# Local Utility and Multivariate Risk Aversion

aa r X i v : . [ ec on . T H ] F e b LOCAL UTILITY AND MULTIVARIATE RISK AVERSION

ARTHUR CHARPENTIER , ALFRED GALICHON , AND MARC HENRY Abstract.

We revisit Machina’s local utility as a tool to analyze attitudes to multivariate risks.We show that for non-expected utility maximizers choosing between multivariate prospects, aver-sion to multivariate mean preserving increases in risk is equivalent to the concavity of the localutility functions, thereby generalizing Machina’s result in [22]. To analyze comparative risk atti-tudes within the multivariate extension of rank dependent expected utility of Galichon and Henry[14], we extend Quiggin’s monotone mean and utility preserving increases in risk and show thatthe useful characterization given in Landsberger and Meilijson [21] still holds in the multivariatecase.

Keywords: local utility, multivariate risk aversion, multivariate rank dependent utility, pes-simism, multivariate Bickel-Lehmann dispersion.

MSC class : 00A06

JEL subject classiﬁcation Arthur Charpentier, UQAM & Univerist´e de Rennes 1, [email protected] Alfred Galichon, Sciences-Po, Paris, [email protected] Marc Henry, The Pennsylvania State University, [email protected]

Introduction

One of the many appealing features of expected utility theory is the characterization of attitudestowards risk through the shape of the utility function. Following extensive evidence of violationsof the independence axiom which delivers linearity in probabilities of the functional characterizingpreferences over risky prospects, most notably the celebrated Allais paradox [1], Machina showed in[22], [23] that smoothness of the preference functional was suﬃcient to recover representability of riskattitudes through a local approximation, which he called local utility function . Parallel to the studyof risk attitudes in generalized expected utility theories, Stiglitz [34] and Kihlstrom and Mirman[20] analyzed attitudes to the combination of income risk and price risk in preferences over multiplecommodities within the expected utility framework. This paper is concerned with non expectedutility analysis of attitudes to multivariate risks. So far, three approaches have emerged to analyzeattitudes to multivariate risks without the independence axiom in [38], [29] and [16]. All three applydimension reduction devices to preferences over multivariate prospects. Yaari [38] considers rank de-pendent utility over multivariate prospects with stochastically independent components only; Safraand Segal [29] show additive separability of the local utility function under a property they call dominance (equivalent to the notion of correlation neutrality in [13]) and Grant, Kajii and Polak[16] show that under a property they call degenerate independence , preferences over uncertain mul-tivariate prospects can be fully recovered from preferences over uncertain income and preferencesover deterministic multivariate outcomes. We consider the general case, where attitudes to incomerisk and price risk cannot be separated in this way and show that in general smooth preferences overmultivariate prospects, the main result of Machina [22] still holds, and aversion to increases in risk isequivalent to concavity of the local utility function. The proof relies on the martingale characteriza-tion of increasing risk in Galichon and Henry [14]. A special case of this result appears in Galichonand Henry [14], who derive the family of local utility functions in a multivariate rank dependentutility model under aversion to multivariate mean preserving increases in risk. Machina also showedin [22] that interpersonal comparisons of risk aversion can be characterized by properties of the localutility function. Karni generalizes in [19] the equivalence between decreasing certainty equivalentsand concave transformations of the local utility functions to smooth preferences over multivariateprospects. To complement this result, we extend the notion of compensated spread to multivariateprospects and generalize the characterization of Quiggin’s monotone increases in risk [26] as meanpreserving comonotonic spreads in [21]. This allows us to recover a multivariate version of Lands-berger and Meilijson’s seminal result on the eﬃciency of partial insurance contracts for monotonemean preserving reductions in risk in [21]. We also generalize Quiggin’s notion of pessimism andcharacterize pessimistic decision functionals by the shape of their local utility function. We apply

OCAL UTILITY AND MULTIVARIATE RISK AVERSION 3 these notions to interpersonal comparison of risk aversion within the multivariate rank dependentmodel of [14] and we show that pessimism is equivalent to weak risk aversion in that framework.The rest of the paper is organized as follows. Section 1 deﬁnes local utility. Section 2 shows thataversion to mean preserving increases in risk is equivalent to concavity of the local utility functionsand Section 3 extends Quiggin’s monotone mean preserving increases in risk and Section 4 applies itto interpersonal comparisons of risk aversion within the multivariate rank dependent utility model.The last section concludes.

Notation and basic deﬁnitions.

Let ( S, F , P ) be a non-atomic probability space. Let X : S → R d be a random vector. We denote the cumulative distribution function of X by F X . E is the expectationoperator with respect to P . For x and y in R d , let x · y be the standard scalar product of x and y ,and k x k the Euclidian norm of x . We denote by X = d µ the fact that the distribution of X is µ andby X = d Y the fact that X and Y have the same distribution. Q X denotes the quantile function ofdistribution X . In dimension 1, this is deﬁned for all t ∈ [0 ,

1] by Q X ( t ) = inf x ∈ R { Pr( X ≤ x ) > t } .In larger dimensions, it is deﬁned in Deﬁnition 5 of Section 3.2.1 below. We call L d the set ofintegrable random vectors of dimension d and L d the set of random vectors X of dimension d suchthat E k X k < ∞ . We denote by D the subset of L d containing random vectors with a densityrelative to Lebesgue measure. A functional Φ on L d is called upper semi-continuous (denoted u.s.c.)if for any real number α , { X ∈ L d : Φ( X ) < α } is open. A functional Φ is lower semi-continuous(l.s.c.) if − Φ is upper semi-continuous. Φ is called law-invariant if Φ( X ) = Φ( ˜ X ) whenever ˜ X = d X .By a slight abuse of notation, when Φ is law invariant, Φ( F X ) will be used to denote Φ( X ). Fora convex lower semi-continuous function V : R d → R , we denote by ∇ V its gradient (equal to thevector of partial derivatives). 1. Local Utility

We consider decision makers choosing among distributions functions on R d with ﬁnite mean. Weassume that the decision makers’ preferences are given as a complete, reﬂexive and transitive binaryrelation represented by a real valued functional Φ, which is continuous relative to the topology ofconvergence in distribution. Suppose further that Φ is Gˆateaux diﬀerentiable. Assumption 1 (Local Utility) . The following properties hold. (1) Φ is continuous with respect the topology of weak convergence of probability measures.

CHARPENTIER, GALICHON, AND HENRY (2)

For each distribution function F on R d , there is a function x U Φ ( x, F ) such that, foreach distribution function G , ddt Φ [(1 − t ) F + tG ] (cid:12)(cid:12) + = Z U Φ ( x, F ) [ dG ( x ) − dF ( x )] . The function U Φ ( x ; F ) thus deﬁned is called local utility function relative to Φ at F . Since expected utility preferences are linear in probabilities, the local utility of an expected utilitydecision maker is constant and equal to her utility function. Theorem 1 in [22] and its extensionto Gˆateaux diﬀerentiability in [8] for the special case of Rank Dependent Utility, show that smoothpreference functionals are monotonic if and only if their local utility functions are increasing. Thiscan be extended to the case of multivariate prospects.

Deﬁnition 1 (Stochastic dominance) . A distribution F is said to dominate stochastically a distri-bution G (denoted F % SD G ) if there exist ˜ X = d F and ˜ Y = d G such that ˜ X ≥ ˜ Y almost surely,where ≥ denotes componentwise order in R d . A preference functional is said to preserve stochastic dominance if stochastically dominant prospectsare always preferred. If the preference functional Φ is law invariant and monotonic, in the sensethat Φ( X ) ≥ Φ( Y ) when X yields larger outcomes than Y in almost all states, then it preservesstochastic dominance. The proof of Theorem 1 of [22] is dimension free and therefore, a Fr´echetdiﬀerentiable preference functional preserves stochastic dominance if and only if the utility functionis non decreasing. The proposed extension to Gˆateaux diﬀerentiable functionals in [8] is speciﬁc toRank Dependent Utility, however.If in addition, the decision maker is indiﬀerent to correlation increasing transfers, or correlationneutral according to the terminology of [13], then Safra and Segal show in [29] that the local utilityfunctions are additively separable, namely that U Φ ( x ; F ) = P dj =1 U j ( x j ; F ), where x j is the j -thcomponent of the outcome x ∈ R d . Yaari’s rank dependent utility maximizers over stochasticallyindependent d -dimensional risks in [38] are represented by(1.1) Φ( X ) = d X i =1 α i Z φ i ( u ) Q X i ( t ) dt, where Q X i is the quantile function of component X i of the risk X , the φ i ’s, i = 1 , . . . , d , are non-negative functions on [0 ,

1] (quantile weights interpreted as probability distortions) and the α i ’s, i = 1 , . . . , d , are positive weights. The local utility of decision maker Φ is given by(1.2) U Φ ( x ; F ) = d X i =1 α i Z x i φ i ( F i ( z )) dz, where F i is the i -th marginal of distribution F (see for instance Section 4 of [31]). OCAL UTILITY AND MULTIVARIATE RISK AVERSION 5 Local utility and mean preserving increases in risk

We now show that attitude to risk with smooth preference over multivariate prospects can becharacterized by the shape of local utilities, as was proved in the case of univariate risks in Theorem 2of [22]. The latter shows that aversion to mean preserving increases in risk is equivalent to concavityof local utility functions. Extending this result to preferences over multivariate prospects calls for ageneralization of the notion of mean preserving increase in risk proposed in [28].

Deﬁnition 2 (Mean preserving increase in risk) . A distribution G is called a mean preservingincrease in risk (hereafter MPIR) of a distribution F , denoted G % MP IR F , if either of the followingequivalent statements hold. (a) For all concave functions f on R d , R f dF ≥ R f dG . (b) There exists Y = d G and X = d F such that ( X, Y ) is a martingale, i.e., E [ Y | X ] = X . The equivalence between (a) and (b), Theorem 7.A.1 in [33], is due to [35] and the interpretationas an increase in risk is the same as in [28] for the univariate case. When the domain is restrictedto D , [14] show that (a) and (b) are also equivalent to (c): For all u.s.c. law invariant concavefunctionals Ψ on D and any X = d F and Y = d G , Ψ( X ) ≥ Ψ( Y ). An immediate corollary of thelatter is that cardinal risk aversion, i.e., concavity of the functional representing preferences, impliesordinal risk aversion, in the sense of aversion to mean preserving increases in risk. We can now statethe main result of this section, which is a direct generalization of Theorems 2 and 3 of [22]. Theorem 1 (Risk aversion and local utility) . Let Φ be a preference functional satisfying Assump-tion 1. Then the following statements are equivalent. (i) Φ is risk averse, i.e., Φ( F ) ≥ Φ( G ) when G is an MPIR of F , (ii) U Φ ( · ; F ) is a concave function for all F and (iii) For arbitrary prospects F and F ∗ and any α ∈ [0 , , Φ( αF + (1 − α ) G µ F ∗ ) ≥ Φ( αF + (1 − α ) F ∗ ) , where µ F is the mean of F and G µ is the degenerate distribution at µ .Proof of Theorem 1. (i) ⇔ (iii): Using the martingale diﬀerence characterization MPIR, it is easyto show that αF + (1 − α ) F ∗ is a mean preserving increase in risk relative to αF + (1 − α ) G µ F ∗ , sothat monotonicity with respect to MPIR implies (iii).The converse is proved in the following way. A probability measure Q on R d is called an elementaryfusion of a probability measure P (in the terminology of [12]) if there is a set A and β ∈ [0 ,

1] suchthat Q = P | A c + βP | A + (1 − β ) P ( A ) δ µ A , where µ A is the mean of P | A . A probability measure P n is called a simple fusion of a probabilitymeasure P if P n can be obtained from P as the result of a sequence of n elementary fusions. CHARPENTIER, GALICHON, AND HENRY

We ﬁrst show that under Condition (iii), Φ( Q ) ≥ Φ( P ) whenever Q is an elementary fusion of P .Take P a probability measure, β ∈ [0 ,

1] and a set A and ﬁnd α, Q and Q ∗ with mean µ ∗ such that P = αQ + (1 − α ) Q ∗ and P | A c + βP | A + (1 − β ) P ( A ) δ µ A = αQ + (1 − α ) δ µ ∗ . This implies αQ = P | A c + βP | A and (1 − α ) δ µ ∗ = (1 − β ) P ( A ) δ µ A . Hence, α = 1 − (1 − β ) P ( A ) ,Q = 11 − (1 − β ) P ( A ) (cid:2) P | A c + βP | A (cid:3) ,Q ∗ = 1(1 − β ) P ( A ) (cid:2) P − P | A c − βP | A (cid:3) . There remains to check that Q ∗ is centered at µ A . Indeed, the mean of Q ∗ is1(1 − β ) P ( A ) [ P ( A ) µ A + P ( A c ) µ A c − P ( A c ) µ A c − βP ( A ) µ A ] = µ A . By (iii), we know that Φ( αQ + (1 − α ) Q ∗ ) ≤ Φ( αQ + (1 − α ) δ µ ∗ ) hence we have Φ( P ) ≤ Φ( Q ) forany Q elementary fusion of any P with ﬁnite mean.By the continuity of Φ from Assumption 1 (1), this implies that Φ( Q ) ≥ Φ( P ), whenever Q is thelimit of a sequence of simple fusions of P . By the equivalence between (i) and (ii) in Theorem 4.1,page 47 of [12], Φ( Q ) ≥ Φ( P ) whenever P is an MPIR of Q . The implication (iii) = ⇒ (i) follows.(ii) = ⇒ (iii): Write F Y = F and F X = F ∗ . Consider the following two lotteries, so that Z = d αF + (1 − α ) F ∗ and ˜ Z = d αF + (1 − α ) G µ F ∗ ,(2.1) Z α րց − α FF ∗ and ˜ X α րց − α FG µ F ∗ Given ε ∈ [0 , Z ε a mixture between Z and ˜ Z , with weights ε and 1 − ε , and let Z ε = d F ε , F ε = αF + (1 − α )[(1 − ε ) G µ F ∗ + εF ∗ ] . For h ≥

0, note that(2.2) F ε + h = F ε + (1 − α ) h [ F ∗ − G µ F ∗ ] . If we substitute G µ F ∗ = 1(1 − α )(1 − ε ) [ F ε − αF − (1 − α ) εF ∗ ] OCAL UTILITY AND MULTIVARIATE RISK AVERSION 7 in equation (2.2), we get F ε + h = (cid:20) − h − ε (cid:21) F ε + h − ε [ αF + (1 − α ) F ∗ ] , so that Φ( F ε + h ) − Φ( F ε ) = Φ (cid:18)(cid:20) − h − ε (cid:21) F ε + h − ε [ αF + (1 − α ) F ∗ ] (cid:19) − Φ ( F ε ) . Let H = [ αF + (1 − α ) F ∗ ], so that this expression becomesΦ((1 − η ) F ε + ηH ) − Φ ( F ε )with η = h/ (1 − ε ). Now,Φ((1 − η ) F ε + ηH ) − Φ ( F ε ) = Z U Φ ( x ; F ε ) d [(1 − η ) F ε + ηH − F ε ] + o ( h )= η Z U Φ ( x ; F ε ) d [[ αF + (1 − α ) F ∗ ] − F ε ] + o ( h )which equals h − ε (cid:20)Z U Φ ( x ; F ε ) d [ αF + (1 − α ) F ∗ ] − Z U Φ ( x ; F ε ) dF ε (cid:21) + o ( h ) . Since ddε Φ( F ε ) = lim h → Φ( F ε + h ) − Φ( F ε ) h , using F ε = αF + (1 − α )[ εF ∗ + (1 − ε ) G µ F ∗ ], we ﬁnd ddε Φ( F ε ) = lim h → h (cid:20) h (1 − α ) (cid:18)Z U Φ ( x ; F ε ) dF ∗ − U Φ ( µ F ∗ ; F ε ) (cid:19)(cid:21) i.e., ddε Φ( F ε ) = (1 − α ) (cid:20)Z U Φ ( x ; F ε ) dF ∗ − U Φ ( µ F ∗ ; F ε ) (cid:21) ≤ U Φ ( · , F ε ) is a concave function. Hence, we obtain that Φ( F ) ≥ Φ( F ),i.e., Φ( αF + (1 − α ) F ∗ ) ≥ Φ( αF + (1 − α ) G µ ∗ F ) . (iii) = ⇒ (ii): By (iii), we have Φ( αF +(1 − α ) G µ F ∗ ) ≥ Φ( αF +(1 − α ) F ∗ ), which yields, by Gˆateauxdiﬀerentiability, R U Φ ( x, F ) dG µ F ∗ ( x ) ≥ R U Φ ( x, F ) dF ∗ ( x ). Hence, U Φ ( µ, F ) ≥ R U Φ ( x, F ) dF ∗ ( x ),which implies concavity of U Φ ( · , F ), as required. (cid:3) Using the local utility, we extend insights from the vast literature on multivariate risk taking (seefor instance [10] and references therein) to non expected utility preference functions. We can alsodeﬁne a full insurance premium for preferences over multivariate prospects. Let F be a prospectevaluated by a decision maker with smooth preferences as Φ( F ). A full insurance premium can be CHARPENTIER, GALICHON, AND HENRY deﬁned as an element of the set of vectors π ∈ R d satisfying Φ( F ) = U Φ ( µ − π ; F ), where µ is themean of F . 3. Multivariate mean preserving increases in risk

In Section 3.1, we consider univariate risks ( d = 1) and characterize local utility of decision makersthat are averse to Quiggin’s monotone mean preserving increases in risk, using a celebrated result ofLandsberger and Meilijson [21]. In Section 3.2, we extend the latter to the multivariate case, orderto provide the multivariate equivalent of aversion to monotone mean preserving increases in risk andits local utility characterization.3.1. Aversion to monotone mean preserving increases in risk.

In [26], Quiggin shows thatthe notion of mean preserving increases in risk is too weak to coherently order rank dependent utilitymaximizers according to increasing risk aversion. Quiggin [26] shows that the notion of monotonemean preserving increases in risk (Monotone MPIR) is the weakest stochastic ordering that achievesa coherent ranking of risk aversion in the rank dependent utility framework. Monotone MPIR is themean preserving version of Bickel-Lehmann dispersion ([3],[4]), which we now deﬁne.

Deﬁnition 3 (Bickel-Lehmann Dispersion) . Let Q X and Q Y be the quantile functions of the randomvariables X and Y . X is said to be Bickel-Lehmann less dispersed , denoted X - BL Y , if Q Y ( u ) − Q X ( u ) is a nondecreasing function of u on (0 , . The mean preserving version is called monotonemean preserving increase in risk (hereafter MMPIR) and denoted - MMP IR . MMPIR is a stronger ordering than MPIR in the sense that X - MMP IR Y implies X - MP IR Y since it is shown in [9] that an MPIR can be obtained as the limit of a sequence of simple meanpreserving spreads Y of X , deﬁned by Q Y ( u ) − Q X ( u ) non-positive below some u ∈ [0 ,

1] andnon-negative above u . [26] relates MMPIR aversion of a rank dependent utility decision maker toa notion he calls pessimism . Aversion to MMPIR is deﬁned in the usual way as follows. Deﬁnition 4 (Risk aversion) . A preference functional Φ over random prospects is called averse tomonotone mean preserving increases in risk if and only if X - MMP IR Y implies Φ( X ) ≥ Φ( Y ) . Consider a decision maker with preference relation characterized by the functional deﬁned foreach prospect distribution F by Φ( F ) = Z f (1 − F ( x )) dx (3.1)with f (0) = 0, f (1) = 1 and f non decreasing. Then Theorem 3 of [7] shows that aversion toMMPIR is equivalent to f ( u ) ≤ u for each u ∈ [0 , OCAL UTILITY AND MULTIVARIATE RISK AVERSION 9 is x U Φ ( x, F ) = R x f ′ (1 − F ( z )) dz , aversion to MMPIR can be characterized with the localutility. We now generalize this local utility characterization of MMPIR aversion beyond rank de-pendent utility functionals to all preference functionals that admit a local utility. For the purposeof this characterization, we strengthen the diﬀerentiability requirement of Assumption 1 to Fr´echetdiﬀerentiability with smooth local utility. Assumption 2 (Smooth local utility) . For each distribution function F on R , there exists a diﬀer-entiable function x U Φ ( x, F ) , such that for all distribution G , Φ( G ) − Φ( F ) = Z U Φ ( x, F )[ dG ( x ) − dF ( x )] + o ( d ( F, G )) , where d is the 1-Wasserstein distance d ( F, G ) := inf { E | X − Y | ; X = d F, Y = d G } , which metrizes the topology of convergence in distribution (see Theorem 6.9 of [37] ). Theorem 2 (Local utility of MMPIR averse decision makers) . Let Φ be a preference functional on L distributions satisfying Assumption 2. Φ is MMPIR averse if and only if Z U ′ Φ ( x, F ) δ ( x ) dF ( x ) ≤ , for all F and all non decreasing functions δ , such that R δdF = 0 and R | δ | dF < ∞ . Remark 1.

Note that δ can be chosen equal to y δ ( y ) = 1 { y > x } − [1 − F X ( x )] for any x ∈ R and in the special case of rank dependent utility functional (3.1), the characterization aboveis equivalent to f (1 − F X ( x )) ≤ − F X ( x ) for all x and X , which is equivalent to f ( u ) ≤ u for all u ∈ [0 , as mentioned previously. In Proposition 2 of [21], Landsberger and Meilijson give a characterization of Bickel-Lehmanndispersion in the spirit of the characterization of MPIR given in the equivalence between (a) and (b)of Proposition 2. In the latter, MPIR increases are characterized by the addition of noise, whereasin the former MMPIR are characterized by the addition of a zero mean comonotonic variable.

Proposition 1 (Landsberger-Meilijson) . A random variable X has Bickel-Lehmann less disperseddistribution than a random variable Y if and only iﬀ there exists Z comonotonic with X such that Y = d X + Z . Using Proposition 1, we can prove Theorem 2.

Proof of Theorem 2.

From Proposition 1, Φ is MMPIR averse if and only if Φ( X + Z ) − Φ( X ) ≤ X, Z ) comonotonic and E Z = 0. Take, therefore, X and Z two comonotonic random variables, with Z in L and centered. Call F the distribution of X and F ε the distribution of X + εZ , for ε >

0. Note that εZ and X are also comonotonic. By Assumption 2, since E | X + εZ − X | = O ( ε ),Φ( F ε ) − Φ( F ) = Z U Φ ( x, F )[ dF ε ( x ) − dF ( x )] + o ( ε )= Z U Φ ( Q X + εZ ( u )) − U Φ ( Q X ( u )] du + o ( ε )= Z U Φ ( Q X + Q εZ ( u )) − U Φ ( Q X ( u )] du + o ( ε )= ε Z U ′ Φ ( Q X ( u ) , F ) Q Z ( u ) du + o ( ε )where the penultimate equation holds because the quantile function is comonotonic additive and thelast equation holds because Z is integrable. Therefore Z U ′ Φ ( Q X ( u ) , F ) Q Z ( u ) du ≤ Z with mean zero is equivalent. After changing variables, we obtain the desiredcharacterization. Conversely, let X and Z be comonotonic. For each n ∈ N and each i = 1 , . . . , n , X + i − n Z and X + in Z are comonotonic. Calling F t the distribution of X + tZ , for any t ∈ [0 , F ) − Φ( F ) = n X i =1 n (cid:26)Z U ′ Φ ( Q X + i − n Z ( u ) , F i − n ) Q Z ( u ) du (cid:27) + o (cid:18) n (cid:19) . The terms in brackets are non positive, hence, letting n → ∞ , we have Φ( F t ) − Φ( F ) ≤

0, whichyields the result by Proposition 1. (cid:3)

We now show how this notion of Bickel-Lehmann dispersion and the Landsberger-Meilijson char-acterization can be extended to multivariate prospects and how it can be applied to the rankingof risk aversion of multivariate rank dependent utility maximizers. To that end, we appeal to themultivariate notions of quantiles and comonotonicity developed in [14], [11] and [25].3.2.

Local utility and multivariate mean preserving increases in risk.

Multivariate quantiles and comonotonicity.

Ekeland, Galichon and Henry [14], [11] deﬁnemultivariate quantiles by extending the variational characterization of univariate quantiles based onrearrangement inequalities of Hardy, Littlewood and P´olya [17]. The following well known equality(3.2) Z Q X ( u ) u du = max n E [ X ˜ U ] : ˜ U uniformly distributed on [0 , o , is extended to the multivariate case in the following way. Let µ is a reference absolutely continuousdistribution on R d with ﬁnite second moment. This could be, for instance, the uniform distributionon the unit hypercube in R d . Let X be a random vector in D . The quantile Q X of X is deﬁned as OCAL UTILITY AND MULTIVARIATE RISK AVERSION 11 the version of X (i.e., random vector with the same distriution as X ), which maximizes correlationwith a random vector U = d µ :(3.3) E [ Q X ( U ) · U ] = max n E [ X · ˜ U ] : ˜ U = d µ o . It follows from the theory of optimal transportation (see Theorem 2.12(ii), p. 66 of [36]) that thereexists an essentially unique convex lower semi-continuous function V : R d → R such that Q X = ∇ V satisﬁes Equation 3.3. Hence the deﬁnition of multivariate quantiles due to [14] and [11]. Deﬁnition 5 ( µ -quantile) . The µ -quantile function of a random vector X in D with respect to anabsolutely continuous distribution µ on R d is deﬁned by Q X in Equation (3.3). This concept of a multivariate quantile is the counterpart of the deﬁnition of multivariate comono-tonicity in [14] and [11], motivated by the fact that two univariate prospects X and Y are comono-tonic if there is a prospect U and non-decreasing maps T X and T Y such that Y = T Y ( U ) and X = T X ( U ) almost surely or, equivalently, E [ U X ] = max n E [ ˜ U X ] : ˜ U = d U o and E [ U Y ] =max n E [ ˜ U Y ] : ˜ U = d U o . Deﬁnition 6 ( µ -comonotonicity) . Random vectors X and Y in D are called µ -comonotonic if thereexists U = d µ such that E [ X · U ] = max n E [ ˜ X · U ] : ˜ X = d X o and E [ Y · U ] = max n E [ ˜ Y · U ] : ˜ Y = d Y o . Two random vectors are µ -comonotonic if they can be rearranged simultaneously so that theyare both equal to their µ -quantile. Another variational notion of multivariate comonotonicity, called c -comonotonicity, is proposed in Puccetti and Scarsini [25]. Deﬁnition 7 ( c -comonotonicity) . Random vectors X and Y in D are called c -comonotonic if thereexists a convex function V such that Y = ∇ V ( X ) . Both µ -comonotonicity and c -comonotonicity will feature in the extension of Bickel-Lehmanndispersion in the following section.3.2.2. Multivariate mean preserving increases in risk.

The Bickel-Lehmann dispersion order and itsmean-preserving version in [26], monotone MPIR, rely on the notion of monotone single crossings,hence on the monotonicity of the function Q Y − Q X . A natural extension of the class of non-decreasing functions to functions on R d is the class of gradients of convex functions, whose deﬁnitiondoesn’t rely on the ordering on the real line. Hence the following deﬁnition of µ -Bickel-Lehmanndispersion, which depends on the baseline distribution µ relative to which multivariate quantiles aredeﬁned. Deﬁnition 8 ( µ -Bickel-Lehmann dispersion) . A random vector X ∈ D is called µ -Bickel-Lehmannless dispersed than a random vector Y ∈ D , denoted X - µBL Y , if there exists a convex function V : R d → R such that the µ -quantiles Q X and Q Y of X and Y satisfy Q Y ( u ) − Q X ( u ) = ∇ V ( u ) for µ -almost all u ∈ [0 , d . As deﬁned above, µ -Bickel-Lehmann dispersion deﬁnes a transitive binary relation, and thereforean order on D . Indeed, if X - µBL Y and Y - µBL Z , then Q Y ( u ) − Q X ( u ) = ∇ V ( u ) and Q Z ( u ) − Q Y ( u ) = ∇ W ( u ). Therefore, Q Z ( u ) − Q X ( u ) = ∇ ( V ( u ) + W ( u )) so that X - µBL Z . When d = 1,this deﬁnition simpliﬁes to Deﬁnition 3.3.2.3. Characterization.

We have the following generalization of the Landsberger-Meilijson charac-terization of Proposition 1.

Theorem 3.

A random vector X ∈ D is µ -Bickel-Lehmann less dispersed than a random vector Y ∈ D if and only if there exists a random vector Z ∈ D such that (i) X and Z are µ -comonotonicand (ii) Y = d X + Z .Proof of Theorem 3. Assume X - µBL Y and call Q X and Q Y the µ -quantiles of X and Y . Let U be a random vector with distribution µ such that X = Q X ( U ). By assumption, ∇ V ( U ) is equalto Q Y ( U ) − Q X ( U ) = Q Y ( U ) − X . Call Z = ∇ V ( U ). By Theorem 2.12(ii), p. 66 of [36], ∇ V isthe µ -quantile Q Z of Z . Hence we have X = Q X ( U ) and Z = Q Z ( U ) and X and Z are therefore µ -comonotonic and we have Y = d Q Y ( U ) = X + Z as required. Conversely, take X and Z µ -comonotonic. Then X = Q X ( U ) and Z = Q Z ( U ) for some U = d µ , where Q X and Q Z are the µ -quantiles of X and Z respectively. Call Y = X + Z and Q Y = Q X + Z the µ -quantile of Y . In theproof of Theorem 1 of [14], it is shown that Q X + Z = Q X + Q Z when X and Z are µ -comonotonic.Hence, we have Q Y = Q X + Q Z , i.e., Q Y − Q X = Q Z , and Q Z is the gradient of a convex functionby Deﬁnition 5. The result follows. (cid:3) The characterization given in Theorem 3 now allows us to generalize our characterization ofMMPIR aversion to the multivariate case.

Proposition 2 (Local utility of multivariate MMPIR averse decision makers) . A decision functional Φ satisfying Assumption 2, is µ -MMPIR averse if and only if its local utility function U Φ satisﬁes E µ [ ∇ V ( U ) · ∇ U Φ ( Q X ( U ); F X )] ≤ for all V convex with E µ V ( U ) = 0 and all X ∈ D with distribution function F X and µ -quantilefunction Q X . OCAL UTILITY AND MULTIVARIATE RISK AVERSION 13

Proof of Proposition 2.

Let Y dominate X with respect to mean preserving µ -Bickel-Lehmann dis-persion, i.e., Y % µ − MMP IR X . This is equivalent to Y = d X + Z with X and Z µ -comonotonic, E Z = 0. For each ε >

0, deﬁne Y ε = X + εZ , which also dominates X with respect to µ -Bickel-Lehmann dispersion. Φ is µ -MMPIR averse if and only if for all ε >

0, Φ( X + εZ ) − Φ( X ) ≤ Q X + εZ and Q X the µ -quantiles of Y ε and X respectively and U = d µ , comonotonicity of X and Z implies Q Y ε ( U ) = Q X + εZ ( U ) = Q X ( U ) + εQ Z ( U ) . Now, calling F ε the distribution function of Y ε and F the distribution function of X , we have byAssumption 2, Φ( F ε ) − Φ( F ) = Z U Φ ( x, F )[ dF ε ( x ) − dF ( x )] + o ( ε )= Z [ U Φ ( Q X + εZ ( u )) − U Φ ( Q X ( u ))] dµ ( u ) + o ( ε )= Z [ U Φ ( Q X + Q εZ ( u )) − U Φ ( Q X ( u ))] dµ ( u ) + o ( ε )= ε Z Q Z ( U ) · ∇ U Φ ( Q X ( u ) , F ) dµ ( u ) + o ( ε ) . Therefore we have E [ ∇ V ( U ) · ∇ U Φ ( Q X ( U ) , F )] ≤ (cid:3) The characterization given in Theorem 3 is also crucial to the results in the next section oncomparative risk attitudes of multivariate rank dependent utility maximizers.3.2.4.

Relation to other multivariate dispersion orders.

We now look at the relation between µ -Bickel-Lehmann dispersion and other generalizations of Bickel-Lehmann dispersion proposed in thestatistical literature. The notion of strong dispersion was proposed by [15]. Deﬁnition 9 (Strong dispersive order) . Y is said to dominate X in the strong dispersive order,denoted Y % SD X if Y = d φ ( X ) , where φ is an expansion, i.e., such that k φ ( x ) − φ ( x ′ ) k ≥ k x − x ′ k for all pairs ( x, x ′ ) . The following Proposition gives conditions under which µ -Bickel-Lehmann implies [15]’s strongdispersion. Proposition 3.

Let X and Y be two random vectors in D . The following propositions hold. Y is more dispersed than X in the strong dispersion order, i.e., Y % SD X , if Y = d X + Z ,where X and Z are c -comonotonic. If Y % µBL X and the µ -quantiles of X and Y are gradients of strictly convex functions,then Y % SD X . Proof of Proposition 3. If Y % µ − BL X , then by Theorem 3, Y = d X + Z , where X and Z are µ -comonotonic. Hence Y = d Q X + Z ( U ) = Q X ( U ) + Q Z ( U ) = d X + Q Z ( Q − X ( X )) , where Q X = ∇ V X and Q Z = ∇ V Z are gradients of convex functions. Therefore, denoting φ ( x ) = x + ψ ( x ) = x + ∇ V Z ◦ ( ∇ V X ) − ( x ), we need to show that φ satisﬁes the condition J Tφ ( x ) J φ ( x ) − I ≥ x as in the characterization of the strong dispersive order in Theorem 2 of [15]. This followsfrom the fact that the jacobian of a gradient of a strictly convex function is symmetric positivedeﬁnite. Now, if two matrices S and S are both symmetric and positive deﬁnite, then, so is S / S S / . The latter is therefore diagonalizable with positive eigenvalues. Since S / S S / x = λx is equivalent to S S y = λy with y = S / x , S S has the same eigenvalues as S / S S / . Hence J ψ ( x ) = (cid:2) J ∇ V X (cid:0) ( ∇ V X ) − ( x ) (cid:1)(cid:3) − (cid:2) J ∇ V Z (cid:0) ( ∇ V X ) − ( x ) (cid:1)(cid:3) has positive eigenvalues (see also Lemma 6.2.8 page 144 of [2]). This completes the proof of (ii).The proof of (i) follows the same lines with Y = d X + Q Z ( X ), where Q Z is the gradient of a convexfunction. (cid:3) Partial insurance and monotone mean preserving decreases in risk.

The characterization ofmonotone mean preserving increase in risk given in Theorem 3 allows us to extend to multivariaterisk sharing a celebrated result of Landsberger and Meilijson in [21] stating that partial insurancecontracts are Pareto eﬃcient relative to second order stochastic dominance if and only if they involvea decrease in Bickel-Lehmann dispersion. Consider an individual A bearing a risk Y that sheconsiders sharing with individual B , in the sense that A would bear X A and B would bear X B with X A + X B = Y . The partial insurance contract is therefore a (potential) decrease in the risk borne by A from Y to X A . The new allocation ( X A , X B ) is shown in [5] (up to technical regularity conditions)to be Pareto eﬃcient (in the sense that it can’t be improved for both parties irrespective of theirmean preserving increase in risk averse preferences) if and only if it is µ -comonotonic in the sense ofDeﬁnition 6. Now, by Theorem 3, µ -comonotonicity of X A and X B , with X A + X B = Y , is equivalentto X A being Bickel-Lehmann less dispersed than Y . We therefore recover the strong relation betweenQuiggin’s notion of monotone mean preserving increases in risk and partial insurance identiﬁed in[21] and extend it to multivariate risk sharing.4. Increasing risk aversion and rank dependent utility

To make interpersonal comparisons of attitudes to multivariate risk, we deﬁne compensated in-creases in risk in the spirit of [9].

OCAL UTILITY AND MULTIVARIATE RISK AVERSION 15

Deﬁnition 10 (Compensated Increases in Risk) . Let Φ be the functional representing a decisionmaker’s preferences over multivariate prospects in D . A prospect Y ∈ D is a compensated increasein risk from the point of view of Φ if X - µBL Y and Φ( Y ) = Φ( X ) . A ranking of risk aversion is then derived in the usual way, except that the ranking of aversionto multivariate risks is predicated on the reference measure µ in the deﬁnition of dispersion. Deﬁnition 11 (Increasing risk aversion) . A decision maker ˜Φ is more risk averse than a decisionmaker Φ if ˜Φ is averse to a compensated increase in risk from the point of view of Φ , i.e., if X - µBL Y and Φ( Y ) = Φ( X ) imply ˜Φ( Y ) ≤ ˜Φ( X ) . In the special case of rank dependent utility maximizers, aversion to monotone MPIR and increas-ing risk aversion take a very simple form. We consider here the multivariate generalization of Yaaridecision makers given in [14]. A multivariate rank dependent utility maximizer is characterized bya functional Φ on multivariate prospects X ∈ D , which is a weighted sum of µ -quantiles, i.e.,Φ( X ) = E [ Q X ( U ) · φ ( U )] , (4.1)where Q X is the µ -quantile of X , U = d µ and φ ( U ) ∈ D . As shown in Theorem 1 of [14], Φ( X + Z ) = Φ( X ) + Φ( Z ) when X and Z are µ -comonotonic. Hence we immediately ﬁnd the followingcharacterization of monotone MPIR aversion and increasing risk aversion. Theorem 4 (Rank dependent utility) . Let Φ and ˜Φ be multivariate rank dependent utility function-als, i.e., Φ and ˜Φ satisfy (4.1). Then the following hold. (a) Φ is averse to a monotone MPIR (i.e., a mean preserving µ -Bickel-Lehmann dispersion) ifand only if for all Z ∈ D , Φ( Z ) ≤ Φ( E Z ) . (b) ˜Φ is more risk averse than Φ iﬀ for all Z ∈ D , Φ( Z ) = 0 ⇒ ˜Φ( Z ) ≤ . It turns out, therefore, that aversion to MMPIR in the multivariate rank dependent utility modelis equivalent to weak risk aversion ( E X preferred to X ). Since Theorem 2 of [14] shows that aversionto MPIR in the multivariate RDU model is equivalent to φ ( u ) = − αu + u , with α > u ∈ R d ,we recover the fact that MPIR averters are also monotone MPIR averters as in the univariate case. Corollary 1. If Φ is averse to mean preserving increases in risk, than it is also averse to monotonemean preserving increases in risk. Yaari’s rank dependent utility maximizers over stochastically independent multivariate risks in[38] are special cases of (4.1) where the reference distribution µ has independent marginals. In that special case, (a) of Theorem 4 is equivalent to concavity of the local utility function in (1.2) (i.e., non-increasing φ i for each i ) and (b) of Theorem 4 is equivalent to ˜ φ i being a decreasing transformationof φ i for each i , so that we recover the classical results of [38]. Conclusion

Attitudes to multivariate risks were characterized using Machina’s local utility in a framework,where objects of choice are multidimensional prospects. Aversion to mean preserving increases inmultivariate risk is characterized by concavity of the local utility function as in the univariate case.Comparative attitudes are characterized within the multivariate extension in [14] of rank dependentutility with the help of a multivariate extension of Quiggin’s monotone mean preserving increase inrisk notion and a generalization of its characterization in [21]. This allows us to extend Landsbergerand Meilijson’s result on the eﬃciency of partial insurance contracts for monotone mean preservingreductions in risk in [21]. Characterization and derivation of risk premia within the multivariaterank dependent utility model is the natural next step in this research agenda.

OCAL UTILITY AND MULTIVARIATE RISK AVERSION 17

References [1] Allais M., “Le comportement de l’homme rationnel devant le risque: Critiques des postulats et axiomes de l’´ecoleam´ericaine,”

Econometrica

21, pp. 503-546, 1953.[2] Ambrosio, L., N. Gigli and G. Savar´e, “Gradient ﬂows in metric spaces and in the space of probability measures,”Basel: Birkhauser, 2000.[3] Bickel, P. and E. Lehmann, “Descriptive statistics for non-parametric models, III: Dispersion,”

Annals of Statis-tics

4, 1139-1158, 1976.[4] Bickel, P. and E. Lehmann, “Descriptive statistics for non-parametric models, IV: Spread,” in: Jureckova, J.(ed.) Contributions to statistics. Boston, MA: Reidel, 1979.[5] Carlier, G., R.-A. Dana and A. Galichon, “Pareto eﬃciency for the concave order and multivariate comonotonic-ity,”

Journal of Economic Theory

SIAM Journal of Mathematical Analysis

41, pp. 2554-2576, 2010.[7] Chateauneuf, A., M. Cohen and I. Meilijson, “Four notions of mean-preserving increase in risk, risk attitudesand applications to the rank-dependent expected utility model,”

Journal of Mathematical Economics

40, pp.547-571, 2004.[8] Chew, S., E. Karni and Z. Safra, “Risk aversion in the theory of expected utility with rank dependent probabili-ties,”

Journal of Economic Theory

42, pp. 370-381, 1987.[9] Diamond, P. and J. Stiglitz, “Increases in risk and in risk aversion,”

Journal of Economic Theory

8, pp. 337-360,1974.[10] Eeckoudt, L., B. Rey and H. Schlesinger, “A good sign for multivariate risk taking, ”

Management Science

Mathematical Finance

Ann. Probab.

Volume 20, Number 1, 421-454, 1992.[13] Epstein, L. and S. Tanny, “Increasing generalized correlation: a deﬁnition and some economic consequences,”

Canadian Journal of Economics

13, pp. 16-34, 1980.[14] Galichon, A. and M. Henry, “Dual theory of choice with multivariate risks,”

Journal of Economic Theory

Statistics and Probability Letters

22, pp.325-332, 1995.[16] Grant, S., A. Kajii and B. Polak, “Many good choice axioms: when can many-good lotteries be treated as moneylotteries,”

Journal of Economic Theory

56, pp. 313-337, 1992.[17] Hardy, G., J. Littlewood and G. P´olya,

Inequalities . Cambridge: Cambridge University Press, 1952.[18] Heath, D., “Interpolation of martingales,”

Annals of Probability

5, pp. 804-806, 1977.[19] Karni, E., “Generalized expected utility analysis of multivariate risk aversion,”

International Economic Review

30, pp. 297-305, 1989.[20] Kihlstrom, R. and L. Mirman, “Risk aversion with many commodities,”

Journal of Economic Theory

8, pp.361-368, 1974.[21] M. Landsberger and I. Meilijson, “Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Prattmeasure of risk aversion,”

Annals of Operation Research

52, pp. 97-106, 1994. [22] Machina, M., “ ‘Expected utility’ analysis without the independence axiom,”

Econometrica

50, pp. 277-323,1982.[23] Machina, M., “ A stronger characterization of declining risk aversion,”

Econometrica

50, pp. 1069-1079, 1982.[24] A. M¨uller, and Scarsini, M., “Fear of loss, inframodularity, and transfers,”

Journal of Economic Theory

Journal of Multivariate Analysis

Continuous martingales and Brownian motion , Third Edition, Springer Verlag, 1999.[28] Rothschild, M. and J. Stiglitz, “Increasing risk: I. A deﬁnition,”

Journal of Economic Theory

2, pp. 225-243,1970.[29] Safra, Z. and U. Segal, “Dominance axioms and multivariate nonexpected utility preferences,”

InternationalEconomic Review

34, pp. 321-334, 1993.[30] Scarsini, M., “Dominance conditions for multivariate utility functions,”

Management Science

34, pp. 454-460,1988.[31] Segal, U., “Some remarks on Quiggin’s anticipated utility,”

Journal of Economic Behavior and Organization

Probability in the Engineering and InformationalSciences

12, pp. 1-23, 1998.[33] Shaked, M. and J. Shantikumar,

Stochastic Orders , New York: Springer, 2007.[34] Stiglitz, J., “Behavior towards risk with many commodities,”

Econometrica

37, pp. 660-667, 1969.[35] Strassen, V., “The existence of probability measures with given marginals,”

Annals of Mathematical Statistics

36, pp. 423-439, 1965.[36] Villani, C.,

Topics in Optimal Transportation , Providence: American Mathematical Society, 2003.[37] Villani, C.,

Optimal Transportation: old and new , Berlin: Springer-Verlag, 2009.[38] Yaari, M., “Univariate and multivariate comparisons of risk aversion: a new approach,”