# More pessimism than greediness: a characterisation of monotone risk aversion in the Rank-Dependent Expected Utility Model

Alain Chateauneuf, Michèle Cohen, Isaac Meilijson

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Alain Chateauneuf, Michèle Cohen, Isaac Meilijson. More pessimism than greediness: a character-isation of monotone risk aversion in the Rank-Dependent Expected Utility Model. 1997. halshs-04654662

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Distributed under a Creative Commons Attribution 4.0 International License
CAHIERS ECO & MATHS

97.53

MORE PESSIMISM THAN GREEDINESS :
A CHARACTERISATION OF MONOTONE RISK AVERSION
IN THE RANK-DEPENDENT EXPECTED UTILITY MODEL

Alain CHATEAUNEUF

Michèle COHEN

Isaac MEILIJSON

CERMSEM & CEME
# More pessimism than greediness :
A characterisation of monotone risk aversion
in the Rank-Dependent Expected Utility Model

Alain Chateauneuf<fnref n="1" />

Michèle Cohen<fnref n="2" />

Isaac Meilijson<fnref n="3" />

May, 10, 1997

## Abstract

We study attitudes towards risk in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, which is characterised by two functions, a *utility function* on outcomes in conjunction with a *probability-perception function*. We use the notion of *monotone risk aversion* introduced by Quiggin ([18]), that is, aversion to monotone mean-preserving increase in risk (monotone increase in risk, for short), based on a notion of co-monotonicity of random variables that has been shown to play a decisive role in this field. This paper presents a characterisation of the pairs $(u, f)$ of utility function and probability perception function of monotone risk averse decision makers. The characterisation is done in terms of an index of *greediness* $G_u$ of the utility function $u$ and a index of *pessimism* $P_f$ of the probability perception function $f$. We prove that a decision maker with utility function $u$ and probability perception function $f$ is monoton risk averse if and only $P_f \ge G_u$. The novel element is that concavity of $u$ is not necessary.

Keywords : Risk aversion, comonotonicity, pessimism, Rank-Dependent Expected Utility.

<footnote n="1">¹CERMSEM, Université de Paris I, Maison des Sciences Economiques, 106 - 112 Boulevard de l'hôpital, 75647 Paris Cedex 13, France. E-mail : chateaun@univ-paris1.fr</footnote>

<footnote n="2">²CEME, Université de Paris I, Maison des Sciences Economiques, 106 - 112 Boulevard de l'hôpital, 75647 Paris Cedex 13, France. E-mail : cohenmd@univ-paris1.fr</footnote>

<footnote n="3">³School of Mathematical Sciences, Raymond and Beverly Sackler, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel</footnote>
Résumé

Nous étudions les attitudes vis-à-vis du risque dans le modèle Espérance d'Utilité Dependant du Rang. Ce modèle remplace l'espérance d'utilité par une autre fonctionnelle, caractérisée par deux fonctions, une fonction d'utilité u sur les résultats et une fonction f de perception des probabilités. Nous utilisons la notion d'aversion pour le risque monotone, introduite par Quiggin [18], qui est l'aversion pour l'accroissement de risque monotone, à moyenne constante, basée sur la notion de comonotonie de variables aléatoires. Dans ce papier, nous donnons une caractérisation des paires (u, f), de fonctions d'utilité et de fonctions de perception des agents ayant de l'aversion monotone pour le risque. Cette caractérisation est donnée à l'aide d'un indice de non-concavité (ou avidité) $G_u$ de la fonction d'utilité et d'un indice $P_f$ de pessimisme de la fonction f : un agent caractérisé par (u, f) aura de l'aversion pour le risque monotone si et seulement si $P_f \ge G_u$. Cette caractérisation ne nécessite pas que u soit concave.

Mots clés : Aversion pour le risque, comonotonie, pessimisme, Espérance d'Utilité Dépendant du Rang.
# 1 Introduction

Under the expected utility (EU) model, a decision maker (DM) is characterised by only one function *u* called utility function.

The crucial criticism of the EU model by Allais [1] from a theoretical point of view, is that in such a model the same function *u* characterises two distinct behaviours - satisfaction from certain wealth and attitude to risk - that have no reason to be linked: a DM with diminishing marginal utility on certain wealth may be risk-seeking, but this is precluded by the EU model.

The rank-dependent expected utility (RDEU) model (see [17, 27, 10, 24, 18, 6]) has been built in part as an attempt to answer the criticism of Allais. A RDEU DM is characterised by two functions: a utility function *u* on outcomes in conjunction with a *probability-perception* function *f*: [0, 1] → [0, 1]. This function is strictly increasing and satisfies *f*(0) = 0, *f*(1) = 1. The RDEU decision maker compares a lottery distributed *F* with another by the expected utility of the lottery evaluated as if the lottery had distribution function 1-*f*(1-*F*) instead of *F* (see (4)).

A risk averse DM is usually defined as one that, for every bounded random variable, prefers the expectation of the random variable to the random variable itself. This notion will be called *weak risk aversion*. Aversion to mean-preserving increase in risk in the sense of Rothschild & Stiglitz (see [13, 21]) will be called *strong risk aversion*.

In the EU model, aversion to risk is a rather clear-cut notion since each of these two variants of aversion to risk is fully characterised by concavity of the utility function *u*.

The situation in the RDEU model is less clear-cut due to the trade-off between risk aversion implications of utility and probability perception. Weak risk aversion under all concave utilities *u* is modeled by perception functions *f* majorised by the identity function, while strong risk aversion under all concave utilities *u* requires the stricter condition that *f* be convex (see Chew, Karni & Safra [10]). Chateauneuf & Cohen [7] proved that majorisation of the probability perception function by the identity is a necessary condition for weak risk aversion, while concavity of *u* is not, but did not characterise weak risk aversion.

Quiggin [18] brought to light that strong risk aversion may be too strong a concept, i.e., that second degree dominance is perhaps too weak an order of risk, and introduced a new notion *monotone (mean-preserving) increase in risk*, defined in terms of co-monotone random variables instead of mean-preserving spreads. It follows from the results of Landsberger & Meilijson [16] that the notion of monotone increase in risk coincides with the statistical notion of Bickel & Lehmann dispersion (see [3, 4]) for random variables with equal means. The notion of *monotone risk aversion* (aversion to monotone increase in risk) is particularly fitted to RDEU (Rank Dependent Expected Utility) theory (see [10, 17, 24, 27]), where co-monotonicity plays a fundamental part at the axiomatic level. Quiggin (see [18]) further showed that monotone risk aversion under all concave utilities *u* is modeled by perception functions *f* majorised by the identity function. Thus, the same probability perception functions that display *weak risk aversion* under all concave utilities also display *monotone risk aversion*. In this paper we go further: A DM may be averse to monotone increase in risk without having a concave utility. Using the co-monotonicity tools developed by Landsberger & Meilijson [16], we characterise the class of pairs (*u*, *f*) of utility and probability perception functions that model aversion to monotone increase in
risk. More specifically, for an increasing probability perception function $f$, let

$$P_f = \inf_{0 < v < 1} \left[ \frac{1 - f(v)}{f(v)} / \frac{1 - v}{v} \right] \quad (1)$$

to be its index of *pessimism*. The probability perception function is majorised by the identity function if and only if $P_f \ge 1$. The identity function $f(v) \equiv v$ is the only convex perception function with $P_f = 1$ but is not the only identity-majorised perception function with this property. For a probability $v$ of winning, $(1 - v)/v$ is the *odds-ratio* against winning. Pessimists exaggerate this odds ratio by amplifying it to $(1 - f(v))/f(v)$. The index of pessimism can be intuitively understood as the minimal such amplification factor. That is, the DM exaggerates odds ratios by *at least* this amount. For an increasing utility function $u$, define

$$G_u = \sup_{x_1 < x_2 \le x_3 < x_4} \left[ \frac{u(x_4) - u(x_3)}{x_4 - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1} \right] \quad (2)$$

to be its index of *greediness*. This index also satisfies $G_u \ge 1$, and the value 1 corresponds exclusively to concavity. When $u$ is differentiable, the index of greediness (or non-concavity) of $u$ is the supremal value of $u'(x)/u'(y)$ taken over $x \ge y$. Intuitively, it is the maximal possible amplification factor of marginal utility from a low level of wealth to a high level of wealth.

The characterisation of monotone risk aversion in the RDEU model will be based on the comparison of these two indices.

In section 2 we recall the basic definitions and some basic properties of the RDEU model, monotone increase in risk and monotone risk aversion. Section 3 is devoted to the proof of the main theorem, and section 4 handles the corresponding case of monotone risk-seeking attitudes and illustrates in the RDEU context the monotone risk seeking behaviour of a DM with diminishing marginal utility.

# 2 The RDEU model, monotone increase in risk and monotone risk aversion

## 2.1 Notation

We assume that risk prevails and describe it through a set of states of nature $\Omega = [0, 1]$ endowed with the uniform probability measure $P$ on the Borel $\sigma$-field. Let $V = \{X, Y, \dots\}$ be the set of bounded $\mathcal{R}$-valued random variables on $\Omega$.

For any $X$ in $V$, we denote by $F_X$ (respectively, $F_X^* = 1 - F_X$) the cumulative (de-cumulative) distribution function of $X$ and by $E(X)$ the expected value of $X$. Let $\mathcal{D}$ be the set of cumulative probability distribution functions on $\mathcal{R}$.

The distribution (or law) of a discrete random variable $Z$ will be denoted by

$$\mathcal{L}(Z) = (x_1, p_1; \dots; x_k, p_k; \dots; x_n, p_n) \quad (3)$$

with $x_1 < x_2 < \dots < x_n, p_i \ge 0$ and $\sum p_i = 1$.
## 2.2 The RDEU model

Variants of the Rank Dependent Expected Utility theory due to Quiggin [17] have been treated by Yaari [27], Chew, Karni & Safra [10], Segal [24, 25] and Allais [2]. General axiomatisation can be found in Wakker [26], Quiggin & Wakker [19], Chateauneuf [6].

**Definition 1** A DM behaves in accordance with the Rank Dependent Expected Utility (RDEU) model if the DM is characterised by a continuous, strictly increasing utility function in conjunction with a probability-perception function $f: [0, 1] \rightarrow [0, 1]$ that is strictly increasing and satisfies $f(0) = 0, f(1) = 1$. Such a DM (weakly) prefers the random variable $X$ to the random variable $Y$ if and only if $V(X) \geq V(Y)$, where the RDEU functional $V$ is given by

$$ 
\begin{aligned}
V(X) &= V_{u,f}(X) = -\int_{-\infty}^{\infty} u(x)df(P(X > x)) \\
&= \int_{-\infty}^{0} [f(P(u(X) > t)) - 1] dt + \int_{0}^{\infty} (P(u(X) > t))dt.
\end{aligned} 
\quad (4) $$

If the support of the random variable $X$ is a finite set, $V(Z)$ can be written as

$$ 
\begin{aligned}
V(Z) &= \sum_{i=1}^{n} u(x_i) \left[ f\left(\sum_{j=i}^{n} p_j\right) - f\left(\sum_{j=i+1}^{n} p_j\right) \right] \\
&= u(x_1) + \sum_{i=2}^{n} (u(x_i) - u(x_{i-1})) \left[ f\left(\sum_{j=i}^{n} p_j\right) \right] \\
&= u(x_1) + \sum_{i=2}^{n} (u(x_i) - u(x_{i-1})) f(v_{i-1}),
\end{aligned} 
\quad (5) $$

where $v_i = \sum_{j=i+1}^{n} p_j = P(Z > x_i)$.

If the perception function $f$ is the identity function $f(v) \equiv v$, then $V(Z) = V_{u,I}(Z)$ is the expected utility $E[u(Z)]$ of the random variable. The Yaari functional (see [27]) is the special case $V(Z) = V_{I,f}(Z)$, in which the utility on outcomes is the identity function $u(x) \equiv x$. If both perception and utility are identity functions, then $V(Z) = V_{I,I}(Z)$ is simply the expected value $E[Z]$ of the random variable.

## 2.3 Monotone increase in risk and monotone risk aversion

Let us first recall the definition of co-monotone random variables (see [22, 23, 27]).

**Definition 2** Two real-valued functions $X$ and $Y$ on $\Omega$ are co-monotone if for any $\omega$ and $\omega' \in \Omega$, $[X(\omega) - X(\omega')] [Y(\omega) - Y(\omega')] \geq 0$.

For example, any random variable in $V$ is co-monotone with any degenerate random variable. For any random variable $U$ and non-decreasing functions $g$ and $h$, the random variables $g(U)$ and $h(U)$ are co-monotone.

**Definition 3** For $X, Y \in V$, the distribution of $Y$ is a monotone increase in risk of the distribution of $X$, or, $Y$ is a monotone spread of $X$, if there exists a mean-zero random variable $Z \in V$ such that $X$ and $Z$ are co-monotone and $Y$ has the same probability distribution as $X + Z$.
In a statistical framework, Bickel & Lehmann (see [3, 4]) define a notion of comparative dispersion:

**Definition 4** For two cumulative distribution functions $F$ and $G$ in $D$, $F$ is Bickel & Lehmann less dispersed than $G$ if for every $0 < x < y < 1$, $F^{-1}(y) - F^{-1}(x) \le G^{-1}(y) - G^{-1}(x)$.

The definition of monotone spread requires the two distributions to have the same mean. The definition of dispersion doesn't require $F$ nor $G$ to possess finite first moments, and even if they do, these moments need not be equal. In fact, $Y \sim G$ is more dispersed than $X \sim F$ if and only if $Y + c$ is more dispersed than $X$, for arbitrary $c \in \mathbb{R}$. Gathering different properties obtained in the literature on this subject (see [8, 16, 18]), it is possible to obtain the following connection between Bickel & Lehmann dispersion and monotone increase in risk.

**Proposition 1** When two random variables $X$ and $Y$ in $\mathbb{V}$ have the same expected value, $Y$ is a monotone spread of $X$ if and only if $F_X$ is less dispersed than $F_Y$ in the sense of Bickel & Lehmann.

The corresponding notion of monotone risk aversion can be viewed as aversion to monotone increase in risk. More formally,

**Definition 5** A DM is monotone risk averse (respectively, monotone risk seeking) if for any $X$ and $Y$ in $\mathbb{V}$ such that $Y$ is a monotone spread of $X$, the DM prefers $X$ to $Y$ (respectively, $Y$ to $X$).

**Example 1.** Spreading out a two-point distribution.

Let $0 < v < 1$ and $-\infty < x_1 < x_2 < x_3 < \infty$, and let $x_4 = x_3 + (x_2 - x_1)v/(1 - v)$. Consider the mean-preserving monotone increase in risk from the two-point distribution $\mathcal{L}(X) = (x_2, 1 - v; x_3, v)$ (see (3)) to the two-point distribution $\mathcal{L}(Y) = (x_1, 1 - v; x_4, v)$. Monotone risk aversion implies that $V(X) \ge V(Y)$ for all choices of $v$ and $x_i$ as above. Using the representation $V(X) = u(x_2) + f(v)[u(x_3) - u(x_2)]$ (see (5)), this inequality is readily seen to be equivalent to

$$ \frac{u(x_4) - u(x_3)}{x_4 - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1} \le \frac{1 - f(v)}{f(v)} / \frac{1 - v}{v}. \quad (6) $$

Hence, a necessary condition for monotone risk aversion should be that the supremum of the LHS of (6), related to the index of greediness $G_u$ (see (2)) of the utility function $u$, be less than or equal to the infimum of the RHS of (6), the index of pessimism $P_f$ (see (1) and (7)) of the probability perception function $f$. However, the supremum of the LHS of (6) should be taken over vectors $(x_1, x_2, x_3, x_4)$ satisfying $x_1 < x_2 < x_3 < x_4$ and also $x_4 = x_3 + (x_2 - x_1)v/(1 - v)$, so this supremum could in principle depend on $v$. By Lemma 4 presented in the sequel, this supremum, independent of $v$, is equal to the index of greediness $G_u$ of the utility function $u$, to be compared with the index of pessimism $P_f$ of the probability perception function $f$.

The next section introduces these indices and analyses some of their main properties.
# 3 Pessimism, optimism, greediness and thriftiness

## 3.1 Index of pessimism

A RDEU DM is pessimistic under risk if the DM assesses lotteries as a EU DM who under-estimates for each $x \in \mathcal{R}$ the probability to have at least x; this is the effect of a probability perception function $f$ such that $f(v) \le v$ for all $v \in [0, 1]$. We repeat the definition (1) of the index of pessimism and add a slightly modified representation that will prove useful for some purposes.

$$P_f = \inf_{0<v<1} \left[ \frac{1-f(v)}{f(v)} / \frac{1-v}{v} \right] = \inf_{0<v<1} \left[ \frac{1-f(v)}{1-v} / \frac{f(v)}{v} \right]. \quad (7)$$

Since for an identity-majorised probability perception function $f$, $1 - f(v) \ge 1 - v$ and $f(v) \le v$, the index of pessimism of such an $f$ satisfies $P_f \ge 1$. As mentioned in the Introduction, the index of pessimism can be interpreted intuitively as the least factor by which the DM exaggerates the odds ratio $(1 - v)/v$ against winning, to its perceived value $(1 - f(v))/f(v)$.

**Example 2.** The index of pessimism of a *kink*.

Let a *kink* function be the pointwise maximum of two increasing linear functions defined on $[0, 1]$: one with slope less than 1 going through $(0, 0)$, the other with slope exceeding 1 going through $(1, 1)$. Claim: if $f$ is a *kink*, then

$$P_f = \min(1/f'(0), f'(1)) > 1. \quad (8)$$

To see this, let $v_0$ be the point at which the two linear sections of the kink intersect. Observe that on $(0, v_0)$, $v/f(v)$ is constant and $(1 - f(v))/(1 - v)$ is increasing, while on $(v_0, 1)$, the former is decreasing and the latter is constant. Their product is thus unimodal, attaining local minimal values only at endpoints.

**Example 3.** The index of pessimism of a *hyperbolic perception function*.

For $P > 1$, the function

$$f_P(v) = \frac{v}{v + (1 - v)P} \quad (9)$$

will henceforth be called the *hyperbolic* perception function with index of pessimism $P$. To justify the name, express the hyperbolic function $f_P$ as $f_P(v) = [P/(P-1)]/[v + (1-v)P] - [1/(P-1)]$ to realise that $f_P$ is a non-linear, convex perception function with index of pessimism $P$. It is an iso-pessimistic perception function in the sense that the expression $[(1 - f_P(v))/(1 - v)]/[f_P(v)/v]$, whose infimum over $v$ generally defines the index of pessimism, is identically equal to the index of pessimism $P$ of $f_P$.

Some basic properties of the index of pessimism $P_f$:

**Lemma 2** (i) For perception functions $f$ and $g$, if $f(v) \le g(v)$ for all $v \in (0, 1)$, then $P_f \ge P_g$.

(ii) If $f$ is a non-linear, convex perception function, then $P_f > 1$.

The following property characterises strict pessimism.
(iii) For a perception function $f$, $P_f > 1$ if and only if $f$ is majorised by some non-linear, convex perception function. Furthermore, every perception function $f$ with $P_f > 1$ is majorised by the hyperbolic perception function with index of pessimism $P_f$ (see (9)).

(iv) Fix $P > 1$ and let $L_P(v) = v/P$ and $R_P(v) = 1 + (v-1)P$ be the linear support functions of the hyperbolic perception function $f_P$ at 0 and 1 respectively. Then $P_f = P$ for every perception function $f$ satisfying $L_P \le f \le f_P$ or $R_P \le f \le f_P$.

The following property gives an alternative definition of $P_f$.

(v) For a pessimistic perception function $f$, $P_f = Q$, where

$$ 
\begin{aligned}
Q &= \inf\{P > 1 | f(v) > f_P(v) \text{ for some } v \in (0, 1)\} \\
  &= \sup\{P \ge 1 | f(v) \le f_P(v) \text{ for all } v \in (0, 1)\}. \qquad (10)
\end{aligned}
$$

**Proof:**

Property (i) follows easily from the definition of $P_f$ (see (7)) as the infimum of $(v/(1-v)) * (1/f(v) - 1)$, that majorises the corresponding expression involving $g$.

Property (ii) follows from Property (i) by observing that every non-linear, convex perception function is separated from the identity function by some kink (see Example 2 and (8)).

One direction of the proof of Property (iii) is clear: If $f \le g$ and $g$ is a non-linear, convex perception function, then by Properties (i) and (ii), $P_f \ge P_g > 1$. As for the opposite direction, assume that $P_f > 1$. Let $g(v) = f_{P_f}(v) = v/(v + (1-v)P_f)$ be the hyperbolic perception function with index of pessimism $P_f$ and re-write the inequality

$$ P_f = \inf_w \left\{ \left[ \frac{(1-f(w))}{(1-w)} \right] / \left[ \frac{f(w)}{w} \right] \right\} \le \left[ \frac{(1-f(v))}{(1-v)} \right] / \left[ \frac{f(v)}{v} \right] \quad (11) $$

simply as $f \le g$.

Property (iv) is proved by first applying Property (i) to obtain that $P_f \ge P_{f_P} = P$, and then observing that $P_f = \inf_{v \in (0,1)} \{\dots\} \le \inf_{v \in I} \{\dots\}$ for every sub-interval $I \subset (0, 1)$. The inequality $P_f \le P$ follows by taking $I = (0, \epsilon)$ and majorising $f$ by $L_P$ on $I$, or by taking $I = (1-\epsilon, 1)$ and majorising $f$ by $R_P$ on $I$.

Proof of Property (v): To see that $P_f \ge Q$, observe that this is trivial if $Q = 1$ and concentrate on the case $Q > 1$. Take any $\epsilon \in (0, Q-1)$. By definition of $Q$, $f_{Q-\epsilon}$ majorises $f$. Hence, by Property (i), $P_f \ge Q - \epsilon$. Since this is true for all sufficiently small $\epsilon$, $P_f \ge Q$. To prove the opposite inequality, take any $P > Q$ and any $v \in (0, 1)$ at which $f(v) > f_P(v)$. Then,

$$ P_f = \inf_w \frac{1 - f(w)}{1 - w} / \frac{f(w)}{w} \le \frac{1 - f(v)}{1 - v} / \frac{f(v)}{v} < \frac{1 - f_P(v)}{1 - v} / \frac{f_P(v)}{v} \equiv P. \quad (12) $$

Since $P_f < P$ for all $P > Q$, the inequality $P_f \le Q$ follows.

**Example 4. The index of pessimism of power functions.**

As an application of Property (iv), we will sketch a proof that for $\alpha > 1$, the perception functions $f(v) = v^\alpha$ and $g(v) = 1 - (1-v)^{1/\alpha}$ have index of pessimism $\alpha$.

Plainly, $f$ is “sandwiched” between $R_\alpha$ and $f_\alpha$, while $g$ is sandwiched between $L_\alpha$ and $f_\alpha$. The details are skipped.
## 3.2 Index of optimism

A RDEU DM is optimistic under risk if the DM assesses lotteries as a EU DM who over-estimates for each $x \in \mathcal{R}$ the probability to have at least $x$; this is the effect of a probability perception function $f$ such that $f(v) \ge v$ for all $v \in [0, 1]$. For a probability perception function $f$, define

$$O_f = \inf_{0<v<1} \left[ \frac{f(v)}{1-f(v)} / \frac{v}{1-v} \right] = \inf_{0<v<1} \left[ \frac{f(v)}{v} / \frac{1-f(v)}{1-v} \right] \quad (13)$$

to be its index of optimism. Since for an identity-minorised perception function $f$,

$1 - f(v) \le 1 - v$ and $f(v) \ge v$, the index of optimism of such an $f$ satisfies $O_f \ge 1$. Parallel to Example 4, the perception functions $f(v) = v^{1/\alpha}$ and $g(v) = 1 - (1 - v)^\alpha$, that display optimism for $\alpha > 1$, have index of optimism $\alpha$.

Since the index of optimism can be analysed and intuitively understood along the same lines as the index of pessimism, further details are omitted.

## 3.3 Index of non-concavity (or greediness)

For a strictly increasing utility function $u$, define $G_u$ (see (2)) to be its index of greediness (or non-concavity). Since non-decreasing functions are differentiable almost everywhere, the ratio in (2) can be made arbitrarily close to 1 by concentrating all four points $x_i$ close to a point of differentiability of $u$. Hence, $G_u \ge 1$. The value 1 corresponds exclusively to concavity. If $u$ is differentiable, then the index of greediness can be expressed in the simpler form

$$G_u = \sup_{y<x} \frac{u'(x)}{u'(y)} \quad (14)$$

Moreover, if $u$ is convex, then $G_u = u'(\infty)/u'(-\infty)$.

In EU theory, a utility function $v$ is said to display (Arrow-Pratt) more risk aversion than a utility function $u$ if there is a concave, strictly increasing function $\eta$ such that for all $x \in \mathcal{R}$, $v(x) = \eta(u(x))$. In this definition, neither of the two utility functions is required to be concave. Since

$$\begin{align*}
\frac{v(x_4) - v(x_3)}{x_4 - x_3} / \frac{v(x_2) - v(x_1)}{x_2 - x_1} &= \frac{\eta(u(x_4)) - \eta(u(x_3))}{x_4 - x_3} / \frac{\eta(u(x_2)) - \eta(u(x_1))}{x_2 - x_1} \\
&= \left[ \frac{\eta(u(x_4)) - \eta(u(x_3))}{u(x_4) - u(x_3)} / \frac{\eta(u(x_2)) - \eta(u(x_1))}{u(x_2) - u(x_1)} \right] \left[ \frac{u(x_4) - u(x_3)}{x_4 - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1} \right] \\
&\le \frac{u(x_4) - u(x_3)}{x_4 - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1}, \tag{15}
\end{align*}$$

it follows directly from the definition (see (2)) of the index of greediness that

**Lemma 3** If the utility function $v$ displays Arrow-Pratt more risk aversion than the utility function $u$, then $G_v \le G_u$.

## 3.4 Index of non-convexity (or thriftiness)

For a strictly increasing utility function $u$, define

$$T_u = \sup_{x_1 < x_2 \le x_3 < x_4} \left[ \frac{u(x_2) - u(x_1)}{x_2 - x_1} / \frac{u(x_4) - u(x_3)}{x_4 - x_3} \right] \quad (16)$$
to be its index of *thriftiness* (or non-convexity). Since non-decreasing functions are differentiable almost everywhere, the ratio in (16) can be made arbitrarily close to 1 by concentrating all four points $x_i$ close to a point of differentiability of $u$. Hence, $T_u \ge 1$. The value 1 corresponds exclusively to convexity. If $u$ is differentiable, then the index can be expressed by

$$T_u = \sup_{y<x} \frac{u'(y)}{u'(x)}. \qquad (17)$$

Moreover, if $u$ is concave, then $T_u = u'(-\infty)/u'(+\infty)$.

In parallel to the decrease of greediness when the Arrow-Pratt index of risk aversion increases, if a utility function $v$ displays Arrow-Pratt more risk aversion than the utility function $u$, then $T_v \ge T_u$.

# 4 Characterisation of monotone risk aversion

Before stating the main theorem, we state a technical lemma needed for its proof. This lemma also clarifies a point left open in Example 1. Appendix 1 contains a proof of the Lemma.

**Lemma 4** Let $u: \mathcal{R} \to \mathcal{R}$ be continuous and strictly increasing and let $\lambda > 0$. Denote

$$E_{\lambda} = \{\bar{x} | \bar{x} = (x_1, x_2, x_3, x_4) \in \mathcal{R}^{\Delta}, \ \S_{\infty} < \S_{\infty} \le \S_{\infty} < \S_{\Delta}, \ \frac{\S_{\Delta} - \S_{\infty}}{\S_{\infty} - \S_{\infty}} = \lambda\} \quad (18)$$

$$G_u(\lambda) = \sup_{\bar{x} \in E_\lambda} \left[ \frac{u(x_4) - u(x_3)}{x_4 - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1} \right]. \quad (19)$$

$$\text{Then } G_u(\lambda) = G_u$$

**Theorem 5** A RDEU DM with probability perception function $f$ and utility function $u$ is monotone risk averse if and only if $P_f \ge G_u$.

Let us first notice that this theorem implies that for a monotone risk averse DM, $P_f \ge G_u \ge 1$, which implies that $f(v) \le v$. In other words, $f(v) \le v$ is a necessary condition for monotone risk aversion. This fact also follows from (i) weak risk aversion implies identity majorisation, and (ii) monotone risk aversion implies weak risk aversion.

# 5 Proof of Theorem

## 5.1 Necessity of the condition $P_f \ge G_u$

Let $v \in (0, 1)$ be given, and consider the setup introduced in Example 1. By Lemma 4, it is enough to prove that for every $(y_1, y_2, y_3, y_4) \in E_\lambda$, where $\lambda = (1 - v)/v$, a monotone risk averse DM characterised by $(u, f)$ verifies

$$\frac{u(y_4) - u(y_3)}{y_4 - y_3} / \frac{u(y_2) - u(y_1)}{y_2 - y_1} \le \frac{1 - f(v)}{1 - v} / \frac{f(v)}{v}. \quad (20)$$

Let $X$ be a random variable with values $y_2 < y_3$ and respective probabilities $1 - v$ and $v$.
The RDEU functional for a DM characterised by a pair $(u, f)$ is (see (5))

$$V(X) = u(y_2) + f(v)[u(y_3) - u(y_2)]. \qquad (21)$$

Let us apply a monotone mean-preserving spread to this distribution in the following way: Since $(y_1, y_2, y_3, y_4) \in E_\lambda$, there exists $\gamma > 0$ such that $y_1 = y_2 - \gamma v$ and $y_4 = y_3 + \gamma(1 - v)$. The random variable $Y$ taking values $y_1 < y_4$ with respective probabilities $1 - v$ and $v$ is then riskier than $X$ in the sense of a monotone increase in risk. Since

$$V(Y) = u(y_1) + f(v)[u(y_4) - u(y_1)], \qquad (22)$$

then $V(Y) - V(X) \le 0$ if and only if $(1 - f(v))(u(y_1) - u(y_2)) + f(v)(u(y_4) - u(y_3)) \le 0$. But $(y_4 - y_3)/(y_2 - y_1) = (1 - v)/v$, so the required inequality (20) follows.

## 5.2 Sufficiency of the condition $P_f \ge G_u$

We must compare $V(X)$ and $V(Y)$ for an arbitrary pair $(X, Y)$ of random variables for which $Y$ is a mean-preserving monotone increase in risk with respect to $X$. The proof will proceed in two steps: (i) sufficiency of the condition for random variables with finitely many values, and (ii) a continuity argument to extend sufficiency of the condition to the set $\mathbb{V}$, composed of bounded random variables in general. (i) As in Landsberger & Meilijson [15] and Chateauneuf, Cohen & Meilijson [9], let the result of an out-stretch $(v, w; a, b)$ (where $0 < v \le w < 1$ and $-\infty < a, b < \infty$) of a distribution function $F$ be the distribution $G$ obtained from $F$ by shifting to the right by an amount $b$ the section of $F$ above height $1 - v$ and by $a$ to the left the section of $F$ below height $1 - w$. (If $F$ is an income distribution, then the poorest $(1 - w)$-quantile of the population got poorer by the constant amount $a$ and the richest $v$-quantile got richer by the constant amount $b$). Clearly, if $vb = (1 - w)a$, then $(v, w; a, b)$ is a mean-preserving out-stretch. Notice that these out-stretches may entail splitting an atom in each side in two, one piece of which gets shifted with the corresponding ray and the other stays in place. It is intuitively true and easy to prove (see [15]) that (1) if $G$ is obtained from $F$ by a mean-preserving out-stretch then $G$ is a mean-preserving monotone increase in risk with respect to $F$, and (2) if $F$ and $G$ are supported by finite sets and $G$ is a mean-preserving monotone increase in risk with respect to $F$, then there is a finite sequence $F = F_1, F_2, \dots, F_k = G$ of distributions such that for each $1 \le i < k$, $F_{i+1}$ is a mean-preserving out-stretch of $F_i$. Due to these facts, it is enough to prove that $V(Y) \le V(X)$ whenever the distribution of $Y$ is a mean-preserving out-stretch of that of $X$. Let $(v, w; a, b)$ be this stretch and let

$$\mathcal{L}(X) = (x_1, 1 - v_1; x_2, v_1 - v_2; \dots; x_i, v_{i-1} - v_i; \dots; x_j, v_{j-1} - v_j; \\ \dots; x_{n-1}, v_{n-2} - v_{n-1}; x_n, v_{n-1}), \qquad (23)$$

be the distribution of $X$, with atoms $x_1 < x_2 < \dots < x_n$ and probabilities $P(X > x_k) = v_k$. Let $i < j$ be chosen so that $v_{i-1} \ge w \ge v_i \ge v_{j-1} \ge v \ge v_j$. Then $V(X) - V(Y)$ is

$$\begin{aligned} \Delta V = & [u(x_1) - u(x_1 - a)]\{1 - f(v_1)\} + [u(x_2) - u(x_2 - a)]\{f(v_1) - f(v_2)\} + \dots \\ & + [u(x_i) - u(x_i - a)]\{f(v_{i-1}) - f(w)\} - [u(x_{j-1} + b) - u(x_{j-1})]\{f(v) - f(v_{j-1})\} \\ & - [u(x_j + b) - u(x_j)]\{f(v_{j-1}) - f(v_j)\} - \dots - [u(x_n + b) - u(x_n)]\{f(v_{n-1})\}. \end{aligned} \qquad (24)$$

To prove non-negativity of (24), it is enough to prove its non-negativity when each term in square brackets with a + sign in front is replaced by the minimum of these terms, and
each term in square brackets with a – sign is replaced by the maximum of these. Let the minimum of $u(x_k) - u(x_k - a)$ over $k \le i$ be attained at $k = l$ and the maximum of $u(x_k + b) - u(x_k)$ over $k \ge j - 1$ be attained at $k = m$. Then, since $(1 - w)a = vb$,

$$
\begin{align*}
\Delta V &\ge [u(x_l) - u(x_l - a)]\{1 - f(w)\} - [u(x_m + b) - u(x_m)]\{f(v)\} \\
&= \left[\frac{u(x_l) - u(x_l - a)}{a}\right] \frac{1 - f(w)}{1 - w} (1 - w)a - \left[\frac{u(x_m + b) - u(x_m)}{b}\right] \frac{f(v)}{v}vb \\
&= \left\{\frac{u(x_l) - u(x_l - a)}{a} \frac{1 - f(w)}{1 - w} - \frac{u(x_m + b) - u(x_m)}{b} \frac{f(v)}{v}\right\}vb \\
&\ge \frac{u(x_l) - u(x_l - a)}{a} \frac{f(v)}{v} \left\{\left[\frac{1 - f(w)}{1 - w} \cdot \frac{f(v)}{v}\right] - G_u\right\}vb. \tag{25}
\end{align*}
$$

If in the last expression in the RHS of (25) we had $w = v$ (instead of $w \ge v$), then the proof would have been finished, since $[(1 - f(v))/(1 - v)]/[f(v)/v] \ge P_f$ by definition of $P_f$ (see (7)), and the pessimism $P_f$ displayed by $f$ (see (7)) exceeds the greediness $G_u$ (see (2)) displayed by $u$, by assumption. Since $P_f \ge G_u$ implies that $f$ is majorised by the identity function, it remains to show

**Lemma 6** *The index of pessimism of a pessimistic DM satisfies*

$$
P_f = \inf_{0 < v_2 \le v_1 < 1} \frac{\frac{1 - f(v_1)}{1 - v_1}}{\frac{f(v_2)}{v_2}}. \qquad (26)
$$

Proof of Lemma 6: It is clear that the RHS of (26) is less than or equal $P_f$, since the infimum over $v_2 \le v_1$ is less than or equal to the infimum over $v_2 = v_1$, that is $P_f$ by definition. To see that the infimum in the RHS of (26) is greater or equal than $P_f$, it is enough to show that for arbitrary $0 < v_2 < v_1 < 1$,

$$
\frac{1 - f(v_1)}{(1 - v_1)} / \frac{f(v_2)}{v_2} \ge \min \left[ \frac{1 - f(v_1)}{(1 - v_1)} / \frac{f(v_1)}{v_1} , \frac{1 - f(v_2)}{(1 - v_2)} / \frac{f(v_2)}{v_2} \right] . \quad (27)
$$

Otherwise, $f(v_2)/v_2 > f(v_1)/v_1$ and $(1-f(v_2))/(1-v_2) > (1-f(v_1))/(1-v_1)$. But
then $f(v_1) > 1-(1-f(v_2))(1-v_1)/(1-v_2) > 1-(1-f(v_1)v_2/v_1)(1-v_1)/(1-v_2)$, implying
that $f(v_1) > v_1$, a contradiction. This completes the proof of Lemma 6 and of part (i) of
the proof of sufficiency of the condition $P_f \ge G_u$, dealing with random variables supported
by a finite set. It remains to present a continuity argument that will extend sufficiency to
general bounded random variables.

(ii) Let us now consider two elements X and Y of V, such that Y is a monotone mean-preserving spread of X. We can assume without loss of generality that Y = X + Z with Z and X co-monotone and E(Z) = 0.

We use the standard uniform approximations of bounded random variables by non-
decreasing step functions, which preserve co-monotonicity.

Thus, let $X_n$ and $Z_n$ be such approximations of X and Z. Explicitly,

$$
X_n = \sum_{i=0}^{n2^n-1} \left[ \frac{i}{2^n} \mathbf{1}_{\left\{ \frac{i}{2^n} \le X < \frac{i+1}{2^n} \right\}} - \frac{i+1}{2^n} \mathbf{1}_{\left\{ -\frac{i+1}{2^n} \le X < -\frac{i}{2^n} \right\}} \right] \quad (28)
$$

where $\mathbf{1}_A$ denotes the characteristic function of the event A. A similar expression defines $Z_n$. $X_n$ and $Z_n$ are co-monotone, since they are non-decreasing functions of co-monotone random variables. (One of the equivalent definitions of co-monotonicity of X and Z is the
existence of a random variable U of which each of X and Z is a non-decreasing function. Obviously, so are $X_n$ and $Z_n$, automatically).

For $n$ large enough so that $X, Y$ and $Z$ are supported by the interval $(-n, n)$,

$$X - \frac{1}{2^n} \le X_n \le X \quad \text{and} \quad Z - \frac{1}{2^n} \le Z_n \le Z. \qquad (29)$$

Hence, with $Y_n = X_n + Z_n$,

$$Y - 1/2^{n-1} \le Y_n \le Y. \qquad (30)$$

The second statement in (29) and the fact that $E(Z) = 0$ entail $E(Z_n) \uparrow 0$. Define then $\alpha_n \downarrow 0$ by $E(Z_n + \alpha_n) = 0$ and set $Z'_n = Z_n + \alpha_n, Y'_n = X_n + Z'_n$.

Clearly, $X_n$ and $Y'_n$ are finitely supported random variables and $Y'_n$ has been obtained from $X_n$ by a monotone mean-preserving spread. It follows from part (i) of the proof that for each $n$,

$$V(X_n) \ge V(Y'_n). \qquad (31)$$

For a random variable $T$, $V(T)$ is the Choquet integral of $u(T)$ with respect to the capacity $c = f \circ P$, i.e., $V(T) = \int u(T)df \circ P$. Since the Choquet integral is monotone and co-monotonically additive (see [5, 12, 22, 23]), (31) will entail the desired result $V(X) \ge V(Y)$. For the sake of completeness, we present a direct argument.

$Y'_n \ge Y_n$ implies $u(Y'_n) \ge u(Y_n)$ and, a-fortiori, $V(Y'_n) \ge V(Y_n)$. By (31),

$$V(X_n) \ge V(Y_n), \forall n \in N. \qquad (32)$$

Since the utility function $u$ is continuous, it is uniformly continuous on any compact subset of $\mathcal{R}$. Hence, from (29) and (30), $\forall \epsilon \ni N(\epsilon)$ s.t. $n \ge N(\epsilon)$ implies that

$$u(X) - \epsilon \le u(X_n) \le u(X), \quad u(Y) - \epsilon \le u(Y_n) \le u(Y). \qquad (33)$$

Monotonicity, co-monotonic additivity and the normalisation property $\int 1_\Omega df \circ P = 1$ lead to the two inequalities: $\forall n \ge N(\epsilon)$,

$$V(X) - \epsilon \le V(X_n) \le V(X), \quad V(Y) - \epsilon \le V(Y_n) \le V(Y). \qquad (34)$$

Therefore, $\lim_{n \to \infty} V(X_n) = V(X)$ and $\lim_{n \to \infty} V(Y_n) = V(Y)$. Hence, by (32), $V(X) \ge V(Y)$.

**Remark 1** Quiggin (see [18]) uses the concept of monotone risk aversion only for DM with concave $u$. The theorem above proves that concavity of $u$ is not a necessary condition for monotone risk aversion. Similarly, convexity is not necessary for monotone risk-seeking attitudes. The following theorem is presented as a corollary since its proof is analogous to that of Theorem 5.

**Corollary 7** A RDEU DM with probability perception function $f$ and utility function $u$ is monotone risk seeking if and only if the DM's index of optimism exceeds the DM's index of non-convexity: $O_f \ge T_u$.
# 6 Examples and discussion

## 6.1 Risk-seeking attitude with diminishing marginal utility

Let a RDEU DM's choices among lotteries be characterised by the concave utility function $u(x) = -ae^{-bx}$ and the identity-minorised perception function $f(v) = 1 - (1-v)^h$, with $a > 0$, $b > 0$ and $h > e^b$. As is easy to ascertain, $O_f = h$ and $T_u = e^b$. Hence, in spite of the concavity of the utility function, the DM is monotone risk seeking.

## 6.2 Different behaviours with the same utility function

Consider a RDEU DM with utility function $u(x) = -ae^{-bx}$ as above and probability perception function $f(v) = v^k$, with $k \ge e^b$. Since $P_f = k$ and $G_u = e^b$, the DM is monotone risk averse.

## 6.3 Comparison with strong risk aversion

Chew, Karni and Safra [10] proved that strong risk aversion (aversion to any mean preserving increase in risk) is characterised by concavity of *u* and convexity of *f*. This shows an essential difference between the Quiggin - Bickel & Lehmann and Rothschild & Stiglitz models of increase in risk. For the latter there is no degree of pessimism that can compensate for departures from concavity of *u* and still make the DM averse to risk. The main theorem of this paper showed that for the stronger co-monotone notion of risk, such a compensation exists and is completely separable in the sense that no joint condition in terms of both *u* and *f* are needed. Rather, the compensation is based on a comparison of intrinsic indices of pure type - one for greediness displayed by *u* and the other for pessimism displayed by *f*.

# 7 Appendix 1: Proof of Lemma 4

Obviously, $G_u(\lambda) \le G_u$. We have to prove the opposite inequality $G_u(\lambda) \ge G_u$, i.e., for any $(x_1, x_2, x_3, x_4) \in \mathbb{R}^4$ such that $x_1 < x_2 \le x_3 < x_4$ and any $\varepsilon > 0$, there exists a $(y_1, y_2, y_3, y_4) \in E_\lambda$ such that

$$ \frac{u(y_4) - u(y_3)}{y_4 - y_3} / \frac{u(y_2) - u(y_1)}{y_2 - y_1} \ge \frac{u(x_4) - u(x_3)}{x_4 - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1} - \varepsilon. \quad (35) $$

By continuity of $u$, there exists some $x_0 \in (x_3, x_4)$ such that for every $x \in (x_0, x_4)$,

$$ \frac{u(x) - u(x_3)}{x - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1} \ge \frac{u(x_4) - u(x_3)}{x_4 - x_3} / \frac{u(x_2) - u(x_1)}{x_2 - x_1} - \varepsilon. \quad (36) $$

Divide the interval $(x_1, x_2)$ into $K$ sub-intervals of equal length $\Delta = (x_2 - x_1)/K$ such that $\lambda\Delta < x_4 - x_0$. This guarantees that the sequence $x_3, x_3 + \lambda\Delta, x_3 + 2\lambda\Delta, x_3 + 3\lambda\Delta, \dots$ has some element $x_3 + k\lambda\Delta$ (call it $x$) in the interval $(x_0, x_4)$. Since $(u(x) - u(x_3))/(x - x_3) = (1/k) \sum_{i=0}^{k-1}[u(x_3 + (i+1)\lambda\Delta) - u(x_3 + i\lambda\Delta)]/\lambda\Delta$, there is a sub-interval $(y_3, y_4) = (x_3 + i\lambda\Delta, x_3 + (i+1)\lambda\Delta)$ of $(x_3, x_4)$ such that $(u(y_4) - u(y_3))/(y_4 - y_3) \ge (u(x) - u(x_3))/(x - x_3)$. Similarly, there exists a sub-interval $(y_1, y_2) = (x_1 + j\Delta, x_1 + (j+1)\Delta)$ of $(x_1, x_2)$ along which $(u(y_2) - u(y_1))/(y_2 - y_1) \le (u(x_2) - u(x_1))/(x_2 - x_1)$. This completes the proof.

**Acknowledgment.** We are grateful to Peter Wakker for helpful suggestions and comments.
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