# Comonotone Random Variables in Economics: A Review of some results

Alain Chateauneuf, Michèle Cohen, Robert Kast

► To cite this version:

Alain Chateauneuf, Michèle Cohen, Robert Kast. Comonotone Random Variables in Economics: A Review of some results. 1997. halshs-04252285

HAL Id: halshs-04252285

https://shs.hal.science/halshs-04252285

Submitted on 20 Oct 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
CAHIERS ECO & MATHS

97.32

COMONOTONE RANDOM VARIABLES IN ECONOMICS :
A REVIEW OF SOME RESULTS

Alain CHATEAUNEUF

Michèle COHEN

Robert KAST

CEME & CERMSEM
# Comonotone random variables in economics: A review of some results.

Alain Chateauneuf*, Michèle Cohen**, Robert Kast***

Février 1997

**Abstract** : Different notions of comonotony dispersed in a wide range of literature are reviewed. We first gather and analyse different definitions and properties that have been given in various contexts, then we emphasize several relevant applications of comonotony in economics. This includes decision theory along the line of the "non-expected" utility models and its applications to the measure of risk, risk aversion and aversion to uncertainty. A characterization of Pareto optimal allocation for risk averse agents, using comonotony, is particularly relevant to the theory of insurance. Inequality measurement is reviewed along its implicit use of comonotony and a generalization is proposed. The hedging effect of non-comonotone random variables is applied to financial assets valuation and to insurance premium.

**Key words** : Comonotony, hedging, risk, uncertainty, dispersion, inequality, sharing risks.

**J.E.L. classification n°** : D61, D63, D71, D80, D81, G12.

**Résumé** : Différentes notions de comonotonie dispersées dans une vaste littérature sont répertoriées. Nous réunissons et analysons tout d'abord différentes définitions et propriétés qui ont été données dans des contextes variés, puis nous nous concentrons sur plusieurs applications pertinentes de la comonotonie à l'économie. Ceci inclue la théorie de la décision dans l'esprit des modèles de l'utilité "non-espérée" et ses applications à la mesure du risque, de l'aversion pour le risque et de l'aversion pour l'incertitude. Une caractérisation par la comonotonie des allocations Pareto-optimales pour des agents adversaires du risque est particulièrement pertinente pour la théorie de l'assurance. Des mesures des inégalités sont passées en revue selon l'usage implicite qu'elles font de la comonotonie et une généralisation est proposée. La propriété de couverture de variables aléatoires non-comonotones est appliquée à l'évaluation d'actifs financiers et au calcul de primes d'assurance.

**Mots clés** : Risque, incertain, dispersion, comonotonie, couverture, inégalités, partage des risques.

<footnote>* CERMSEM, Université de Paris I, 90 avenue de Tolbiac, F 75013 Paris.</footnote>

<footnote>** CEME, Université de Paris I, 12 Place du Panthéon, F 75005 Paris.</footnote>

<footnote>*** GREQAM, CNRS, 2 rue de la charité, F 13002 Marseille.</footnote>

<footnote>We wish to thank Rose-Anne Dana, Jean-Yves Jaffray and André Lapied for helpful comments.</footnote>
Introduction

A very stimulating paper by Landsberger and Meilijson (1994) aroused our interest for different notions of comonotony dispersed in a wide range of literature. Here, we gather and analyse different definitions and properties that have been given in various contexts, in particular in Denneberg's (1994) seminal book. We then emphasize several relevant applications of comonotony in economics.

The general concept of comonotony concerns measurable functions without reference to a probability distribution ; it was introduced by Dellacherie<fnref n="1" /> (1970). Schmeidler (1986) rediscovered both the Choquet integral and the comonotony concept which he used to characterize this integral in a generalization of Riesz' representation theorem. Moreover, Schmeidler (1989) applied his representation theorem to construct a new model of decision under uncertainty. We connect the previous concept of comonotony to the one established on probability distributions as in Yaari (1985) or Landsberger and Meilijson (1994).

Random variables that are not comonetone can have a hedging effect. This property was put forward by Yaari (1985), then by Roëll (1987) and re-examined by Wakker (1990), it was applied to risk premium calculation by Denneberg (1990) and to financial assets by Chateauneuf, Kast and Lapied (1996).

The notion of comonotony has enriched the domain of inequality measurement. This had already been done by Weymark (1981) who didn't make the notion explicit, it was independently developed by Yaari (1988), Ben Porath and Gilboa (1994) and Chateauneuf (1996).

The same tool was used (non-explicitly) to characterize Pareto-optimal allocations of random endowments in Borch (1962) and in Wilson (1968); Landsberger and Meilijson (1994) mention these results and apply them to risk sharing through insurance, we bring in light that their result is model-free, for any strong risk averse decision maker.

Perhaps the most striking point made by Landsberger and Meilijson (1994) is the connexion between the dispersion order defined by Bickel and Lehmann (1979), and comonotony. We extend this connexion to the comonetone dispersion order used by

<footnote n="1">¹ Denneberg mention in his book(1994) that the first emergence of this notion seems to be in Hardy, Littlewood and Polya (1934) under the name of "similarly ordered" functions.</footnote>
Quiggin (1981) who defined a new concept of risk aversion: monotone risk aversion. This concept fits naturally in the framework of the new models of non-expected utility under risk (whereas, in expected utility theory, Quiggin's concept reduces to the classical one which is based on second order stochastic dominance).

Definitions and properties are presented in the first section, applications to decision theory, in the second section. The third section deals with measure of increasing risk, risk aversion and aversion to uncertainty. Characterizations of Pareto optimality are derived in section four and applications to inequality measurements in section five. Section six deals with applications of the specific hedging effects of non comonotone random variables to finance and insurance. In each section we try to distinguish between results obtained under risk, i.e. where there is a given probability distribution, and the ones which are valid under non-probabilized uncertainty.

# 1. Comonotony : definitions and properties.

## 1.1 Non-probabilized uncertainty.

Uncertainty is described by a set of states, $\Omega$, and a set of events, $\mathcal{A}$. We assume $\mathcal{A}$ to be a $\sigma$-algebra and we define $\mathcal{F}$ as the set of $\mathcal{A}$-measurable numerical functions $X: (\Omega, \mathcal{A}) \to (\mathbb{R}, \mathcal{B})$ where $\mathcal{B}$ is the $\sigma$-algebra of Borelians.

**Definition 1.0:** Two measurable functions $X$ and $Y$ are comonotone if and only if $\forall \omega, \omega' \in \Omega, [X(\omega) - X(\omega')][Y(\omega) - Y(\omega')] \ge 0$.

More generally,

**Definition 1.1:** A family $(X_i)_{i \in I}$ of measurable functions is a class of comonotone functions if and only if $\forall i, j \in I, \forall \omega, \omega' \in \Omega, [X_i(\omega) - X_i(\omega')][X_j(\omega) - X_j(\omega')] \ge 0$.

Notice that, by definition, pairwise comonotony is equivalent to global comonotony (in contrast with, say, the notion of stochastic independence). The binary relation
"are comonotone", on a set of measurable functions is reflexive, symmetric, but not transitive in general; indeed, we have the straightforward:

**Property 1.1:** Any constant measurable function is comonotone to every measurable function.

As an illustration, consider a complete order $\geq_{\Omega}$ on $\Omega$, then: the set of weakly increasing functions is a comonotone class; the same is true for weakly decreasing functions; no strictly increasing function is comonotone to a strictly decreasing function but both are comonotone to any constant.

For example, in $\mathbb{R}^n$, let $x_i$, $i = 1, ..., n$ be $n$ distinct real numbers, let $p$ be a permutation of $\{1, ..., n\}$ and consider $C_p$ the class of vectors comonotone with $x_p = (x_{p(1)}, x_{p(2)}, ..., x_{p(n)})$. The union of $C_p$'s is $\mathbb{R}^n$ but their intersections are not empty since comonotony is not transitive.

Another straightforward result is:

**Property 1.2:** If $X$ and $Y$ are comonotone, $\{X, Y, X + Y, f(X, Y)\}$, for any non-decreasing numerical function $f$ (in particular any $f(X, Y) = \lambda X + \mu Y$, for $\lambda, \mu \geq 0$), is a class of comonotone measurable functions

Many equivalent definitions of comonotony can be given, as shown in the following proposition.

**Proposition 1.1:** For a family $(X_i)_{i \in I}$ of measurable functions the following conditions are equivalent:

1) $(X_i)_{i \in I}$ is a class of comonotone functions.

2) Defining, for any measurable function $X: \Omega \to R$,

$$m(X) = \{A \in \mathcal{A} / \exists x \in R; [X > x] = A \text{ or } [X \geq x] = A\}, \\ \bigcup_{i \in I} m(X_i) \text{ is a chain, i.e.: } \forall A_i \in m(X_i) \quad \forall A_j \in m(X_j), A_1 \supseteq A_j \text{ or } A_j \supseteq A_i, \quad (1.1)$$

3) There exists a measurable function $Z: \Omega \to R$ such that: $\forall i \in I \exists V_i: R \to R$ non decreasing such that $X_i = V_i(Z)$.
4) If $I$ is finite, $\forall i \in I \exists V_i : R \to R$ non decreasing and continuous and such that
$$ \forall z \in R \sum_{i \in I} V_i(z) = z, \text{ with } X_i = V_i(\sum_{j \in I} X_j). $$

Denneberg (1994) gives the same definitions and a proof of their equivalence in the case of two variables, instead of a family.

Proof: $1 \Leftrightarrow 2$: assume 1 is false and two variables $X$ and $Y$ in $(X_i)_{i \in I}$ are not comonotone, then, applying Denneberg's proof, $\bigcup_{i \in I} m(X_i)$ is not a chain because $m_X \cup m_Y$ is not. The reciprocal follows the same argument.
$4 \Rightarrow 3$ and $3 \Rightarrow 1$ are obvious.

$1 \Rightarrow 4$ is true, according to Denneberg, for $X_1$ and $X_2$, so take $X_1 = u(X_1 + X_2)$ and $X_2 = v(X_1 + X_2)$ $u$ and $v$ non decreasing. Assume $X_1, X_2$ and $X_3$ are comonotone, then, from proposition 2, $X_1 + X_2$ and $X_3$ are comonotone and there exist two non increasing functions $f$ and $g$ such that: $X_1 + X_2 = f(X_1 + X_2 + X_3)$ and $X_2 = g(X_1 + X_2 + X_3)$. Then take $u_1 = uog$, $u_2 = vog$ and $u_3 = f$ in definition 4. The case of a finite family derives from induction. ♦

**Remark:** The definition of comonotony can be extended to sets of measurable functions with values in any totally preordered set $C$ endowed with the $\sigma$-algebra $\mathcal{C}$ induced by the preorder $\ge_C$. Then, definition 1.0 becomes:

Two measurable functions $X$ and $Y$ are comonotone if and only if $\forall \omega, \omega' \in \Omega$, $[X(\omega) >_C X(\omega')] \Rightarrow [Y(\omega) \ge_C Y(\omega)]$.

Other wise stated, two measurable functions $X$ and $Y$ are comonotone if and only if $\forall \omega, \omega' \in \Omega$, $[X(\omega) >_C X(\omega')]$ and $[Y(\omega) <_C Y(\omega')]$ is impossible.

For instance, $C$ can be the outcome set of a set $\mathcal{F}$ of acts, applications from $\Omega$ to $C$, where there exists a preference relation (total preorder) on $\mathcal{F}$ inducing a total preorder on $C$.

There is an important link between the property of comonotony and the linearity of a non necessarily additive functional: the Choquet integral (Choquet (1953)). Let us recall the definition of a capacity and of a Choquet integral:

**A capacity $\mu$** on the space $(\Omega, \mathcal{A})$ is a non necessarily additive measure, monotone for inclusion and such that $\mu(\Omega) = 1$ and $\mu(\emptyset) = 0$.
**A Choquet integral** with respect to a capacity $\mu$, is a function $I$ on the set $B$ of bounded real valued $\mathcal{A}$-measurable functions on $\Omega$ defined by:

$$ \forall X \in \mathcal{B} \quad I(X) = \int_{-\infty}^{0} (\mu[X>x] - 1)dx + \int_{0}^{+\infty} \mu[X>x] dx \quad (1.1). $$

For any set $A$ in $\mathcal{A}$, calling $A^*$ its characteristic function, we have $\mu(A) = I(A^*)$ and the Lebesgue notation is generalized: $I(X) = \int_{\Omega} X d\mu$.

**Theorem 1.1 (Schmeidler's representation theorem):** A functional $I: B \to R$ satisfying $I(\Omega^*) = 1$, is a Choquet integral with respect to the unique capacity $\mu$ such that $\forall A \in \mathcal{A} \ \mu(A) = I(A^*)$, if and only if it satisfies:

$$ (i) \textit{Monotonicity :} \quad \forall X, Y \in B \ X \geq Y \Rightarrow I(X) \geq I(Y). $$

$$ (ii) \textit{Comonotone additivity :} \quad \forall X, Y \in B \ X \text{ and } Y \text{ comonotone } \Rightarrow I(X + Y) = I(X) + I(Y). $$

The "if" part was proved by Dellacherie (1970), the "only if" part is in Schmeidler (1986). In section 2, we shall see Schmeidler's application of his theorem to decision under uncertainty (Schmeidler (1989)).

As an illustration, consider a finite set $\Omega$ and a measurable function $X$. Call $i = 1, ..., n$ the states in such a way that $X(1) \leq X(2) \leq ... \leq X(n)$ are the ranked values taken by $X$, we have:

**Canonical decomposition:** $X$ has a unique non negative linear decomposition in characteristic comonotone functions:

$$ X = X(1) \Omega^* + [X(2) - X(1)] \{2, ..., n\}^* + [X(3) - X(2)] \{3, ..., n\}^* + ... + [X(n) - X(n-1)] \{n\}^*, $$

As a result, the finite expression of a Choquet integral is:

$$ I(X) = X(1) \cdot 1 + [X(2) - X(1)] \mu([2, ..., n]) + [X(3) - X(2)] \mu([3, ..., n]) + ... + [X(n) - X(n-1)] \mu([n]). $$
Or else:

$$I(X) = (1 - \mu[\{2, \dots, n\}) X(1) + (\mu[\{2, \dots, n\}] - \mu[\{3, \dots, n\}]) X(2) \\ + \dots + \mu[n] X(n).$$

Notice that, setting $P(\{1\}) = 1 - \mu[\{2, \dots, n\})$, $P(\{2\}) = \mu[\{2, \dots, n\}] - \mu[\{3, \dots, n\}]$, ..., $P(\{n\}) = \mu(\{n\})$, $I(X)$ is the mathematical expectation of $X$ with respect to the probability measure $P$. This is a special case of the following general result which is in Wakker (1989) and Denneberg (1994) p. 82 and 83.

**Proposition 1.2 (Denneberg (1994) and Wakker (1989))** : Let $\mu$ be a capacity on a measurable space $(\Omega, \mathcal{A})$ and $D = (X_i)_{i \in I}$ a class of comonotone bounded functions in $\mathcal{F}$, then there exists a (simply) additive probability measure $P_D$ on $(\Omega, \mathcal{A})$ such that $\forall i \in I$, $\int X_i d\mu = \int X_i dP_D$.

The following proposition relates the marginal and the joint image set functions of two comonotone measurable functions.

**Proposition 1.3: Two measurable functions X and Y on $(\Omega, \mathcal{A})$ are comonotone if and only if, for any monotone bounded set function $\mu$ on $(\Omega, \mathcal{A})$ we have :**
$$\forall x, y \in \mathbb{R} \quad \mu([X \le x] \cap [Y \le y]) = \min\{ \mu([X \le x], \mu([Y \le y])) , \text{ and also} \\ \forall x, y \in \mathbb{R} \quad \mu([X < x] \cap [Y < y]) = \min\{ \mu([X < x], \mu([Y < y])) .$$

**Proof:** If $X$ and $Y$ are comonotone, from proposition 1.1 (ii), either $[X \le x] \supseteq [Y \le y]$ or $[Y \le y] \supseteq [X \le x]$ stands; then, for any monotone bounded set function $\mu$ we have the relation: $\mu([X \le x] \cap [Y \le y]) = \min\{ \mu([X \le x], \mu([Y \le y]))$. Reciprocally, it is sufficient that the latter relation be satisfied by all set functions such that, for any $\omega_1$ and $\omega_2$ of $\Omega$, $\mu = \delta_{\omega_1} + \delta_{\omega_2}$

Then we cannot have $X(\omega_1) > X(\omega_2)$ and $Y(\omega_1) < Y(\omega_2)$ because there would exist a $\mu$ such that $\mu([X \le X(\omega_2)] \cap [Y \le Y(\omega_1)]) = 0$ when, in fact, $\min\{\mu([X \le X(\omega_2)] , \mu([Y \le Y(\omega_1)])\} = \min\{\mu(\omega_1) , \mu(\omega_2)\} \neq 0$. ♦

Notice that this relation between the joint and the marginal image set functions characterizes comonotony of the measurable functions only if it is satisfied for every
monotone bounded set functions on $\mathcal{A}$. But, for the reciprocal, the relation will be
sufficient if it is satisfied for :

all capacities, or for :

all probability distributions, or even for :

all positive measures putting non-zero weights on two distinct singletons.

Let us now relate comonotony with the more common notion of covariance, the
following result was established in Chateauneuf, Kast, Lapied (1994).

**Proposition 1.4:** *Two bounded measurable functions X and Y on (Ω, Σ, P), Σ containing all singletons, are comonotone if and only if their covariance is positive for any additive probability distribution on (Ω, Σ).*

This proposition emphasizes the fact that two comonotone functions cannot have a
hedging effect.

**1.2 Probabilized uncertainty.**

Here, uncertainty is described by a probability space $(\Omega, \mathcal{A}, P)$ where probability
distribution $P: \mathcal{A} \to [0, 1]$ is given. In decision theory, such a formalization is
referred to as *situations of risk*. Any random variable $X: \Omega \to \mathbb{R}$ defines a
probability $P_X$ on $\mathbb{R}$ and we shall call $F_X(x) = P[X \le x]$ and $G_X(x) = P[X > x]$ the cumulative and decumulative distributions of $X$.

We will define $F^{-1}_Z$ by : $F^{-1}_Z(t) = \sup \{x \in \mathbb{R} / F_Z(x) \le t\}$

In this context, all definitions can be given *P-almost everywhere*. We shall use the
notation $\forall_p \omega$ for "for P -almost every $\omega$".

**Definition 2.1:** A family $(X_i)_{i \in I}$ of random variables is a class of $P$-comonotone functions if and only if $\forall i, j \in I, \forall_p \omega, \omega' \in \Omega, [X_i(\omega) - X_i(\omega')] [X_j(\omega) - X_j(\omega')] \ge 0$.

As is noticed in Röell (1987), all results in Yaari (1987) hold with this definition
instead of our definition 1.1 which was proposed in Yaari's paper.

Our previous properties 1.1 and 1.2 can straightforwardly be restricted to P-
comonotony. Furthermore (see, for instance Roëll (1987), Landsberger and
Meilijson (1994)) the following result proves to be very useful in many applications:
**Property 2.3:**

If X and Y are two P-comonotone random variables, then $F^{-1}_{X+Y} = F^{-1}X + F^{-1}Y$.

Notice that a crucial point in the proof as well as in many others, is the following:

**Canonical identification of bounded random variables and distributions:** Given a probability distribution $\Pi$ on a bounded interval in $\mathbb{R}$, there exists a $(\Omega, \mathcal{A}, P) = ([0, 1], \mathfrak{B}_{[0, 1]}, \lambda)$ where $\lambda$ is the uniform distribution on $[0, 1]$ and a family of random variables $X$ on $(\Omega, \mathcal{A}, P)$ such that $P_X = \Pi$. However, there exists only one $P$-almost everywhere increasing random variable $X$ such that $P_X = \Pi$, namely $X = F^{-1}$ where $F$ is the cumulative distribution of $\Pi$.

Similarly, the following proposition is a straightforward application of proposition 1.1.:

**Proposition 2.1:** For a family $(X_i)_{i \in I}$ of measurable functions on $(\Omega, \mathcal{A}, P)$ the following conditions are equivalent:

i) $(X_i)_{i \in I}$ is a class of P-comonotone functions.

ii) There exists a measurable function $Z : \Omega \to \mathbb{R}$ such that: $\forall i \in I \exists V_i : \mathbb{R} \to \mathbb{R}$ non-decreasing such that $X_i = V_i(Z)$ $P$-almost everywhere.

iii) If $I$ is finite, $\forall i \in I \exists V_i : \mathbb{R} \to \mathbb{R}$ non-decreasing and continuous and such that
$$ \forall z \in \mathbb{R} \sum_{i \in I} V_i(z) = z, \text{ with } X_i = V_i(\sum_{i \in I} X_i) \text{ P-almost everywhere.} $$

In contrast, property 2) of proposition 1.1 is not interesting any more because, given $P$, the family $m(X)$ defines $F_X$ or $G_X$. Instead we have the following proposition which yields the joint distribution of $n$ P-comonotone random variables:

**Proposition 2.2:** A finite family $(X_i)_{i=1...n}$, of random variables is a class of P-comonotone functions if and only if

$$ P([X_1 \le x_1] \cap [X_2 \le x_2] \cap \dots \cap [X_n \le x_n]) = \min\{ P([X_i \le x_i]) , i = 1 \dots n \} \text{ and} \\ \text{also } P([X_1 < x_1] \cap [X_2 < x_2] \cap \dots \cap [X_n < x_n]) = \min\{ P([X_i < x_i]) , i = 1 \dots n \}. $$

The proof is in Landsberger and Meilijson (1994) who call comonotony what we call P-comonotony.
The following representation theorem, (a special case of which is implicit in the proof of proposition 3 in Yaari's dual theory), corresponds, in the case of probabilized uncertainty, to Schmeidler's representation theorem 1.2 in the case of non probabilized uncertainty.

**Theorem 1.2:** Assume that P is non-atomic and σ-additive, a functional I on the set B of bounded random variables on (Ω, Q, P) such that I(Ω*) = 1 satisfies P-almost everywhere monotony and P-comonotone additivity if and only if there exists a unique continuous non-decreasing function f: [0, 1] → [0, 1], with f(0) = 0 and f(1) = 1, such that:

$$ \forall X \in \mathcal{B} \ I(X) = \int_{-\infty}^{0} (f(P[X \ge x]) - 1) dx + \int_{0}^{+\infty} f(P[X \ge x])dx \quad (1.2). $$

For a proof, see, e.g., Chateauneuf (1994).

## 2. Decision theory.

The first economic explicit application of the notion of comonotony appeared in decision theoretic models extending expected utility theory. In the classical theory of expected utility, the crucial axiom is Savage's sure thing principle. Numerous experimental violations of this axiom are related to non comonotony of acts. Indeed, as have been noticed before, mixing two acts which are not comonotone can have a hedging effect, a property which is neglected by the sure thing principle. Schmeidler (1989) and Yaari (1987), each one in his own axiomatic, use the property that comonotone acts cannot have a hedging effect when combined with each other.

See Klibanoff (1996) for an interesting discussion on this subject.

### 2.1 Decision under uncertainty

Decision theory under uncertainty deals with a preference relation ≽ on the set $\mathcal{F}$ of "acts" which are measurable functions X: (Ω, Q) → (C, G) where C is a set of consequences and G is the σ-algebra defined by the preorder on C.
- Schmeidler's Subjective Choquet Expected Utility model (1989), where C is the set of distributions with finite support over a set of outcomes and $\mathfrak{F}$ the corresponding set of acts, states:

There exists a utility function $u$ on consequences, defined up to an affine transformation, and a unique capacity $\mu$ on $(\Omega, \mathcal{A})$ such that the value of an act is the Choquet integral with respect to $\mu$, of the utility of its consequences.

$$ \forall X \in \mathfrak{F} \quad I(X) = \int_{\Omega} u(X) \, d\mu $$

This result obtains under usual standard axioms on preferences and the central axiom of comonotone independence:

$$ \forall X, Y, Z, \text{pairwise comonotone, and } \forall \alpha \in (0, 1] $$

$$ X \succcurlyeq Y \Rightarrow \alpha X + (1-\alpha)Z \succcurlyeq \alpha Y + (1-\alpha)Z. $$

- In the case of a general set of consequences, the crucial axiom, weaker than the sure thing principle for Subjective Choquet Expected Utility is the comonotone sure thing principle:

Let $A_i, i=1...n$ be a partition of $\Omega$ and $X, Y$ be two comonotone finite valued acts,

$$ \begin{gathered} [X = x_1 A_1^* + \dots + x_n A_n^* \geq Y = y_1 A_1^* + \dots + y_n A_n^*] \\ \Rightarrow [X' = x_1 A_1^* + \dots + x_n A_n^* \geq Y' = y_1 A_1^* + \dots + y_n A_n^*] \end{gathered} $$

where:

$$ x_1 \leq \dots \leq x_n, y_1 \leq \dots \leq y_n $$

$$ x_1 \leq \dots \leq x'_n, y_1 \leq \dots \leq y'_n $$

Unfortunately this axiom (see for instance Gilboa (1987), Chew and Wakker (1996), jointly with standard ones, merely characterizes a more general model than Choquet expected utility.

## 2.2 Decision under risk (probabilized uncertainty)

In this section and for the sake of simplicity, let lotteries be expressed in terms of distribution on set $C = [0, 1]$ of consequences but, equivalently, let
($\Omega$, $\mathcal{A}$, $P$) =([0, 1], $\mathfrak{B}[0, 1]$, $\lambda$) with $\lambda$ uniform and consider $B$ the set of random variables from($\Omega$, $\mathcal{A}$, $P$) to $C$.

- Yaari's dual choice theory (1987) establishes that a preference relation on $B$ is represented by a functional $I$ such that:

$\exists f:[0, 1]\rightarrow[0, 1]$ increasing continuous, with $f((0)=0, f(1)=1$ and:

$$I(X) = \int_{0}^{1} f(P[X > x])dx \quad (2.2).$$

Notice that $f(P)$ is a capacity and then $I$ is a Choquet integral.

This result is obtained under the standard axioms for the preference relation and the central axiom of comonotone additivity:

$$\forall X, Y, Z \text{ in } B, \text{ pairwise } P\text{-comonotone, } X \sim Y \Rightarrow X+Z \sim Y+Z.$$

- In order to obtain the general Rank Dependent Expected Utility model,

$$RDEU(X) = \int_{0}^{1} f(P[u(X)>x])dx \quad (2.3)$$

where the cardinal utility $u: [0, 1] \rightarrow \mathbb{R}$, increasing and continuous, has been normalized: $u(0) = 0$ and $u(1) = 1$, and the perception $f:[0, 1]\rightarrow[0, 1]$ increasing continuous, with $f((0)=0, f(1)=1$, the crucial axiom, here expressed in terms of lotteries, is again the comonotone sure thing principle, for known probabilities :

$$\forall x, y, \epsilon \in [0, 1] \\ p_1 \delta x_1 + \dots + p_i \delta x_i + \dots + p_n \delta x_n \geq p_1 \delta y_1 + \dots + p_i \delta y_i + \dots + p_n \delta y_n \Rightarrow \\ p_1 \delta x_1 + \dots + p_i \delta y_i + \dots + p_n \delta x_n \geq p_1 \delta y_1 + \dots + p_i \delta y_i + \dots + p_n \delta y_n .$$

$$\text{where} \quad \begin{array}{l} x_1 \le \dots \le x \le \dots \le x_n, \quad y_1 \le \dots \le x \le \dots \le y_n \\ x_1 \le \dots \le x' \le \dots \le x'_n, \quad y_1 \le \dots \le x' \le \dots \le y_n \end{array}$$

Clearly, the comonotone sure thing principle is much weaker than Savage's sure thing principle. It can be shown that it allows changes in the direction of preferences in case of extreme reductions of risk.

Unfortunately this axiom (see for instance Green and Julien (1988), Segal (1989), Chew and Epstein (1989), Wakker (1993) and Chateauneuf (1996)) jointly with
standard ones merely characterizes a more general model than Choquet expected
utility in the case of risk.

**3. Increasing Risk, risk aversion and uncertainty aversion**

The notion of risk aversion has been more studied than the notion of uncertainty
aversion. One reason for this, is that there are several straightforward candidates for
a measure of risk, of increasing risk, (in the case of probabilized uncertainty) while
there is nothing similar to measure uncertainty. We shall therefore invert our usual
plan and present definitions of risk aversion first, followed by an intermediate
situation that we call statistical uncertainty , before we deal with non-probabilized
uncertainty.

**3.1. Probabilized uncertainty**

The classical and general notion of increasing risk is expressed in terms of spread for
random variables with same means.

**Definition 3.1:** Let X and Y be two random variables from $(\Omega, \mathcal{A}, P) \to \mathbb{R}$
$Y$ Mean Preserving Spread (MPS) X if and only if and $\exists \theta$ random variable on $\Omega$
with $E(\theta/X) = 0$ such that : Y has the same probability distribution as $X + \theta$.

We can also say that $E(X) = E(Y)$ and $X$ Stochastically Dominates $Y$ at the Second Order ($X \text{ SSD } Y$). This notion of MPS leads to a general (in the sense that it doesn't depend on a theory of preference representation) definition of risk aversion.

**Definition 3.2:** A Decision Maker is strongly risk averse (SRA) if and only if, for any X, Y, s.t. Y MPS X, he prefers X to Y.

Some criticisms have been made about this notion of *MPS* since, in some cases of *Y*
*MPS X*, *Y* is not considered as really more "risky" than *X*.

Based on some counter-examples, Quiggin (1991) has defined a new notion of
increasing risk which cannot be questionned :
**Definition 3.3:** Let X and Y be two random variables from $(\Omega, \mathcal{A}, P) \to \mathbb{R}$, Y is a monotone mean preserving spread of X (Y MMPS X) iff $E(X) = E(Y)$ and $\exists Z, Z$ comonotone with X, such that Y has the same distribution as $X + Z$.

With such a notion, adding Z to X can, in any case, have a hedging effect and Y is really "more risky" than X.

In terms of distributions with equal means, this definition implies, but is not equivalent to Second Order Stochastic dominance (SSD) (see, for instance Cohen (1995)).

The notion of MPS is based on a characterization of dispersion preserving the mean. This is in accordance with the basic idea that behavior toward risks takes both return and risk into account; an idea formalized by Markowitz' (1952) Mean-Variance analysis. If one forgets this idea of arbitrating between risk and return and looks at characterizing dispersion only, variance is not the most general risk measure (furthermore, it is not always finite).

Instead, Bickel and Lehmann (1976) defined a dispersion order on a set of probability distributions, based on the differences between quantiles, which can be applied even though variances are not finite.

**Definition 3.4 Bickel-Lehmann's (1976) (B-L) Dispersion Order:**

A random variable X with cumulative distribution function F is less dispersed than a random variable Y with cumulative distribution function G in the sense of Bickel-Lehmann, (B-L less dispersed) if and only if:

$$G^{-1} - F^{-1} \text{ is increasing or: } \forall u,v \in ]0,1[, F^{-1}(v) - F^{-1}(u) \le G^{-1}(v) - G^{-1}(u).$$

The surprising link between MMPS and B-L dispersion is found in Landsberger and Meilijson (1994):

**Proposition 3.1 (Landsberger-Meilijson (1994):** Let two random variables, X with distribution F, Y with distribution G, F is less dispersed than G according to Bickel-Lehmann's Dispersion Order if and only if there exists a random variable Z, comonotone with X, s.t. Y has the same distribution as $X + Z$.
From this proposition, we deduce that *Y MMPS X* iff *E(X) = E(Y)* and *X* is B-L less dispersed than *Y*.

The associated notion of risk aversion is the following :

**Definition 3.5:** *A Decision Maker is monotone risk averse (MRA) if and only if, for any X,Y, s.t. ∀X, Y, *Y MMPS X*, (or *Y is mean preserving B-L more dispersed than X*), he prefers *X* to *Y*.*

Clearly SRA implies MRA (For more details see Cohen (95)).

Notice that, whereas under Expected Utility, these two concepts are both characterized by a concave *u*, their characterization in more general models can be completely different and is consistent with more diversified behaviors.

For instance, in RDEU model, while SRA is characterized by a concave utility function *u* and a convex perception function *f* (Chew Karni Safra (1987), MRA is characterized by a coefficient of pessimism associated to the perception function *f*, greater than a coefficient of non-concavity associated to the utility function *u* (see Chateauneuf, Cohen, Meilijson (1997))<fnref n="1" />.

**3.2. Statistical uncertainty**

A transition between situations of probabilized uncertainty and situations of non-
propabilized uncertainty is the situation of a statistical decision problem<fnref n="2" />. In
statistics, uncertainty is assumed to be probabilized, but the distribution is unkown,
all that is known is that it belongs to a family of probability laws, this family is
indexed by a subset of ℝⁿ in parametric statistics, by some general set in non-
parametric statistics. In Bayesian statistics, the parameter set is probabilized by an "a
priori" probability distribution. All these set-ups can be considered as special cases
of what we call non-probabilized uncertainty.

In a given problem, let us call $\mathcal{O}$ the set of relevant probability distributions<fnref n="3" /> on our
uncertainty space $(\Omega, \mathcal{A})$ and, for $\theta \in \mathcal{O}$, $P^{\theta}$ any relevant probability distribution.

<footnote n="1">¹Quiggin (1991) proved only that under RDEU : *u concave and f(p) ≤ p* implies MRA.</footnote>

<footnote n="2">²See Kast (1991) chapter 3, for instance.</footnote>

<footnote n="3">³Notice that, in some problems, there is no such set of distribution that a decision maker would think "relevant".</footnote>
For any $\theta \in \Theta$, given two random variables $X$ and $Y$ with respective cumulative distribution functions $F_{\theta}$ and $G_{\theta}$, we can use Bickel-Lehmann's definition :

$X$ is $P^{\theta}$-more dispersed than $Y$ iff $G_{\theta}^{-1} - F_{\theta}^{-1}$ is increasing or
$$ \forall u, v \in ]0, 1[, F_{\theta}^{-1}(v) - F_{\theta}^{-1}(u) \leq G_{\theta}^{-1}(v) - G_{\theta}^{-1}(u). $$

From there, it is natural to define a notion of dispersion related to situation of statistical uncertainty:

$X$ is $\Theta$-more dispersed than $Y$ iff $\forall \theta \in \Theta$, $G_{\theta}^{-1} - F_{\theta}^{-1}$ is increasing.

As a straightforward extension of proposition 3.1, we have:

**Proposition 3.2:** *X is $\Theta$-more dispersed than $Y$ iff $\forall \theta \in \Theta$, there exists a random variable $Z_{\theta}$, $P_{\theta}$-comonotone to $X$, such that $Y = X + Z_{\theta}$.*

This last definition and its characterization by the proposition is an intermediate notion of dispersion order that could be developed in the more general context of non-probabilized uncertainty.

### 3.3. Non probabilized uncertainty

The most straightforward notion of uncertainty aversion is captured by the convexity of preferences (Arrow (1953)). Schmeidler (1989) gives a characterization of the convexity of preferences in terms of convexity of the capacity<fnref n="1" />.

$\mu(A \cup B) + \mu(A \cap B) \geq \mu(A) + \mu(B)$ which implies superadditivity of the integral.

Wakker (1990), used Schmeidler's theory to characterize two types of behaviors with respect to uncertainty. Replacing the comonotone independence axiom by the following:

<footnote n="1">¹Convexity is more rightly called supermodularity (Denneberg(1994)).</footnote>
**Pessimism:** $\forall X, \forall Y, \forall Z$ comonotone to $Y$

$$X > Y \Rightarrow \alpha X + (1-\alpha) Z > \alpha Y + (1-\alpha) Z.$$

This axiom, in which $Z$ is not assumed to be comonotone with $X$, can be interpreted as a preference for a hedge : $\alpha X + (1-\alpha) Z$ can have less variability than $\alpha Y + (1-\alpha) Z$, hence, a pessimistic decision maker (in the sense that uncertainty is expected not to turn out favorable) will prefer the former to the latter. Under this axiom, if the preference order is represented by a Choquet Expected Utility with $u$ linear and not constant, the capacity $\mu$ is convex and the integral is super-additive :

$$\int u(X) d\mu = \text{Min} \{ \int u(X) dP / P \text{ proba } P \geq \mu \}.$$

Conversely, assume the comonotone independence axiom is replaced by the following:

**Optimism:** $\forall Y, \forall X, \forall Z$ comonotone to $Y$

$$X \succ Y \Rightarrow \alpha X + (1-\alpha) Z \succ \alpha Y + (1-\alpha) Z.$$

Optimism can be interpreted as the characterization of the preference order of a decision maker who expects uncertainty to turn out favorable. Then, if the preference order is represented by Choquet Expected Utility with a linear and not constant utility, the capacity is submodular and the integral is subadditive :

$$\int u(X) d\mu = \text{Max} \{ \int u(X) dP / P \text{ proba } P \leq \mu \}$$

**4. Pareto-optimal allocations.**

The first welfare theorem says that, the general equilibrium allocations are Pareto-optimal. In a context of uncertainty with only one consumption good, this result is interpreted as an optimal allocation of insurance and reinsurance contracts over the risky initial endowments. However, results presented here are only obtained in the context of probabilized uncertainty and we assume that decision makers are risk
averters. In the context of risk sharing, Pareto optimality is a characterization of an
optimal allocation of risks among risk-averse agents.

It has been put forward by Borch (1962) and by Wilson (1968) (in the context of
Expected Utility Theory) that Pareto-optimal allocations of risks (described by
random variables) are characterized by the equality of the ratios of marginal utilities
and the property that every individual allocation is an increasing function of the sum
of all allocations. This last property is our property iv in proposition 1.1.. In the
context of insurance, its implication is a justification of the mutualization of risks (in
the sense of sharing the sum of all risks, as is done in a mutual fund framework).

In the following proposition we show that reference to the Expected Utility Model is unessential; indeed, only the satisfaction of second order stochastic dominance is relevant.

In this context, let $(S, 2^S, \Pi)$ be a finite probability space, $S = \{s_1, \dots, s_n\}$ and for any $i = 1 \dots n$, $p_i = \Pi(\{s_i\}) > 0$. Let $V$ be the set of all measurable real random variables. An uncertain payoff $D$ is an element of $V$ and an allocation of this payoff among agents $i = 1 \dots m$ is a random vector $X = (X_1, \dots, X_m)$ with the $X_i$'s in $V$ and $\sum_{i=1}^m X_i = D$. Let $\mathcal{D}$ be the set of all allocations of $D$.

**Proposition 4.1:** *Assume every agent i has a preference relation $\succ_i$ on V, which is consistent with Second Order Stochastic Dominance (SSD) i. e. :*

$\forall X, Y \in V, Y \text{ SSD } X \Rightarrow \forall i = 1 \dots m, Y \succ_i X \text{ and the same for strict stochastic dominance and strict preference.}$

Then, if an allocation $X = (X_1, \dots, X_m)$ is Pareto-optimal in $\mathcal{D}$, $X$ is a comonotone family, or, otherwise stated, any non-comonotone allocation is strictly dominated by a comonotone one.

*Proof:* Let us show that any non-comonotone allocation $X$ is strictly dominated.

Without loss of generality let us assume $X_1(s_1) = x_1 > x_2 = X_1(s_2)$, $X_2(s_1) = y_1 < y_2 = X_2(s_2)$ and $X_1(s_1) + X_2(s_1) \le X_1(s_2) + X_2(s_2)$.

Then, let us modify, $(x_1, x_2)$, $(y_1, y_2)$ into $(x'_1, x'_2)$, $(y'_1, y'_2)$ such as :

$$x'_1 = x'_2 = \frac{p_1 x_1 + p_2 x_2}{p_1 + p_2}, \quad y'_1 = y_1 + \frac{p_2}{p_1 + p_2} (x_1 - x_2) \quad \text{and} \\ y'_2 = y_2 - \frac{p_1}{p_1 + p_2} (x_1 - x_2)$$
It is straightforward to verify that $x'_1 + y'_1 = x_1 + y_1$, $x'_2 + y'_2 = x_2 + y_2$, that $y_1 < y'_1 \le y'_2 < y_2$ and that $p_1 x'_1 + p_2 x'_2 = p_1 x_1 + p_2 x_2$, $p_1 y'_1 + p_2 y'_2 = p_1 y_1 + p_2 y_2$.

So, if we change $X_1$ and $X_2$ into $X'_1$ and $X'_2$ with $X'_1(s_i) = x'_i$ for $i = 1, 2$, $X'_2(s_i) = x'_i$ for $i = 1, 2$, $X'_2(s_i) = X_2(s_i)$ for $i > 2$, it is straightforward that $X' = (X'_1, X'_2, X_3, \dots, X_m)$ is an allocation of $D$ and that $X'_1$ and $X'_2$ strictly second order stochastic dominate $X_1$ and $X_2$ respectively. Hence, $X'$ strictly dominates allocation $X$. ♦

This proof is simpler than Landsberger and Meilijson's (1994), but, in fact, they got a stronger result: since they proved in a *constructive* way, through an algorithm of at most $\frac{n(n-1)}{2}$ stages, the existence of a dominating comonotone allocation when starting from a non comonotone one.

**5. Inequality measures**

Let us consider an homogeneous population of $n$ individuals, each of them with an income $x_i$, $i = 1, \dots, n$. Without loss of generality, we assume incomes to be ranked: $x_1 \le \dots \le x_i \le \dots \le x_n$ and we call $x = (x_1, \dots, x_i, \dots, x_n)$ an income profile.

An important property for an inequality measure to satisfy is usually recognized as the Pigou-Dalton principle of transfers (or, equivalently, the Lorenz' criterion): A transfer of income from a richer person to a poorer person preserving the rank order of incomes, will decrease inequality.

Conservation of the rank order is so important in the previous principle as well as in the inequality measurement in general, that it led several authors to make an explicit or implicit reference to the comonotony notion. For instance, Weymark (1981) generalizes the Gini index (a linear function with the weights being the odd numbers) by allowing any monotone increase of weights. His generalized Gini indices are functions which satisfy comonotone additivity. Ben Porath and Gilboa (1991) explicitly refer to comonotone additivity and Yaari (1988) uses it. Accordingly, each of these models makes a more or less explicit use of Choquet integral. If $f$ is a strictly increasing and continuous function from $[0, 1]$ into $[0, 1]$ with $f(0) = 0$ and $f(1) = 1$, these models yield relative inequality indices $I_n$ defined for any profile $x$, by:
$$I_n(x) = 1 - \frac{J(x)}{\bar{x}} \quad \text{where } J(x) = x_1 + \sum_{i=1}^{n} f(\frac{n-i}{n}) (x_i+1 - x_i) \quad \text{and } \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i.$$

Furthermore the Pigou-Dalton principle is equivalent to f convex. It is noteworthy that the Gini index is the special case where $f(p) = p^2$; this was put forward by Blackorby and Donaldson (1978) and by Yaari (1988).

Once the role of comonotone transfers is made clear, Chateauneuf (1994) notices that Lorenz' criterion may be unsatisfactory. Indeed, consider the following reallocation of profile x into profile y:

$$x = (7000, 8000, 9000, 10000)$$

$$y = (7500, 7500, 9500, 9500).$$

It is straightforward that inequality has been reduced, according to Lorenz' criterion. However, individual n°2 has seen his income decrease while n°3, who already had more, has seen his increase. The reason is that $y = x + \varepsilon$ with $\varepsilon$ non comonotone to x. In contrast, if y had been obtained by means of a vector of transfers $\varepsilon'$ comonotone to x the previous "paradox" would not have taken place: all the income spreads would have been reduced. $\forall i, y'_i = x_i + \varepsilon'_i, y'_i \le y'_{i+1}$, with $\varepsilon'_i \ge \varepsilon'_{i+1}$ and $\sum_{i=1}^{n} \varepsilon'_i = 0$.

For instance: $y' = (8000, 8500, 8500, 9000)$ or even, $(8500, 8500, 8500, 8500)$.

It must be noted that this new inequality order is a special case of Lorenz' and is nothing else than a transposition of Quiggin's monotone spread (section 3.1) to inequalities. Along this line, recall that Lorenz' order was identified as the second order stochastic dominance in Atkinson (1970) famous paper.
# 6. Portfolio and financial valuation theories.

It has been emphasized several times in previous sections, that adding a comonotone random variable yields an increase in risk. Furthermore, adding a non-comonotone random variable may (but need not) provide an hedge to the risk which is held. We provide two examples of theoretic analysis of this refinement of the hedging property. The first one is done in a context of non-probabilized uncertainty, and it leads to an explicit assets' valuation formula in the case where the selling price may differ from the buying price. The second is in the case of risk, it derives an insurance premium.

## 6.1. Bid-ask spreads valuation under uncertainty.

Markets where some frictions prevail show some non-linearity in prices. Chateauneuf, Kast and Lapied (1994) and (1995) propose a valuation formula in markets in which a dealer sets prices. Each marketed asset, defined by its payoffs random variable $X$, has a buying price $q(X)$ and a selling price $-q(-X)$, the difference between the two is the dealer's profit: $q(X-X) = 0 \le q(X) + q(-X)$.

Valuation theory answers the following question: at what price should the dealer sell a non-marketed asset that could be replicated by a portfolio of marketed assets? The answer is based on a no-arbitrage condition which allows the dealer to take a premium on assets that are comonotone:

### Comonotony premium :

> $\forall X \text{ and } Y: q(X + Y) \le q(X) + q(Y)$; the equality holds if $X$ and $Y$ are comonotone.

The idea is the following: a portfolio $X + Y$ of non-comonotone assets may be less risky (in the sense of reducing the dispersion) to hold, than each of $X$ and $Y$ separately. Hence, the dealer may charge less on the portfolio than on the sum of the two separate assets.

If $Y$ is comonotone to $X$ the increase of dispersion of portfolio $X + Y$ depends only on $Y: F_Y^1(u) - F_Y^1(v)$.
Hence, in this case the difference $q(X + Y) - q(X)$, which only depends on $q(Y)$, is assumed to be equal to $q(Y)$.

This condition, together with other standard no-arbitrage conditions in valuation theory, yields a valuation formula: the selling price of an asset is the Choquet integral of its discounted payoffs with respect to a concave capacity. Hence (see Denneberg (1994)) the selling price of an asset is the maximum of a family of expected discounted payoffs, that is, prices which could be obtained if there were no dealer.

## 6.2. Insurance premium under probabilized uncertainty

Denneberg (1990), also used the notion of comonotony to construct a new functional for risk premium calculation (see also Wang (1995)). He suggests that the insurance premium function, $I$, should respect $I(X + Y) \le I(X) + I(Y)$ for any two claims derived from insurance contracts with equality if $X$ and $Y$ are comonotone. This condition, with usual minimal requirements for premium calculations (such as: $\text{Sup}(X) \ge I(X) \ge E_P(X)$) leads to a premium formula $I(X) = \int_X d\text{foP}$ with $f$ concave.

## Conclusion

The object of this paper was to point out that, more or less explicitly, the notion of comonotony had been used and defined in many different contexts. It appears that, from statistics to inequality measurements, from individual decision theory and valuation of risks to financial valuation, this notion is relevant to compare the dispersion of random variables, with or without reference to a probability distribution.

The surprising links that have been put forward in this paper between definitions and properties developed in different contexts show that the notion of comonotony can yield powerful tools in many areas. For instance, this has been the case in Chateauneuf (1996) for inequality measurements or in Kast and Lapied (1996a and b) for bid and ask price processes which amount to a generalization of the martingale equivalent result of finance. Another direction where the notion of comonotony could help to generalize economic results is indicated by the new light shed on the
characterization of Pareto-optimal allocations which we would like to extend in two directions (i) to weaker form of risk aversion, under risk, (ii) to uncertainty aversion, under non-probabilized uncertainty.

References

Allais, M. "Le comportement de l'homme rationnel devant le risque: critique des postulats et axiomes de l'école américaine", *Econometrica*, 21, pp. 503-546, 1953.

Arrow, K.J. "Le rôle des valeurs boursières dans l'allocation optimale des risques" in *Econométrie* 40, pp 41-47, Cahiers du CNRS, Paris, 1953.

Atkinson, A. "On the measurement of inequality", *Journal of Economic Theory*, 2, pp.244-263, 1970.

Ben Porath, E. and Gilboa, I. "Linear measures, the Gini index and the income-equality tradeoff", *Journal of Economic Theory*, 64, pp. 443-467, 1994.

Bickel, P.J. and Lehmann, E.L. "Descriptive statistics for nonparametric model IV. Spread", in *Contributions to Statistics*, Jane Juresckova ed., Reidel publishing c, Amsterdam, 1979.

Borch, K. "Equilibrium in reinsurance markets" *Econometrica*, 30, 1962, p. 424-444.

Borch, K. *Economics of Insurance* North-Holland publishing C°, Amsterdam 1990.

Chateauneuf, A. "On the use of comonotony in the axiomatization of EURDP theory for arbitrary consequences", mimeo CERMSEM(first version 1990), under revision for *Journal of Mathematical Economics* 1996.

Chateauneuf, A. "Decreasing inequality: an approach through non-additive models", mimeo CERMSEM(first version 1994) 1996.

Chateauneuf, A. "Modelling attitudes towards uncertainty and risk through the use of Choquet integral", *Annals of Operations Research*, 52, p. 3-20, 1994.

Chateauneuf, A., Kast, R. and Lapied, A. "Choquet pricing in financial markets with frictions", *Mathematical Finance*, vol. 6 n°3, p. 323-330, 1996.

Chateauneuf, A., Cohen, M. and Meilijson, I. "More pessimism than greediness : a characterization of monotone risk aversion in RDEU model", D.T., 1997.

Chateauneuf, A., Kast, R. and Lapied, A. "Market preferences revealed by prices: non-linear pricing in slack markets", in *Models and experiments in Risk and Rationality*, M. Machina and B. Munier eds., Kluwer Acad. Pub., Dordrecht, 1994.
Chew, S.H. and Epstein, L. "A unifying approach to axiomatic non expected utility theories", *Journal of Economic Theory*, 49, p. 207-240, 1989.

Chew, S.H and Wakker, P.P. "The comonotonic Sure-Thing Principle" *Journal of Risk and Uncertainty*, 12, p5-27, 1996

Choquet, G. "Théorie des capacités", *Annales de l'institut Fourier*, V, Grenoble, pp. 131-295, 1953.

Cohen, M. "Risk aversion concepts in expected and non-expected utility models", *Geneva papers on risk and insurance theory*, vol 20, pp. 73-81, 1995.

Dekel, E. "Asset demands without the independence axiom", mimeo Harvard, 1985.

Dellacherie, C. "Quelques commentaires sur les prolongements de capacités", *Séminaire de probabilités 1969/70*, Strasbourg, L.N. in Math. 191, Springer-Verlag, p. 77-81.

Denneberg, D. *Non-additive measures and integral*, Kluwer Acad. Pub., Dordrecht, 1994.

Denneberg, D. "Premium calculation: why standard deviation should be replaced by absolute deviation", *ASTIN bulletin*, 20, p. 181-190.

Fishburn, P. C. "On Handas' new theory of cardinal utility and the maximization of expected return", *Journal of pMMolitical economy*, 86, pp. 321-324, 1978.

Gilboa, I. "Expected utility with purely subjective non-additive probabilities", *Journal of Mathematical Economics*, 16, 65-88, 1987.

Green, J and Julien, B. "Ordinal independence in non-linear utility theory", *Journal of Risk and Uncertainty*, 1, p. 355-387, 1988.

Hardy, G.H., Littlewood J.E. and Polya G. *Inequalities*, Cambridge University Press, Cambridge, 1934 (2nd edition 1952).

Henriet, D. and Rochet J-C. *Microéconomie de l'assurance*, Economica, Paris, 1991.

Kast, R. *Rationalité dans les marchés financiers*, Economica, Paris, 1991.

Kast, R. and Lapied, A. "Term Structure of Interest Rates with Frictions: a Discrete Time Approach", D.T. GREQAM, 1996 a.

Kast, R. and Lapied, A. "Discrete time generalized option pricing with bid-ask spreads", D.T. GREQAM, 1996 b.

Klibanoff, P. : "Characterizing Uncertainty Aversion through Preferences for Mixtures", W.P., 1996.
Landsberger, M. and Meilijson, I. "Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion", *Annals of Operations Research*, 52, pp 97-105, 1994.

Markowitz, H. "Portfolio selection", *Journal of Finance*, 1952.

Quiggin, J. "A theory of anticipated utility", *Journal of economic behavior and organization*, 3, pp. 323-343, 1982.

Quiggin, J. *Increasing risk, another definition in : Progress in decision, utility and risk theory* , Kluwer, 1991.

Rothschild, M. and Stiglitz, J. "Increasing risk : I. A definition", *Journal of Economic Theory*, 2, pp 245-243, 1970.

Segal, U. "Anticipated utility: a measure representation approach", *Annals of Operations Research*, 19, p. 359-373, 1989.

Schmeidler, D. "Integral representation without additivity", *Proceedings of the American Mathematical Society*, 97, 1986, p 255-261.

Schmeidler, D. "Subjective probability and expected utility without additivity", *Econometrica*, Vol 57 N° 3, pp.571-587, 1989.

Röell, E. "Risk aversion in Quiggin and Yaari's rank order model of choice under uncertainty", *The economic journal, vol. 97, conference papers*, pp. 143-159, 1987.

Wakker, P. "Characterizing optimism and pessimism through comonotonicity", *Journal of Economic Theory*, 52, 1990, p. 453-463.

Wakker, P. "Separating utility and probabilistic risk aversion", *Theory and Decision*, 36, 1, pp. 1-44, 1994.

Weymark, J. "Generalized Gini inequality indices", *Mathematical Social Sciences*, 1, 1981, p. 409-430.

Wilson, R. "The theory of syndicates", *Econometrica*, 36, pp 119-132, 1968.

Yaari, M. E. "The dual theory of choice under risk", *Econometrica*, 55, 1987, P. 119-132.

Yaari, M.E. "A controversial proposal concerning inequality measurement", *Journal of Economic Theory*, 44, 1988, p. 381-397.

Wang, S. "Premium calculation by transforming the layer premium density", W.P. University of Waterloo, Ontario.