 1/3
det (M ) < 0 det (M ) < 0
det (M ) = 0.
det (M ) > 0 det (M ) < 0

3.8. Appendix
In particular, the determinant is positive when β1 , β2 > (1 + β1 )/(3β1 − 1). since det (M ) = 3(β1 −1)(β2 −1)+2(β1 +β2 )−4. We focus here our analysis on the case β1 > 1/3 and β2 > (1+β1 )/(3β1 −1).
Indeed, we are interested on the strictly concave problem conditioned by det(M ) > 0.
In the case of non-zero determinant and positive βi > 0, we have


1
2β2
1 + β2−1M =2β1
3β1 β2 − 1 − β1 − β2 1 + β1!


2β1 β2 π1 +(1+β2 )β2 π212β2
1 + β2β1 π 1?
3β1 β2 −1−β1 −β2⇒x == (1+β1 )β1 π1 +2β1 β2 π22β1
β2 π 23β1 β2 − 1 − β1 − β2 1 + β1
3β1 β2 −1−β1 −β2This result is in line with Proposition 3.4.4. Furthermore
∂x?1
∂β2

=

N um
(3β1 β2 − 1 − β1 − β2 )2The numerator is
N um ==(2β1 π1 + β2 π2 + (1 + β2 )π2 )(3β1 β2 − 1 − β1 − β2 ) − (2β1 β2 π1 + (1 + β2 )β2 π2 )(3β1 − 1)(2β1 π1 + (1 + 2β2 )π2 )β2 (3β1 − 1) + (2β1 π1 + (1 + 2β2 )π2 )(−1 − β1 )−(2β1 β2 π1 + (1 + β2 )β2 π2 )(3β1 − 1)

= β22 π2 (3β1 − 1) − (2β1 π1 + (1 + 2β2 )π2 )(1 + β1 )= β22 π2 (3β1 − 1) − 2β1 π1 (1 + β1 ) − (1 + 2β2 )π2 (1 + β1 )= β22 π2 (3β1 − 1) − 2β2 π2 (1 + β1 ) − 2β1 π1 (1 + β1 ) − π2 (1 + β1 )We want to determinate the derivative sign. We thus compute the roots of the polynom∆=(−2π2 (1 + β1 ))2 + 4(2β1 π1 + π2 )(1 + β1 )π2 (3β1 − 1)

We have ∆ > 0, ∀β1 > 1/3. Thus the polynom admits two real roots: β˜21 , β˜22 .
√
2π2 (1 + β1 ) − ∆˜
1β2 =2π2 (3β1 − 1)√
However, we have ∀β1 > 1/3, β˜21 < 0 ⇔ 2π2 (1 + β1 ) − ∆ < 0. Indeed,s√(2β1 π1 + π2 )(3β1 − 1)
∆ = 2π2 (1 + β1 ) 1 +.
π2 (1 + β1 )We previously show that to admit a maximum equilibrium we should have β1 > 1/3, β2 > (1 +β1 )/(3β1 −1) > 0. Thus β˜21 is never considered here. We now must determine if β˜22 > (1+β1 )/(3β1 −1)
in order to know if there exist a threshold for which the sign of the derivative of price equilibrium of
i with respect to β−i is changing. We have:s
2π2 (1 + β1 )(2β1 π1 + π2 )(3β1 − 1)
(1 + β1 )(1 + β1 )
˜
2
β2 >
⇔
(1 + 1 +)>(3β1 − 1)
2π2 (3β1 − 1)
π2 (1 + β1 )(3β1 − 1)s(2β1 π1 + π2 )(3β1 − 1)⇔
1+>0
π2 (1 + β1 )2Thus, there always exist a threshold of β˜2 for which the derivative of firm i price equilibrium change
sign with respect to the price sensitivity of competitor −i. This mean that when (1 + β1 )/(3β1 − 1) <151

Chapter 3.

A game-theoretic analysis of insurers pricing strategies2β2 < β˜2 , an increase of price sensitivity of consumers with respect to competitor −i lead to a decreaseof price equilibrium for both insurers. Intuitively, the decrease of price competitor −i over pass the2
price elasticity advantage of i. However, when β2 > β˜2 , an increase of β2 lead to a decrease of x?2
and an increase of x?1 . Hence, when β2 is high enough a marginal increase of this parameter lead to aimportant comparative advantage for firm 1.

3.8.3

Proofs of Section 3.5

3.8.3.1

Properties of the multinomial logit function

We recall that the choice probability function is defined as
pj→k (x) = pj→j (x) δjk + (1 − δjk )efj (xj ,xk ) , and pj→j (x) =

1+

P

1
,
efj (xj ,xl )

l6=jwhere the summation is over l ∈ {1, . . . , J} − {j} and fj is the price function. The price function 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
Derivatives of higher order use the same notation principle.
The lg function has the good property to be infinitely differentiable. We haveX
1
∂pj→j (x)∂ efj (xj ,xl ) 
=−
!2 .
∂xi
∂xi
P f (x ,x )l6=j
1+
ej j ll6=jSince we haveX
∂ X fj (xj ,xl )
0
0
e
= δji
fj1
(xj , xl )efj (xj ,xl ) + (1 − δji )fj2
(xj , xl )efj (xj ,xi ) ,
∂xi
l6=j

l6=j

we deduce00
X fj1(xj , xl )efj (xj ,xl )fj1(xj , xi )∂pj→j (x)0= −δji!2 − (1 − δji )fj2 (xj , xl )!2 .
∂xi
P f (x ,x )P f 0 (xj ,xl )l6=j
1+
ej j l
1+
e j1
l6=j

l6=j

This is equivalent toX∂pj→j (x)
0
0
= −
fj1
(xj , xl ) lglj (x) pi→j (x)δij − fj2(xj , xl )pj→i (x)pj→j (x)(1 − δij ).
∂xi
l6=j

Furthermore,∂pj→j (x) δjk + (1 − δjk )efj (xj ,xk ) =
∂xi



X
0
0
−
fj1
(xj , xl )pj→l (x) pj→k (x)δij − fj2(xj , xi )pj→i (x)pj→k (x)(1 − δij ).
l6=j

and also
pj→j (x)

152

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

3.8. Appendix
Hence, we get∂pj→k (x)= −δij 
∂xi


X

0
0
fj1
(xj , xl )pj→l (x) pj→k (x) − (1 − δij )fj2
(xj , xi )pj→i (x)pj→k (x)

l6=j



0
0
+ (1 − δjk ) δij fj1(xj , xk )pj→k (x) + δik fj2(xj , xk )pj→k (x) .
Similarly, the second order derivative is given by ∗XX
∂ 2 pj→k (x)
∂p
j→l 
00
00
0
= −δij δjm
pj→k
fj11
(xj , xl )pj→l + (1 − δjm )fj12(xj , xm )pj→m +fj1(xj , xl )∂xm ∂xi
∂xm
l6=j
l6=j


X
∂pj→k
0
− δij fj1
(xj , xl )pj→l 
∂xm
l6=j



∂pj→i
∂pj→k
00
00
0
0
−(1−δij ) δjm fj21
(xj , xi ) + δim fj22
(xj , xi ) pj→i pj→k + fj2(xj , xi )pj→k + fj2(xj , xi )pj→i
∂xm
∂xm



∂pj→k
00
00
0
+ (1 − δjk )δij fj11
(xj , xk )δjm + fj12
(xj , xk )δkm pj→k + fj1(xj , xk )
∂xm



∂pj→k
00
00
0
+ (1 − δjk )δik fj21
(xj , xk )δjm + fj22
(xj , xk )δim pj→k + fj2(xj , xk ).
∂xm
In particular

2
X
X

∂ 2 pj→j (x)200= − lgjj
fj1(xj , xl ) pj→l + 2 
fj1(xj , xl )pj→l  pj→j .
∂x2j
l6=j

3.8.3.2

l6=jProperties of the transition probability

Proposition 3.8.2. Transition probability pl→j (x) is a strictly decreasing function of xj given x−j
and verifies 0 < pl→j (x) < 1.
Proof. Differentiating the transition probability yields to the following result for i, j, k ∈ {1, . . . , J}X
∂pj→k (x)00= −δij 
fj1(xj , xl )pj→l (x) pj→k (x) − (1 − δij )fj2
(xj , xi )pj→i (x)pj→k (x)
∂xi
l6=j


0
0
+ (1 − δjk ) δij fj1(xj , xk )pj→k (x) + δik fj2(xj , xk )pj→k (x) .
Let φl be the family function xj 7→ pl→j (x) for l = 1, . . . , J. φj has the following derivativeX0φ0j (xj ) = − fj1(xj , xl )pj→l (x) pj→j (x),l6=j∂f (t,u)00(xj , xl ) = αj /xl > 0
where fj1
(t, u) = j∂t . Since for the two considered price function, we have f¯j10
˜
and fj1 (xj , xl ) = α̃j > 0, then the function φj is strictly decreasing. For l 6= j, the function φl has thefollowing derivative
0(xl , xj )pl→j (x)(1 − pl→j (x)),φ0l (xj ) = fj2∗. We remove the variable x when possible.

153Chapter 3.

A game-theoretic analysis of insurers pricing strategies∂f (t,u)x00where fj2
(t, u) = j∂u . Again, for the two considered price function, we have fj2(xj , xl ) = −αj x2j <l0 and f˜0 (xj , xl ) = −α̃j < 0. So, the function φl is strictly decreasing.
j2

Futhermore, the function φl decreases from 1 to 0 such that φl (xj ) → 1 (resp. φl (xj ) → 0) when
xj → −∞ (resp. xj → −∞). When i 6= j, functions xi 7→ pl→j (x) are also strictly increasing. Let
xj = (x, . . . , x, x, x, . . . , x) and xj = (x, . . . , x, x, x, . . . , x). We have∀x ∈ [x, x]J , 0 < pl→j (xj− ) < pl→j (x) < pl→j (xj− ) < 1.

3.8.3.3

Expected portfolio size function

The portfolio size xj 7→ N̂j (x) function has the following derivative∂E(Nj (x))= −nj 
∂xj


X

0
fj1
(xj , xl ) lglj (x) lgjj (x) +X0nl fj2(xl , xj ) lgjl (x)(1 − lgjl (x)).

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 both aquasiconcave and a quasiconvex function. Therefore, using the C 2 characterization of quasiconcaveand quasiconvex functions, we have that∂ 2 E(Nj (x))∂E(Nj (x))
=0⇒
= 0.
∂xj
∂x2j
Note that the function E(Nj (x)) has two asymptotes when either xj → 0 or xj → +∞ for fixed x−j .

3.8.3.4

Partial derivatives of the objective function

Consider a general objective function
Oj (x) =1
where mj (x) = J−1

aj − bjxj
mj (x)

(xj − cj ),P

i6=j xi . Simple manipulations yield to∂  xj 
δij−xj+ (1 − δij )
,
P
=
∂xi
xk
mj (x)
mj (x)2 (J − 1)
k6=j





2

∂
−δij (1 − δjm )−(1 − δij )δjm
2xj (1 − δij )(1 − δjm ) Pxj 
+
+
.

=
2
2
∂xi ∂xmxk
mj (x) (J − 1) mj (x) (J − 1)
mj (x)3 (J − 1)2
k6=j

So, we have∂Oj(x) = −bj
∂xi

154



δij
−xj
+ (1 − δij )mj (x)
mj (x)2 (J − 1)
(xj − cj ) + aj − bjxj
mj (x)

δij .

3.8. Appendix∂ 2 Oj(x)∂xm ∂xi

3.8.3.5−δij (1 − δjm )−(1 − δij )δjm
2xj (1 − δij )(1 − δjm )(xj − cj )++mj (x)2 (J − 1) mj (x)2 (J − 1)
mj (x)3 (J − 1)2




δij
−xj
δjm
−xj (1 − δjm )−bj
δij+ (1 − δij )
δ
+
−b
−
b
jm
j
j
mj (x)
mj (x)2 (J − 1)
mj (x)(J − 1)mj (x)2

2xj (1 − δij )(1 − δjm )2δij δjm
δij (1 − δjm ) + (1 − δij )δjm−b
(xj − cj ) − bj=bjj(J − 1)mj (x)2
mj (x)3 (J − 1)2
mj (x)(1 − δij )δjm + (1 − δjm )δij
+bj xj(J − 1)mj (x)2
(1 − δij )δjm + (1 − δjm )δij
2bj xj (1 − δij )(1 − δjm )2δij δjm= bj (2xj − cj )−(xj − cj ) − bj.
23
2
(J − 1)mj (x)mj (x) (J − 1)
mj (x)
= −bj

Proofs of the repeated game

Proof of Prop. 3.5.1. As reported in Appendix 3.8.3.3, insurer choice probability functions xj 7→
pl→j (x) are (strictly) decreasing functions from 1 to 0. Note that pl→j (x) = 0 (respectively pl→j (x) =1) is only attained when xj tends to +∞ (−∞). When i 6= j functions xi 7→ pl→j (x) are strictlyincreasing. Let xj− = (x, . . . , x, x, x, . . . , x) and xj− = (x, . . . , x, x, x, . . . , x). We have0 < pl→j (xj− ) < pl→j (x) < pl→j (xj− ) < 1,
for all x ∈ [x, x]J . Taking supremum and infimum on player j, we get
0 < pl = inf pl→j (xj− ) and suppl→j (xj− ) = pl < 1.
jj

Using the definition of portfolio size Nj,t (x) given in Subsection 3.2.2.1 as a sum of binomial randomvariables Blj,t (x), we have
P (Nj,t (x) = mj |Nj,t−1 > 0, Card(Jt−1 ) > 1)


X
= P
Blj,t (x) = mj Nj,t−1 > 0, Card(Jt−1 ) > 1l∈Jt−1X=Y

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

P

l m̃l =mjX=

Y nl,t−1 n−m̃
pl→j (x)m̃l (1 − pl→j (x)) l,t−1 jm̃lm̃1 ,...,m̃Jt−1 ≥0 l∈Jt−1s.t.

P

l m̃l =mjX>

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

m̃1 ,...,m̃Jt−1 ≥0 l∈Jt−1s.t.

P

l m̃l =mj

155

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Therefore,
P (Card(Jt ) = 0| Card(Jt−1 ) > 1)=

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

Yi | Card(Jt−1 ) > 1

i=1


≥



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

Yi | Card(Jt−1 ) > 1

i=1

≥

N
X

mj
X

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

mj =1

>

N
X

!
Yi

i=1

ξP

Kj,t−1 + mj x?j,t (1 − ej ) <

mj =1

mj
X

!
= ξ¯ > 0.

Yi

i=1

Thus, we have
P (Card(Jt ) > 1| Card(Jt−1 ) > 1)=

1 − P (Card(Jt ) = 0| Card(Jt−1 ) > 1) − P (Card(Jt ) = 1| Card(Jt−1 ) > 1)≤

1 − P (Card(Jt ) = 0| Card(Jt−1 ) > 1)<

1 − ξ¯ < 1.

By successive conditioning, we get
P (Card(Jt ) > 1) = P (Card(J0 ) > 1)tYt
P (Card(Js ) > 1| Card(Js−1 ) > 1) < 1 − ξ¯ .

s=1

So, the probability P (Card(Jt ) > 1) decreases geometrically as t increases.
Proof of Prop. 3.5.2. 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 3.8.3.1), pk→j (x) > pk→l (x) forl 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 3.8.1.7 and 3.8.1.8 respectively. Using the convolution property of the stochastic order J times, we can show a stochastic order of the portfoliosize
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  n , . . . , n , 0,
.
|
{z
}
ñ
ñ
| {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

156

ñ

Yi ≤cx

n
X
1
i=1

n

Yi .

P

i añ,i Yi ≤cx

P

i an,i Yi i.e.

3.8. Appendix
For all increasing convex functions φ, the function x 7→ φ(x + a) is still increasing and convex. Thus
for all random variables X, Y such that X ≤icx Y and real numbers a, b, a ≤ b, we haveE(φ(X + a)) ≤ E(φ(X + b)) ≤ E(φ(Y + b)) ⇔ 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.
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 .

3.8.4

Details of Section 3.6

3.8.4.1

Computation details

Computation is based on a Karush-Kuhn-Tucker (KKT) reformulation of the generalized Nashequilibrium 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 (2013). In our setting
we have J players and three constraints for each player. For each j of the J subproblems, the KKT
conditions are∇xj Oj (x) −X

λjm ∇xj gjm (x) = 0,1≤m≤3

0 ≤ λj ⊥ gj (x) ≥ 0.
The inequality part is called the complementarity constraint. The reformulation proposed uses acomplementarity 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 objectivefunctions 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 forlinear constraints, or strictly monotone functions, see Theorem 2 of Arrow and Enthoven (1961), whichis 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 FischerBurmeister 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 versionof φ. So, subproblem j reduces to solving a so-called nonsmooth equation. In this paper, we use the
Fischer-Burmeister complementarity function. This method is implemented in the R package GNE of
Dutang (2015), whereas the game functions are implemented in the R package NLIG of Dutang andMouminoux (2018).

157

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

3.8.5

Appendix on the loss model

3.8.5.1

Reminders from probability theory

3.8.5.2

A Poisson-Gamma mixture

Consider a random variable M . Assume that M |Λ = λ ∼ P(λ) and Λ ∼ G(r, p/(1 − p)). Let
q = p/(1 − p). The probability generating function of M isrq
PM (t) = E(tM ) = E(E(tM |Λ)) = E(eΛ(t−1) ) = MΛ (t − 1) =q − (t − 1)

r 
r 
r
p/(1 − p)pp===p/(1 − p) + 1 − t
p + 1 − p − (1 − p)t
1 − (1 − p)t
Thus, we find the well known result that M ∼ N B(r, p).

3.8.5.3

Scale parameters for usual discrete distributions

Consider two independent random variables M1 , M2 .
— if Mi ∼ P(λ), then M1 + M2 ∼ P(2λ) sincePM1 +M2 (t) = PM1 (t)PM2 (t) = eλ(t−1) eλ(t−1) = e2λ(t−1) .
— if Mi ∼ N B(r, p), then M1 + M2 ∼ N B(2r, p) since
r 
r 
2r

p
p
p
=
.
PM1 +M2 (t) = PM1 (t)PM2 (t) =1 − (1 − p)t
1 − (1 − p)t
1 − (1 − p)t— if Mi ∼ B(n, p), then M1 + M2 ∼ B(2n, p) sincennPM1 +M2 (t) = PM1 (t)PM2 (t) = (1 − p + pt)) (1 − p + pt)) = (1 − p + pt))

3.8.5.4

2n.

Scale parameters for multinomial distributions

Consider two independent multinomial vectors M1 , M2 : M1 ∼ Mk (n, p1 , . . . , pk ) and M2 ∼Mk (m, q1 , . . . , qk ). We have
!n
!m
X
X
M1 +M2E(z)=
pi z iqi zi
ii

Only when ∀i, pi = qi , we get a multinomial vector
!n+m
E(z

M1 +M2)=X

pi z i.

i

3.8.5.5

Fréchet copula

Consider the multivariate Fréchet copula defined as
CaF (u1 , . . . , uJ ) = (1 − a)C ⊥ (u1 , . . . , uJ ) + aC + (u1 , . . . , uJ ).
where C ⊥ is the independence copula and C + is the Fréchet upper bound, i.e.
C ⊥ (u1 , . . . , uJ ) =

J
Y
i=1

ui , C + (u1 , . . . , uJ ) = min ui .
i=1,...,J

Setting a = 1, we get the Fréchet upper bound, whereas setting a = −1, we get the independence
copula.
A simple algorithm to simulate is simply

1583.8. Appendix
1. simulate V ∼ U(0, 1)
2. if V ≤ a, then simulate Ua ∼ U(0, 1), ∀i, Ui ← Ua .
i.i.d.
3. otherwise, simulate (Ui )i ∼ U(0, 1).
i.i.d.
In other words, Ui = 11V ≤a Ua + 11V >a Vi , where V, Ua , (Vi )i ∼ U(0, 1). Indeed,
P (U1 ≤ u1 , . . . , UJ ≤ uJ )= P (U1 ≤ u1 , . . . , UJ ≤ uJ |V ≤ a)P (V ≤ a)
+P (U1 ≤ u1 , . . . , UJ ≤ uJ |V > a)P (V > a)
= P (Ua ≤ u1 , . . . , Ua ≤ uJ )a + P (V1 ≤ u1 , . . . , VJ ≤ uJ )(1 − a)= aC + (u1 , . . . , uJ ) + (1 − a)C ⊥ (u1 , . . . , uJ ).

3.8.6

On the sensitivity analysis

3.8.6.1

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[, suchthat
ϕ(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)3.8.6.2.

On the sensitivity analysis

Using the implicit function theorem see Appendix 3.8.6.1, we define F : RJ × RJ 7→ RJ
F (x, y) = M (y)x − v(y),where y is a parameter of interest. The following hypothesis are verified— H1: ∃x? , y ? , F (x? , y ? ) = 0.
— H2/H3: F has C 1 component in the neighborhood of (x? , y ? ).
— H4: Jac x F (x, y) = M (y) has non-zero determinant.
Thus, there exists g : RJ 7→ RJ such that x? = g(y ? ), g is C 1 around (x? , y ? ) andJac g(y) = −(Jac x F (x, y))−1 Jac y F (x, y)|x=g(y) = −(M (y))−1 Jac y F (x, y)|x=g(y) .
Using Appendix 3.8.6.2, there exists g : RJ 7→ RJ such that x? = g(y ? ), g is C 1 around (x? , y ? )andJacg(y) = −(Jac x F (x, y))−1 Jac y F (x, y)|x=g(y) = −(M (y))−1 Jac y F (x, y)|x=g(y) .
1. Let y = (β1 , . . . , βJ ).
2y1 −1−y2
 J−1
F (x, y) = M (y)x − v(y), M = 


−1−y1
J−1

2y2

−1−y2
J−1

..
...

We have

...
.

−1−yJ
J−1





y1 π1

...
 . 
 , v =  ..  .

yJ πJ
2yJ


1 + yi X
F (x, y) i = Mi,. x − vi = 2yi xi −xk − yi πi
J −1
k6=i

159

Chapter 3.

A game-theoretic analysis of insurers pricing strategies
if j 6= i
if j = i(0F (x, y) iP1⇒=xk − πi2xi − J−1
∂yj
k6=i
So


1
2x1 − J−1Jac y F (x, y) = 


P

xk

0

...



k6=1

..

.
0

...

1
2xJ − J−1

P

xk






k6=J

Hence
P

1
2x1 − J−1
xk
k6=1


Jac g(y) = −(M (y))−1 

...

0

...



..

.
0

1
2xJ − J−1

P
k6=J

xk






.
x=g(y)

2. Let y = (π1 , . . . , πJ ).


−1−β1
J−1

2β1

 −1−β2
 J−1
F (x, y) = M x − v(y), M = 





...



β1 y1

... 
. 
,v =  ..  .


βJ yJ
2βJ

−1−β2
J−1

2β2

..

.

−1−βJ
J−1

...
We have
F (x, y) i = Mi,. x − vi = Mi,. x − yi βi .
So



−β1


Jac y F (x, y) = 
...

0
..
.

...

0

−βJ





Hence Jac g(y) is constant:2β1

 −1−β2 J−1
Jacg(y) = − 



−1−β1
J−1

2β2

−1−β2
J−1

..
...

−1

...
.

−1−βJ
J−1


... 



2βJ

3.8.7

Other figures and tables of Section 3.6

3.8.7.1

Two-player market – sensitivity analysis

160


−β1


...

0
..
.

...

0

−βJ



.

3.8. Appendix

Figure 3.14 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – NBLN

161

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.15 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – PG

162

3.8. Appendix

Figure 3.16 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – MLN PR

163

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.17 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – MLN PD 2164

3.8. Appendix

Figure 3.18 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – equalled portfolio sizes

165

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.19 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – new capital

1663.8. Appendix

Figure 3.20 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – doubled portfolio sizes

167

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.21 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – new price sensitivity

168

3.8. Appendix

Figure 3.22 – Some indicators of the repeated game for a single run, black solid line for Insurer
1 and red dotted line for Insurer 2 – new credibility factor

169

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.23 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – NBLN

170

3.8. Appendix

Figure 3.24 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – PG

171

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.25 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – MLN PR

172

3.8. Appendix

Figure 3.26 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – MLN PD 2173

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.27 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – equalled portfolio sizes

1743.8. Appendix

Figure 3.28 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – new capital

175

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.29 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – doubled portfolio sizes

176

3.8. Appendix

Figure 3.30 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – new price sensitivity

177

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.31 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – new credibility factor

1783.8. Appendix
3.8.7.2

Three-player market – sensitivity analysis

Figure 3.32 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – NBLN loss model

179

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Period
1
2
3
4
5
6
7
8
9
10

Ruin probabilities
Insurer 1 Insurer 2 Insurer 3
0.01
0.00
0.10
0.00
0.00
0.14
0.00
0.00
0.19
0.01
0.01
0.22
0.01
0.01
0.24
0.01
0.01
0.24
0.01
0.01
0.25
0.01
0.01
0.26
0.01
0.01
0.24
0.01
0.02
0.24

Leadership probabilities
Insurer 1 Insurer 2 Insurer 3
1.00
0.00
0.00
0.80
0.04
0.03
0.45
0.08
0.26
0.34
0.13
0.22
0.31
0.17
0.17
0.28
0.22
0.14
0.30
0.23
0.10
0.31
0.15
0.13
0.27
0.15
0.13
0.24
0.16
0.12

Table 3.23 – Empirical probabilities of ruin and leadership over 100 runs, 3-player game withsimple market proxy – NBLN loss model

Period
1
2
3
4
5
6
7
8
9
10

Ruin probabilities
Insurer 1 Insurer 2 Insurer 3
0.01
0.00
0.04
0.01
0.01
0.08
0.00
0.01
0.12
0.01
0.04
0.15
0.01
0.04
0.18
0.02
0.05
0.22
0.04
0.04
0.24
0.04
0.06
0.26
0.05
0.07
0.27
0.05
0.07
0.29

Leadership probabilities
Insurer 1 Insurer 2 Insurer 3
1.00
0.00
0.00
0.80
0.03
0.08
0.41
0.10
0.27
0.32
0.20
0.21
0.25
0.20
0.18
0.19
0.23
0.16
0.21
0.21
0.15
0.22
0.21
0.12
0.24
0.20
0.11
0.23
0.19
0.10

Table 3.24 – Empirical probabilities of ruin and leadership over 100 runs, 3-player game withweigthed market proxy – NBLN loss model

Period
1
2
3
4
5
6
7
8
9
10

Ruin probabilities
Insurer 1 Insurer 2 Insurer 3
0.00
0.00
0.00
0.00
0.00
0.05
0.00
0.00
0.06
0.00
0.00
0.09
0.00
0.00
0.11
0.00
0.00
0.11
0.00
0.00
0.11
0.00
0.00
0.11
0.00
0.00
0.10
0.00
0.00
0.11

Leadership probabilities
Insurer 1 Insurer 2 Insurer 3
1.00
0.00
0.00
0.95
0.00
0.00
0.87
0.01
0.00
0.78
0.05
0.02
0.71
0.08
0.06
0.68
0.10
0.07
0.64
0.14
0.05
0.61
0.16
0.04
0.60
0.13
0.08
0.61
0.16
0.04

Table 3.25 – Empirical probabilities of ruin and leadership over 100 runs, 3-player game withsimple market proxy – MLN PR lapse
1803.8. Appendix

Figure 3.33 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – NBLN loss model

181

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.34 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – MLN PR lapse

1823.8. Appendix

Figure 3.35 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles, 3-player game with weigthed market
proxy – MLN PR lapse

183

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Period
1
2
3
4
5
6
7
8
9
10

Ruin probabilities
Insurer 1 Insurer 2 Insurer 3
0.00
0.00
0.04
0.00
0.01
0.07
0.00
0.01
0.10
0.00
0.01
0.10
0.00
0.01
0.10
0.00
0.01
0.10
0.00
0.01
0.10
0.00
0.01
0.10
0.00
0.01
0.10
0.00
0.01
0.10

Leadership probabilities
Insurer 1 Insurer 2 Insurer 3
1.00
0.00
0.00
0.99
0.00
0.00
0.92
0.00
0.01
0.83
0.03
0.02
0.79
0.07
0.02
0.74
0.10
0.04
0.75
0.08
0.04
0.73
0.11
0.03
0.66
0.15
0.06
0.65
0.17
0.05

Table 3.26 – Empirical probabilities of ruin and leadership over 100 runs, 3-player game withweighted market proxy – MLN PR lapse

1843.8. Appendix

Figure 3.36 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – new credibility factor

185

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.37 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles, 3-player game with weighted market
proxy – new credibility factor

1863.8. Appendix

Figure 3.38 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles – new sensibility

187

Chapter 3.

A game-theoretic analysis of insurers pricing strategies

Figure 3.39 – Quartiles of some indicators of the repeated game for 100 runs, black solid line
for Insurer 1 and red dotted line for Insurer 2; bottom-up triangles for 1st quartiles, numbers
for medians and top-down triangles for 3rd quartiles, 3-player game with weighted market
proxy – new sensibility

188

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Conclusion

193

Conclusion et perspectives
Cette thèse aborde différentes composantes du marché de l’assurance non-vie à l’aide d’approches
expérimentales, théoriques et de simulations numériques. Les résultats démontrent encore une fois lacomplexité du marché de l’assurance. Au-delà de l’aléa inhérent aux risques, les biais cognitifs, éléments caractéristiques de la psychologie humaine, complexifient d’autant plus la rencontre de l’offre
et de la demande.
D’un côté les assureurs essaient de modéliser et d’anticiper au mieux les comportements desconsommateurs, de l’autre les assurés font eux-même face à leur propre complexité, les conduisantsouvent à prendre des décisions non-rationnelles ou inefficientes au sens de la théorie économique.
Dès lors, l’économie comportementale joue un rôle essentiel à l’explication et à la compréhension desinteractions.
L’objectif de cette conclusion n’est pas d’énumérer une nouvelle fois les résultats détaillés danscette thèse, mais de proposer et justifier l’intérêt de nouvelles perspectives de recherche qui pourraientaméliorer notre compréhension du marché de l’assurance et les comportements de ses acteurs. Depuis
longtemps la théorie comportementale de la décision s’est attachée à l’étude des incohérences dans leschoix des agents économiques. Son objectif est de mettre en évidence les "anomalies" dans les comportements réels des agents, rarement conforme à la théorie des choix rationnels, en considérant par
exemple, l’altruisme, les préférences pour le présent, l’aversion aux risques ou encore les capacités limitées de calcul et de traitements de l’information. Ce n’est pourtant que récemment que cette science,
souvent expérimentale, fait l’objet d’attention particulière des entreprises, mais aussi des régulateurs,souhaitant améliorer l’anticipation des conséquences de leurs choix.
Le marché de l’assurance est un terrain idéal à l’analyse des comportements. L’aléa, souvent non
exogène, donc l’asymétrie d’information en résultant, sont les raisons pour lesquelles les assureurs ont
besoin d’analyser et de comprendre le comportement des assurés, tant au niveau du risque souscritqu’au niveau de leur décision d’achat. Nous avons montrer dans cette thèse en quoi le processus desouscription est révélateur des caractéristiques des individus. Ces caractéristiques sont souvent difficiles à mesurer pour l’assureur, tel que l’aversion au risque, ou encore la confiance. Elles sont pourtantdéterminantes afin d’expliquer la rencontre de l’offre et de la demande car elles permettent de comprendre les effets des biais cognitifs sur les prises de décision.

195

Conclusion et perspectives
À l’aide de la mesure d’honnêteté détaillée dans le premier chapitre de cette thèse, il serait par
exemple intéressant d’analyser les comportements de fraudes en assurance. En effet, il existe une réelle
hétérogénéité de l’honnêteté à travers les individus. De plus, les incitations financières et le sentimentd’injustice sont déterminants dans la décision d’être plus ou moins honnête. Ces résultats font naturellement penser à l’effet de licence, où les sujets justifient un comportement négatif (la fraude) pardes actes positifs effectués par le passé. En assurance automobile par exemple, nous observons que lesassurés n’ayant pas déclaré de sinistres depuis longtemps ont tendance à avoir des coûts de sinistresplus élevés. L’explication pourrait porter sur l’effet de licence. Les assurés n’ayant pas eu de sinistresdepuis des années, estiment que les primes payées durant toute ces années sont en fait un coût et nonl’achat d’un service : la tranquillité. Ainsi, lorsque qu’il ont à déclarer un sinistre, ils pourraient êtreamener à sur-déclarer afin de compenser ses coûts. Sans une approche comportementale de la question,il est difficile de déterminer la vraie raison de cette augmentation des coûts de sinistres. Une nouvelle
expérimentation économique, dans un environnement contrôlé, serait donc un moyen efficace pour yrépondre.
Concernant le deuxième chapitre de cette thèse, je me suis intéressée à l’achat d’assurance obligatoire et l’impact de la quantité d’informations sur les prises de décision. Cependant, il serait également
intéressant d’analyser les comportements face à une trop grande quantité d’information lorsque l’assurance n’est plus obligatoire. En effet, il est naturel de penser que les consommateurs, perdus en
raison de l’effet d’obfuscation, soient découragés à l’achat d’un produit d’assurance optionnel. Dansce cas, l’information rendue disponible aurait une conséquence, non seulement sur les caractéristiques
du contrat choisi mais aussi, sur la décision de souscrire ou non. On peut donc imaginer de multiples implications pour les assureurs. Ces comportements pourraient expliquer d’éventuels effets deselection des risques. En effet, malgré la possibilité que l’acte d’achat soit découragé par une quantitéd’information trop importante, on peut imaginer que les consommateurs souscrivant tout de mêmeà une assurance facultative ont des caractéristiques propres à ce comportement (importante aversion
aux risques, meilleure connaissances ou compréhension de l’assurance,...). Encore une fois, la théorie
économique comportementale se révèle être un outil nécéssaire à l’analyse de tels biais.
Au-delà de l’utilité incontestée de la compréhension des comportements des agents pour les assureurs, il est aussi important de souligner qu’elle peut être un outil de prise de décision pour lesrégulateurs. Dans le dernier chapitre de cette thèse, j’ai abordé la modélisation du marché à l’aide dela théorie des jeux. Celle-ci montre une nouvelle fois l’importance des comportements, tel que l’inertie
des consommateurs, sur les décisions stratégiques des assureurs. Etant donnée l’importance apportéepar les régulateurs à garantir la solvabilité des assureurs, il serait également intéressant d’étendre lesrésultats de ce chapitre afin d’expliquer les phénomènes de ruines. En effet, bien que les chocs sur lasinistralité soient un des éléments majeurs à la survenance de ruine chez les assureurs, les parts demarché, sur le long-terme, sont aussi très dépendante de la sensibilité des consommateurs. Dans ce
sens, il est donc intéréssant pour le régulateur de tenir compte de la compétition entre assureurs lors del’étude des risques de ruine. En outre, la ruine peut survenir à cause de sinistres trop importants, ellepeut également être le résultat d’un volume d’affaires trop faible pour garantir le principe de mutualisation et d’absorption des coûts fixes. Une nouvelle fois, le comportement d’achat et la sensibilité desconsommateurs jouent un rôle non négligeable à la survenance de faillite sur le marché de l’assurance.
L’économie comportementale permet donc de produire des connaissances théoriques nouvelles tout
en confrontant les prédictions des modèles aux observations de terrain et de laboratoire. Elle permet,
grâce à son côté expérimental, de collecter des données difficilement accessibles comme par exemple
l’aversion aux risques. Elle est aussi un outil pour la prise de décision publique, permettant de mesurer
les impacts de politiques économiques et d’en évaluer leur efficacité. Au-delà de son intérêt pour larecherche, l’économie expérimentale est aussi un outil pédagogique pour l’enseignement de l’économie,
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