Dual theory of choice with multivariate risks
aa r X i v : . [ ec on . T H ] F e b DUAL THEORY OF CHOICE WITH MULTIVARIATE RISKS (1)
ALFRED GALICHON AND MARC HENRY Abstract.
We propose a multivariate extension of Yaari’s dual theory of choice underrisk. We show that a decision maker with a preference relation on multidimensionalprospects that preserves first order stochastic dominance and satisfies comonotonic in-dependence behaves as if evaluating prospects using a weighted sum of quantiles. Boththe notions of quantiles and of comonotonicity are extended to the multivariate frameworkusing optimal transportation maps. Finally, risk averse decision makers are characterizedwithin this framework and their local utility functions are derived. Applications to themeasurement of multi-attribute inequality are also discussed.
Keywords : risk, rank dependent utility theory, multivariate comonotonicity, optimal transportation,multi-attribute inequality, Gini evaluation functions.
JEL subject classification : D63, D81, C61
Introduction
In his seminal paper [35], Menahem Yaari proposed a theory of choice under risk, whichhe called “dual theory of choice,” where risky prospects are evaluated using a weighted sumof quantiles. The resulting utility is less vulnerable to paradoxes such as Allais’ celebratedparadox [1]. The main ingredients in Yaari’s representation are the preservation of first Journal of EconomicTheory
Symposium on Inequality and Risk for helpful discussions and comments. The authors are gratefulto two anonymous referees and particularly an Associate Editor for their careful reading of the manuscriptand insightful suggestions. order stochastic dominance and insensitivity to hedging of comonotonic prospects. Bothproperties have strong normative and behavioral appeal once it is accepted that decisionmakers care only about the distribution of risky prospects. The preservation of stochasticdominance is justified by the fact that decision makers prefer risky prospects that yieldhigher values in all states of the world, whereas comonotonicity captures the decision maker’sinsensitivity to hedging comonotonic prospects, that is to say, the fact that the decisionmaker who is indifferent between two prospects that yield their higher and lower returns inthe same states of the world, is also indifferent between any convex combination of thoseprospects. The dual theory has been used extensively as an alternative to expected utilityin a large number of contexts. The main drawback of the dual theory is that it does notproperly handle the case in which the prospects of consumptions of different natures arenot perfect substitutes. The assumption of law invariance of the decision functional (calledneutrality in [35] and by which the decision maker is insensitive to relabelings of the statesof the world) is easier to substantiate when several dimensions of the risk are considered inthe decision functional.To handle these situations, we need to be able to express utility derived from monetaryconsumption in different num´eraires , which is easily done with Expected Utility Theory, butso far not covered by Yaari’s dual theory. Indeed, the latter applies only to risky prospectsdefined as univariate random variables, thereby ruling out choice among multidimensionalprospects which are not perfect substitutes for each other, such as risks involving botha liquidity and a price risk, collection of payments in different currencies, payments atdifferent dates, prospects involving different goods of different natures such as consumptionand environmental quality, etc. Yaari [34] proposes a multivariate version of his dual theory,but it involves independence of the risk components and an axiom of separability (AxiomA in [34]), which essentially removes the multidimensional nature of the problem.We propose to remove this constraint with a multivariate extension of the dual theory torisky prospects defined as random vectors that is applicable as such to the examples listedabove. The main challenge in this generalization is the definition of quantile functions andcomonotonicity in the multivariate setting. Another challenge is to preserve the simplicityof the functional representing preferences, so that they can be parameterized and can becomputed as efficiently as in the univariate case. Both challenges are met with an appeal tooptimal transportation maps that allow for the definition of “generalized quantiles,” their UAL THEORY OF CHOICE 3 efficient computation, and the extension of comonotonicity as a notion of distribution freeperfect correlation. There are many ways of extending the notion of comonotonicity to amultivariate framework consistently with the univariate definition. Our proposed extensionhas the added property of preserving the equivalence between comonotonicity and Paretoefficiency of allocations (see [15] for the original result and [6] for the multivariate exten-sion). With these notions of quantiles and comonotonicity in hand, we give a representationof a comonotonic independent preference relation as a weighted sum of generalized quan-tiles. The main difference between the univariate case and the multivariate case is thatcomonotonicity and generalized quantiles are defined with respect to an objective referencedistribution, which features in the representation. The reference distribution is shown to beequal to the distribution of equilibrium prices in an economy with at least one risk averseYaari decision maker.We then turn to the representation of a risk averse decision makers’s preferences withinthis theory. Risk aversion is defined in the usual way as a preference for less risky prospects,where the notion of increasing risk is suitably generalized to multivariate risky prospects.We show, again in a direct generalization of the univariate case, that risk aversion is charac-terized by a special form of the quantile weights defined above: risk averse decision makersgive more weight to low outcomes (low quantiles) and less weight to high outcomes (highquantiles). As a result, given the reference distribution with respect to which comonotonic-ity is defined, risk averse decision makers are characterized by further simple restrictionson their utility functionals, which makes this model as simple and as tractable as expectedutility. A further advantage of our decision functional is the simple characterization of thelocal utility function and its close relation to the multivariate quantile function.The risk averse Yaari decision functional is a version of the Weymark social evaluationfunction (in [32]) with a continuous state space. Indeed, the formal equivalence betweenthe evaluation of risky prospects and the measurement of inequality noted in [2] and [14]allows us to draw implications of our theory for the measurement of inequality of allo-cations of multiple attributes, such as consumption, education, environment quality, etc.Seen as a social evaluation function, our decision functional provides a compromise be-tween the approach of [11] and [29] in that it allows a flexible attitude to correlations
ALFRED GALICHON AND MARC HENRY between attributes without necessarily imposing correlation aversion and thereby circum-venting the Bourguignon-Chakravarty [4] critique of the assumption that attributes aresubstitutes rather than complements.The paper is organized as follows. The next section gives a short exposition of thedual theory. The following section develops the generalized notion of comonotonicity thatis necessary for the multivariate extension, which is given in Section 3. Risk aversion ischaracterized in Section 4. The economic interpretation of the reference measure is givenin Section 5 and the application to multi-attribute inequality measurement is discussed inSection 6. The final 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 probability distribution of X by L X . E is the expectation operator with respect to P . For x and y in R d , let x · y be thestandard scalar product of x and y , and k x k the Euclidian norm of x . We denote by X = d L X the fact that the distribution of X is L X and by X = d Y the fact that X and Y have the same distribution. The equidistribution class of X = d L X , denoted indifferentlyequi( L X ) or equi( X ), is the set of random vectors with distribution with respect to P equalto L X (reference to P will be implicit unless stated otherwise). F X denotes the cumulativedistribution function of distribution L X . Q X denotes its quantile function. 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 3 of Section 2.1 below. We call L d the set of random vectors X in dimension d such that E h k X k i < + ∞ . We denote by D the subset of L d containingrandom vectors with a density relative to Lebesgue measure. A functional Φ on L d is calledupper 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.For a convex lower semi-continuous function V : R d R , we denote by ∇ V its gradient(equal to the vector of partial derivatives). A doubly stochastic matrix is a square matrixof nonnegative real numbers, each of whose rows and columns sum to 1.1. Dual theory of decision under risk
In this section, we first revisit Yaari’s “Dual theory of choice under risk” presented in theeponymous paper [35]. As in [35], we consider a problem of choice among risky prospects
UAL THEORY OF CHOICE 5 as modeled by random variables defined on an underlying probability space. The riskyprospect X is interpreted as a gamble or a lottery that a decision maker might considerholding and the realizations of X are interpreted as payments.1.1. Representation.
We suppose that the decision maker is characterized by a preferencerelation % on the set of risky prospects. X % Y indicates that the decision maker prefersprospect X to prospect Y , X ≻ Y stands for X % Y and not Y % X , whereas X ∼ Y standsfor X % Y and Y % X . We first introduce the set of axioms satisfied by the preferencerelation that were proposed by Yaari in [35].With the first axiom (which corresponds to Axioms A2 and A3 in Yaari [35]), we takethe standard notion of preference as a continuous pre-order (reflexive and transitive binaryrelation) which is complete. Continuity of the preference relation is required relative to thetopology of weak convergence: a sequence of random prospects X n converges weakly to X if E f ( X n ) converges to E f ( X ) for all continuous bounded functions f on R d . Then, % canbe represented by a continuous real valued function γ in the sense that X % Y if and onlyif γ ( X ) ≥ γ ( Y ). Axiom 1.
The preference relation % is reflexive, transitive, complete and continuous rela-tive to the topology of weak convergence. A prospect X is said to first order stochastically dominate a prospect Y if there exist˜ X = d X and ˜ Y = d Y such that X ( s ) ≥ Y ( s ) for almost all states of the world s ∈ S . Thefollowing axiom requires that whenever one prospect first order stochastically dominates asecond, then the former is preferred to the latter. This is formally stated as follows. Axiom 2.
The preference % preserves first order stochastic dominance in the sense that ifprospect X first order stochastically dominates prospect Y , then X % Y , and if X strictlyfirst order stochastically dominates prospect Y , then X ≻ Y . Two prospects with the same distribution first order stochastically dominate one another.Hence, Axiom 2 implies law invariance of the preference relation, or what [35] calls neutrality ,i.e., X = d Y implies X ∼ Y . Neutrality can be interpreted as the fact that the decisionmaker is indifferent to relabelings of the states of the world. Once neutrality is accepted,then Axiom 2 is reasonable as it is equivalent to requiring that the decision maker prefers ALFRED GALICHON AND MARC HENRY prospects that yield a higher value in every state of the world. We shall see below that withsuitable extensions of the concepts of monotonicity and stochastic dominance, this axiomremains reasonable in the multivariate extension of Yaari’s representation theorem.Finally, the third axiom is the crucial one in this framework, as it replaces indepen-dence by comonotonic independence. Recall that X and Y are comonotonic if ( X ( s ) − X ( s ′ ))( Y ( s ) − Y ( s ′ )) ≥ s, s ′ ∈ S . The absence of a hedging opportunity betweencomonotonic prospects justifies the requirement below. Axiom 3. If X, Y and Z are pairwise comonotonic prospects, then for any α ∈ [0 , , X % Y implies αX + (1 − α ) Z % αY + (1 − α ) Z . We can now state Yaari’s representation result.
Proposition 1 (Yaari) . A preference % on [0 , -valued prospects satisfies Axioms 1-3 ifand only if there exists a continuous non-decreasing function f defined on [0 , , such that X % Y if and only if γ ( X ) ≥ γ ( Y ) , where γ is defined for all X as γ ( X ) = R f (1 − F X ( t )) dt . This result is interpretable in terms of weighting of outcomes (through the weighting ofquantiles). Assume that each of the functions that we consider satisfy the invertibility andregularity conditions needed to perform the following operations. By integration by parts Z f (1 − F X ( t )) dt = Z f (1 − u ) dQ X ( u ) = Z f (1 − u ) ddu Q X ( u ) du = Z f ′ (1 − u ) Q X ( u ) du. Hence, calling φ ( u ) = f ′ (1 − u ), we have the representation of % with the functional R φ ( t ) Q X ( t ) dt . Hence, increasing f corresponds to positive φ , which can be interpreted asa weighting of the quantiles of the prospect X . As noted in [35], the functional γ satisfies γ ( γ ( X )) = γ ( X ), so that γ ( X ) is the certainty equivalent of X for the decision makercharacterized by % .1.2. Risk aversion.
We now turn to the characterization of risk averse decision mak-ers among those satisfying Axioms 1-3. We define increasing risk as in Rothschild andStiglitz [21]. The formulations in the first part of the definition below are equivalent by theBlackwell-Sherman-Stein Theorem (see, for instance, Chapter 7 of [25]).
UAL THEORY OF CHOICE 7
Definition 1 (Concave ordering, risk aversion) . a) A prospect Y is dominated by X in the concave ordering , denoted Y ≤ cv X , when the equivalent statements (i) or (ii) hold: (i) for all continuous concave functions, E f ( Y ) ≤ E f ( X ) . (ii) Y has the same distribution as ˆ Y where ( X, ˆ Y ) is a martingale, i.e., E ( ˆ Y | X ) = X ( ˆ Y is sometimes called a mean-preserving spread of X ).b) The preference relation % is called risk averse if X % Y whenever X ≥ cv Y . Notice that x → x and x → − x are both continuous and concave function. Therefore,condition (i) in the first part of the definition implies that E [ X ] = E [ Y ] is necessary for aconcave ordering relationship between X and Y to exist. With this definition, we can recallthe characterization of risk averse preferences satisfying Axioms 1-3 as those with convex f (see Section 5 of [35] or Theorem 3.A.7 of [25]). Proposition 2.
A preference relation % satisfying Axioms 1-3 is risk averse if and only ifthe function f in Theorem 1 is convex. This monotonicity of the derivative of f has the natural interpretation that risk aversedecision makers evaluate prospects by giving high weights to low quantiles (corresponding tolow values of the prospect) and low weights to high quantiles. Indeed, with the formulation γ ( X ) = R φ ( u ) Q X ( u ) du and the identification φ ( u ) = f ′ (1 − u ), an increasing convex f corresponds to positive decreasing φ , and therefore to a weighting scheme in which lowquantiles (corresponding to unfavorable outcomes) receive high weights and high quantiles(corresponding to favorable outcomes) receive low weights.2. Multivariate quantiles and comonotonicity
The main ingredients in our multivariate representation theorem are the multivariateextensions of quantiles and comonotonicity. As we shall see, the two are intimately related.2.1.
Multivariate quantiles.
We first note that the quantile of a random variable can becharacterized as an increasing rearrangement of the latter. Hence, by classical rearrange-ment inequalities, quantiles are solutions to maximum correlation problems. More precisely,by the rearrangement inequality of Hardy, Littlewood and P´olya [12], we have the following
ALFRED GALICHON AND MARC HENRY well known equality: Z tQ X ( t ) d ( t ) = max { E [ XU ] : U uniformly distributed on [0 , } , (2.1)where the quantile function Q X has been defined above. This variational characterization iscrucial when generalizing Yaari’s representation theorem to the multivariate setting. Indeed,consider now a random vector X on R d and a reference distribution µ on R d , with U distributed according to µ . We introduce maximum correlation functionals to generalizethe variational formulation of (2.1). Definition 2 (Maximal correlation functionals) . A functional ̺ µ : L d → R is called amaximal correlation functional with respect to a reference distribution µ if for all X ∈ L d , ̺ µ ( X ) := sup n E [ X · ˜ U ] : ˜ U = d µ o . It follows from the theory of optimal transportation (see Theorem 2.12(ii), p. 66 of [30])that if µ is absolutely continuous with respect to Lebesgue measure (which will be assumedthroughout the rest of the paper), then there exists a convex lower semi-continuous function V : R d → R and a random vector U distributed according to µ such that X = ∇ V ( U ) holds µ -almost surely, and such that ̺ µ ( X ) = E [ ∇ V ( U ) · U ]. In that case, the pair ( U, X ) is saidto achieve the optimal quadratic coupling of µ with respect to the distribution of X . Thefunction V is called the transportation potential of X with respect to µ or the transportationpotential from µ to the probability distribution of X . This shows that the gradient of theconvex function V thus obtained satisfies the multivariate analogue of equation (2.1). Wetherefore adopt ∇ V as our notion of a generalized quantile. Definition 3 ( µ -quantile) . The µ -quantile function of a random vector X on R d with respectto an absolutely continuous distribution µ on R d is defined by Q X = ∇ V , where V is thetransportation potential of X with respect to µ . This concept of a multivariate quantile is the counterpart of our definition of multivariatecomonotonicity in the representation theorem, and the latter, introduced in the following V is convex and hence differentiable except on set of measure zero by Rademacher’s Theorem (Theorem2.4 in [19]), so that the expression E [ ∇ V ( U ) · U ] above is well defined. UAL THEORY OF CHOICE 9 section has strong economic underpinnings, as discussed in Section 5, where we give theeconomic interpretation of the reference measure µ .2.2. Multivariate comonotonicity.
Two univariate prospects X and Y are comonotonicif 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 . Comonotonicity is hence characterized by maximal correlationbetween the prospects over the equidistribution class. This variational characterization(where products will be replaced by scalar products) will be the basis for our generalizednotion of comonotonicity. Definition 4 ( µ -comonotonicity) . Let µ be a probability measure on R d with finite secondmoments. A collection of random vectors X i ∈ L d , i ∈ I , are called µ -comonotonic if onehas ̺ µ X i ∈ I X i ! = X i ∈ I ̺ µ ( X i ) . When µ is absolutely continuous with respect to Lebesgue measure, it follows from therepresentation of ̺ µ that the family X i is µ -comonotonic if and only if there exists a vector U distributed according to µ such that U ∈ argmax ˜ U { E [ X i · ˜ U ] , ˜ U = d µ } for all i ∈ I .In other words, the X i ’s can be rearranged simultaneously so that they achieve maximalcorrelation with U . When the distributions of the random vectors are absolutely continuouswith respect to Lebesgue measure, the concept of µ -comonotonicity is transitive. Proposition 3.
Suppose that X and Y are µ -comonotonic and that Y and Z are µ -comonotonic, with the distribution of Y assumed to be absolutely continuous with respect toLebesgue measure. Then X and Z are µ -comonotonic.Comonotonic allocations and Pareto efficiency. It is worth discussing this definition ofcomonotonicity as generalizations of the classical univariate notion of comonotonicity are notunique. The main motivation for introducing it is to generalize the univariate equivalencebetween comonotonic and Pareto efficient allocations in a risk-sharing economy. Consideran Arrow-Debreu economy with n agents, and with an aggregate endowment which is arandom vector X . Thus the i -th dimension of the realization X i ( ω ) in state ω ∈ Ω of this random vector is the quantity of good i ∈ { , ..., d } produced in this state. An allocation (or risk-sharing allocation ) of X is a sharing rule of this aggregate endowment among the n agents, hence it is the specification of n random vectors X , ..., X n such that ∀ ω ∈ Ω , n X k =1 X k ( ω ) = X ( ω )where X ik ( ω ) is the quantity of good i allocated to agent k ∈ { , ..., n } in state ω . Anallocation is called (Pareto) efficient if no other allocation dominates the former, agent byagent, in the sense of the concave ordering (as defined in Proposition 4 below).In dimension one, it is known since the seminal paper of Landsberger and Meilijson [15]that a risk-sharing allocation is Pareto efficient with respect to the concave order if andonly if it is comonotonic. That is, given any comonotonic allocation, it is not possible tofind another allocation such that each risky endowment would be preferred under the newallocation by every risk-averse decision maker to the endowments in the original alloca-tion. Multivariate generalization of this equivalence is not obvious, but it turns out that, asrecently shown by Carlier, Dana and Galichon [6], this result can be extended to the multi-variate case, with comonotonicity replaced by multivariate comonotonicity , if one defines anallocation to be comonotonic in the multivariate sense if and only if it is µ -comonotonic forsome measure µ with enough regularity. In our view, this result strongly supports the claimthat our notion of comonotonicity is in some sense the “natural” multivariate extension ofcomonotonicity. Relation with other multivariate notions of comonotonicity.
Puccetti and Scarsini [18] havealso applied the theory of optimal transportation to generalize the notion of comonotonicityto the multivariate setting. They review possible multivariate extensions of comonotonicity,including the notion of µ -comonotonicity that we propose. But the concept they favor differsfrom ours in the sense that according to their notion of multivariate comonotonicity (whichthey call c -comonotonicity ), two vectors X and Y are c -comonotonic if and only ( X, Y )is an optimal quadratic coupling. That is, X and Y are c-comonotonic if and only ifthere is a convex function V such that Y = ∇ V ( X ) holds almost surely. However, unlike µ -comonotonicity, c -comonotonicity is in general not transitive, and does not seem to berelated to efficient risk-sharing allocations and equilibrium. UAL THEORY OF CHOICE 11
Schmeidler [24] introduces an internal notion of comonotonicity: if a decision makerevaluates prospects according to % , then Schmeidler-comonotonicity of two prospects X and Y means that for all pairs of states of the world ( s, t ), X ( s ) % X ( t ) implies Y ( s ) % Y ( t ),i.e., prospects X and Y are more desirable in the same states of the world. In contrast,we extend the Weymark [32] - Yaari [35] motivation in our definition of comonotonicity,which can be related to the state prices in the economy, as explained in Section 5. The twonotions have no obvious relation, as we see by considering two µ -comonotonic prospects X and Y and imposing Schmeidler comonotonicity. By µ -comonotonicity, there exists U = d µ and generalized quantile functions Q X and Q Y such that X = Q X ( U ) and Y = Q Y ( U ).Schmeidler-comonotonicity of X and Y would require that the univariate random variables( E U ) · X and ( E U ) · Y are comonotonic in the usual sense. Although they are equivalent indimension one, in higher dimensions, neither of these two concepts implies the other.3. Multivariate Representation Theorem
Now that we have given a formalization of the notion of maximal correlation in a lawinvariant sense that is suitable for a multivariate extension of Yaari’s dual theory, we canproceed to generalize Yaari’s representation result to the case of a preference relation amongmultivariate prospects. We consider prospects, which are elements of L d . Axiom 1 ′ belowis a mild smoothness requirement for the preference relation. A functional γ is calledFr´echet differentiable in X relative to the L d metric if there is a linear functional L suchthat | γ ( X + h ) − γ ( X ) − L ( h ) | / p E [ h ] →
0. As in [7], the functional will not be Fr´echetdifferentiable at all points; we only require differentiability at one point.
Axiom ′ . The preference % is represented by a continuous functional γ on L d such thatat at least one point its Fr´echet derivative exists and is non-zero.Given sufficient regularity, first order stochastic dominance can be characterized equiva-lently by pointwise dominance of cumulative distribution functions or pointwise dominanceof quantile functions. It is the latter that we adopt for our multivariate definition. Definition 5 ( µ -first order stochastic dominance) . A prospect
X µ -first order stochasticallydominates prospect Y relative to the componentwise partial order ≥ on R d if Q X ( t ) ≥ Q Y ( t ) for almost all t ∈ R d , where Q X and Q Y are the generalized quantiles of X and Y withrespect to a distribution µ on R d . For any U = d µ , we have Q X ( U ) = d X and Q Y ( U ) = d Y . If X µ -first order stochasticallydominates Y , then Q X ( U ) ≥ Q Y ( U ) almost surely. Hence, ˆ X ≥ ˆ Y almost surely for someˆ X = d X and ˆ Y = d Y , which is the “usual multivariate stochastic order” (see [25], p. 266).The converse does not hold in general.The remaining two axioms require fixing an absolutely continuous reference probabilitydistribution µ on R d . Axiom ′ . The preference % preserves µ -first order stochastic dominance in the sense thatif prospect X µ -first order stochastically dominates prospect Y , then X % Y , and if Xµ -first order strictly stochastically dominates prospect Y , then X ≻ Y .The extension of the comonotonicity axiom is the key to the generalization of the dualtheory to multivariate prospects. The statement of Axiom 3 is unchanged, but the conceptof comonotonicity is now dependent on a reference distribution µ . The prospects X, Y and Z are comonotonic, or more precisely µ -comonotonic, if they are all maximally correlated in the law invariant sense of Definition 4 with a reference U (where U has distribution µ ). Axiom ′ . If X, Y and Z are µ -comonotonic prospects, then for any α ∈ [0 , X % Y implies αX + (1 − α ) Z % αY + (1 − α ) Z .We are now in a position to state the multivariate extension of Yaari’s representationtheorem. Theorem 1 (Multivariate Representation) . A preference relation on multivariate prospectsin L d satisfies Axioms ′ , ′ and ′ relative to a reference probability measure µ if and onlyif there exists a function φ such that for U = d µ , φ ( U ) ∈ L d , φ ( U ) ∈ ( R − ) d almost surelyand such that for all pairs X, Y , X % Y if and only if γ ( X ) ≥ γ ( Y ) , where γ is defined forall X by γ ( X ) = E [ Q X ( U ) · φ ( U )] , where Q X is the µ -quantile of X . When d = 1, the representation is independent of µ and we recover the result of Propo-sition 1. As in the univariate case, the decision maker assesses prospects with a weightingscheme φ of quantiles of the prospects. Because γ in Theorem 1 satisfies γ ( γ ( X )) = γ ( X ), γ ( X ) is the certainty equivalent of X as in the univariate case. Furthermore, % satisfies linearity in payments , i.e., for any positive real number a and any b ∈ R d (identified with aconstant multivariate prospect), γ ( aX + b ) = aγ ( X ) + b . UAL THEORY OF CHOICE 13
It should be noted that Choquet expected utility [24] handles multivariate prospectsunder Schmeidler comonotonicity (defined in Section 2.2). As shown in [31], under Ax-iom 2, Choquet expected utility is identical to the functional of Proposition 1. Hence, whenrestricted to decision under risk, Choquet expected utility aggregates the multiple dimen-sions of the prospects with the utility function and then considers univariate quantiles ofthe resulting utility index. This is in contrast with the functional of Theorem 1, whichdirectly evaluates multivariate quantiles of the prospects and thereby models attitudes tosubstitution risk between the dimensions of the prospect.4.
Risk aversion and the local utility function
In this section, we consider the question of representing those decision makers satisfyingAxioms 1 ′ , 2 ′ and 3 ′ that are risk averse in the sense of Definition 1. We then show that the local utility function in the sense of [17] is easily computable and provides an interpretationof the reference distribution µ .4.1. Risk aversion.
For our characterization of risk averse Yaari decision makers, we needto generalize the concept of a mean-preserving spread to the multivariate setting.
Proposition 4 (Concave ordering) . For any prospects X and Y whose respective distribu-tions are absolutely continuous with respect to Lebesgue measure, the following propertiesare equivalent. (a) For every bounded concave function f on R d , E f ( X ) ≥ E f ( Y )(b) Y = d ˆ Y , with E [ ˆ Y | X ] = X . (c) ̺ µ ( X ) ≤ ̺ µ ( Y ) for every probability measure µ . (d) X belongs to the closure of the convex hull of the equidistribution class of Y . (e) Φ( X ) ≥ Φ( Y ) for every u.s.c. law-invariant concave functional Φ : L d → R .When any of the properties above hold, one says that Y is dominated by X in the concaveordering , denoted Y ≤ cv X . Statements (a) and (b) are identical in the multivariate case as in [21]. The equivalencebetween the two is a classical result that can be traced back at least to [27] (see section A.2for details). The interpretation of the ordering as a preference ordering for all risk aversionexpected utility maximizers (a) and as an ordering of mean-preserving spreads (b) also carry over to the multivariate case. Statement (d) is the continuous equivalent to multiplicationby a doubly stochastic matrix.As in the univariate case and Definition 1(b), risk averse decision makers will be definedby aversion to mean-preserving spreads. It turns out that imposing risk aversion on apreference relation that satisfies axioms 1 ′ , 2 ′ and 3 ′ is equivalent to requiring the followingproperty, sometimes called preference for diversification . Axiom 4.
For any two preference equivalent prospects X and Y (i.e., such that X ∼ Y ), convex combinations are preferred to either of the prospects, (i.e., for any α ∈ [0 , , αX + (1 − α ) Y % X ). This is formalized in the following theorem, which gives a representation for risk averse
Yaari decision makers . Theorem 2.
In dimension d ≥ , for a preference relation satisfying Axioms ′ , ′ and ′ ,the following statements are equivalent: (a) % is risk averse, namely X % Y whenever X ≥ cv Y . (b) % satisfies Axiom 4. (c) The function φ involved in the representation of the preference relation in Theorem 1satisfies − φ ( u ) = αu + u for α > and u ∈ R d . So, in the multivariate setting the functional γ is convex if and only if φ ( x ) = − αx − x for some α real positive and x ∈ R d . This is a major difference with dimension one,where the functional is convex if and only if − φ is a non-decreasing map. This implies thata multivariate Yaari risk averse decision maker is entirely characterized by the referencedistribution µ .4.2. Local Utility Function.
Throughout the rest of the paper, we shall assume that theconditions in Theorem 2 are met. Hence, our discussion of local utility functions will belimited to the case of risk averse decision makers. By law-invariance, we denote γ ( P ) := γ ( X ), where X = d P . Without loss of generality, we shall also assume that φ ( u ) = − u , thus γ ( P ) := − E [ ∇ V P ( U ) · U ], where V P = V X is the transportation potential (see section 2.1for the definition) from the reference probability distribution µ of U to the probability UAL THEORY OF CHOICE 15 distribution P of X . As we have seen, the gradient ∇ V P of this transportation potential isthe µ -quantile function of distribution P .As shown in [17], when smoothness requirements are met, a local analysis can be carriedout in which a (risk-averse) non-Expected Utility function behaves for small perturbationsaround a fixed risk in that same way as a (concave) utility function. Formally, the localutility function is defined as u ( x | P ) = D P γ ( x ), where D P γ is the Fr´echet derivative of γ at P (see Section 3 for the definition). Denoting by V ∗ ( x ) = sup u [ u · x − V ( u )] the Legendre-Fenchel transform of a convex lower semicontinuous function V , we have: γ ( P ) = E [ ∇ V P ( U ) · U ] = max { E [ X · U ] : X = d P, U = d µ } = min (cid:26)Z V ( u ) dµ ( u ) + Z V ∗ ( x ) dP ( x ) : V convex and l.s.c. (cid:27) = Z V P ( u ) dµ ( u ) + Z V ∗ P ( x ) dP ( x )by the duality of optimal transportation (see, for instance, Theorem 2.9, p. 60 of [30]).Defining f ( V, Q ) = R V dµ + R V ∗ dQ , we have γ ( P ) = − inf V f ( V, P ). Hence, an envelopetheorem argument formally yields u ( x | P ) = D P γ ( x ) = − V ∗ P ( x ). Therefore, the local utilityfunction is − V ∗ P , the (negative of the) Legendre-Fenchel transform of the transportationpotential V P . This point sheds light on the economic interpretation of this potential, thanksto Machina’s theory of local utility. The function − V ∗ P is concave, which is consistentwith the risk aversion of a Yaari decision maker given the assumptions of Theorem 2. Forunivariate prospects, u ( x | P ) = − V ∗ P ( x ) = R x −∞ F X ( z ) dz , so that we recover the fact thatwhen X = d P is a mean-preserving spread of Y = d Q , u ( z | P ) ≤ u ( z | Q ) for all z .5. Economic interpretation of the reference measure
We now discuss the behavioral interpretation of µ . As we saw in Theorem 1, the general-ization of the Yaari preferences to the multivariate case led us to define a utility functional γ over prospects such that γ ( X ) = E [ X · φ ( ˜ U )] for some prospect ˜ U which is correlated to X . ˜ U is an index such that X · ˜ U measures how favorable the outcome is for the decisionmaker. φ ( ˜ U ) is a weighting of the contingent outcome X , so that γ over- or under-weightsprospects in each state using weights φ ( ˜ U ). Hence, the dispersion of µ induces a departurefrom risk-neutrality. In the special case in which µ is the distribution of a constant u , γ ( X ) = E [ X · φ ( u )] = E [ X ] · φ ( u ) and one recovers the case of a risk-neutral decision maker. On the contrary, when µ exhibits considerable dispersion, then the variance of φ ( ˜ U )is large in general, so that the “favorable” outcomes (in the sense that X · ˜ U is high) areweighted less, at least if φ ( ˜ U ) = − α ˜ U . This induces risk aversion. When φ ( ˜ U ) differs froma rescaling of ˜ U , there may be some discrepancy between the weighting of a given stateand how favorable it is. Hence, the variance of µ is no longer directly associated with riskaversion.We now turn to the equilibrium implications of the reference measure µ and show how it isrelated to the distribution of the state prices in an economy in equilibrium when a decisionmaker with risk averse decision functional as in Theorem 2 is present in the economy.Consider an economy where one of the agents (whom we shall refer to as “Yaari”) haspreferences as in Theorem 2, with reference measure µ . Assume that there is a risk sharingequilibrium in this economy, which is supported by the stochastic discount factor ξ , meaningthat if the original risky endowment of the agent is X , then the agent’s budget set is { X : E [( X − X ) · ξ ] = 0 } . The demand for risk X of Yaari is therefore max ˆ X γ ( ˆ X ) subject to E [( ˆ X − X ) · ξ ] = 0 . Since Yaari is assumed risk-averse, γ is concave, and the demand for risk X satisfies the local optimality condition max ˆ X E [ u ( ˆ X | P )] subject to E [( ˆ X − X ) · ξ ] = 0. Thefirst order conditions yield ∇ u ( X | P ) = λξ , where λ is the Lagrange multiplier associatedwith the budget constraint, where λ = 0 unless there is no trade in equilibrium. Now, asexplained above, u ( x | P ) = − V ∗ P ( x ), hence ∇ u ( X | P ) = −∇ V ∗ P ( X ). Now, by definition ofthe transportation potential V p from µ to P , ∇ V ∗ P ( X ) = d µ . Hence − λξ = d µ which impliesthat µ is (up to scale) the distribution of the stochastic discount factor ξ . Therefore, whenthere is a Yaari decision maker with reference measure µ in the economy, the stochasticdiscount factor should be distributed according to µ . This result is an extension of the well-known result that states that when there is a risk-neutral decision maker in the economy,the stochastic discount factor should equal one, that is, the risk-neutral probability shouldcoincide with the actuarial probability. To summarize, if a risk-sharing equilibrium existswith a Yaari risk-averse decision maker with reference measure µ , then µ coincides with thedistribution of the stochastic discount factor. Thereby, µ is related to the distribution ofthe state prices. UAL THEORY OF CHOICE 17 Relation with multi-attribute inequality measurement
The theory developed here has implications for inequality rankings of allocations of mul-tiple attributes (such as income, education, environmental quality, etc.) in a population.Atkinson [2] recognized the relevance of stochastic orderings to the measurement of inequal-ity and its foundation on principles such as the desirability of Pigou-Dalton transfers (alsoknown as Pigou-Dalton Majorization). Weymark [32] added to Pigou-Dalton Majorizationa principle of comonotonic independence, which he interpreted as neutrality to the sourceof variation in income, and obtained a class of social evaluation functions, which he calledgeneralized Gini evaluation functions. The functional form is identical to the decision func-tional derived independently on a continuous state space by [35]. Indeed, [14] notes theformal equivalence between the problem of decision under risk and the measurement ofinequality. The random vector of risks or prospects that we consider in the present workcan be interpreted as an allocation of multiple attributes over a continuum of individuals.With this interpretation, states of the world are identified with individuals in the populationand the decision function γ is interpreted as a social evaluation function. Law invariance(Yaari neutrality, i.e., insensitivity to relabelings of the states of the world) of the decisionfunctional is thus equivalent to anonymity of the social evaluation function. The rankingof ordinally equivalent allocations obtained through Pigou-Dalton Majorization (see[14])corresponds to the concave ordering discussed in Proposition 4. More precisely, the mean-preserving spread characterization (b) in Proposition 4 is equivalent to (d), which is theinfinite-dimensional analogue of multiplication by a doubly stochastic matrix. Hence, ourrisk averse multivariate Yaari decision functional can be interpreted as a social evaluationfunction for allocations of multiple attributes, which satisfies anonymity, monotonicity andPigou-Dalton Majorization in the sense of Theorem 3 in [14].The inequality literature achieves functional forms for social evaluation functions in themulti-attribute case by adding two distinct types of majorization principles that allow thecomparison of non-ordinally equivalent social evaluations. Tsui [28], [29] considers corre-lation increasing transfers. Gajdos and Weymark [11] extend generalized Gini social eval-uation functions to the multivariate case with a comonotonic independence axiom. Twoallocations are said to be comonotonic if all individuals are ranked identically in all at-tributes (i.e., the richest is also the most educated etc.), and the ranking between two comonotonic allocations is not reversed by the addition of a comonotonic allocation. Theyuse an attribute separability axiom (Axiom A in [34]) to reduce the dimensionality of theproblem via independence of the attributes. Specifically, Theorem 4 of [11] is a specialcase of our Theorem 2 when the attribute vector X and the reference distribution µ bothhave independent marginals. Our representation can also incorporate trade-offs between at-tributes and attitudes to correlations between attributes of the kind that are entertained in[29], but is not restricted to the latter. Correlation aversion would correspond to perceivedsubstitutability, but perceived complementarity can also be entertained in our approach,thereby circumventing Bourguignon and Chakravarty’s critique of correlation increasingmajorization (in [4]) based on the observation that “there is no a priori reason for a personto regard attributes as substitutes only. Some of the attributes can as well be complements”(p. 36). 7. Conclusion
We have developed concepts of quantiles and comonotonicity for multivariate prospects,thus allowing for the consideration of choice among vectors of payments in different curren-cies, at different times, in different categories of goods, etc. The multivariate concepts ofquantiles and comonotonicity were used to generalize Yaari’s dual theory of choice underrisk, where decision makers that are insensitive to hedging of comonotonic risks are shownto evaluate prospects using a weighted sum of quantiles. Risk averse decision makers wereshown to be characterized within this framework by a reference distribution, making thedual theory as readily applicable as expected utility. Risk attitudes were also analyzedfrom the point of view of a local utility function. Implications for the ranking of increasingrisk aversion is the topic of further research. Applications of the representation theoremto the measurement of multi-attribute inequality were also discussed. The flexibility in itshandling of attitudes to correlation between attributes is a promising feature of the decisionfunctional.
Appendix A. Proof of results in the main text
A.1.
Proof of Proposition 3.
By definition, there are two convex lower semi-continuous functions V and V and arandom vector U = d µ such that X = ∇ V ( U ) and Y = ∇ V ( U ) almost surely. Similarly, there are convex functions V and V and a random vector ˜ U such that Y = ∇ V ( ˜ U ) and Z = ∇ V ( ˜ U ). Now the assumptions on the absoluteUAL THEORY OF CHOICE 19continuity of µ and the distribution of Y imply that ∇ V is essentially unique. Hence, ∇ V = ∇ V and, therefore, U = ˜ U holds almost surely. It follows that X and Z are µ -comonotonic. (cid:3) A.2.
Proof of Proposition 4.
The equivalence between (a) and (b) is a famous result stated and extended bymany authors, notably Hardy, Littlewood, P´olya, Blackwell, Stein, Sherman, Cartier, Fell, Meyer and Strassen. SeeTheorem 2 of [27] for an elegant proof. We now show that (b) implies (c). Suppose (b) holds. As explained in Section2.1, there exists a map ζ such that ̺ µ ( X ) = E [ ζ ( X ) · X ] and ζ ( X ) = d µ . Now, E [ ζ ( X ) · X ] = E [ ζ ( X ) · E [ ˆ Y | X ]] = E [ ζ ( X ) · ˆ Y ] , which is less than ̺ µ ( Y ) . Next, we show that (c) implies (d). Indeed, the convex closure co ( equi ( Y ))of the equidistribution class of Y is a closed convex set and hence characterized by its support functional ̺ µ ( Y ).Therefore, X ∈ co ( equi ( Y )) is equivalent to E [ Z · X ] ≤ ̺ µ ( Y ) for all Z , which in turn is equivalent to ̺ µ ( X ) ≤ ̺ µ ( Y ).Now, we show that (d) implies (e). Indeed, if X ∈ co ( equi ( Y )), then there is a sequence ( Y nk ) k ≤ n of random vectorseach distributed as Y and positive weights α nk such that P nk =1 α nk = 1 and X = lim n →∞ P nk =1 α nk Y nk . Then,for any law invariant concave functional, we have Φ (cid:0)P nk =1 α nk Y nk (cid:1) ≤ P nk =1 α nk Φ (cid:0) Y nk (cid:1) = Φ( Y ) and the conclusionfollows by upper semi-continuity. Finally, (e) implies (a) because when L X is absolutely continuous with respect toLebesgue measure, for any bounded concave function f , X E f ( X ) is a law invariant concave upper semi-continuousfunctional. (cid:3) A.3.
Proof of Theorem 1.
Note first that γ defined for all prospects X by γ ( X ) = E [ Q X ( U ) · φ ( U )] for a function φ such that φ ( U ) ∈ ( R − ) d is Lipschitz and monotonic, so that Axioms 1 ′ and 2 ′ are satisfied for a preference relationrepresented by γ . Finally, comonotonic independence follows directly from the fact that for any two prospects X and Y , the generalized quantile functions Q X , Q Y and Q X + Y satisfy Q X + Y ( U ) = Q X ( U ) + Q Y ( U ). We nowshow this fact. By the definition of the generalized quantile functions, we have E [ Q X + Y ( U ) · U ] = sup ˜ U = d U E [( X + Y ) · ˜ U ] ≤ sup ˜ U = d U E [ X · ˜ U ] + sup ˜ U = d U E [ Y · ˜ U ] = E [ Q X ( U ) · U ] + E [ Q Y ( U ) · U ]. On the other hand, we alsohave E [ Q X + Y ( U ) · U ] = sup ˜ Z = d X + Y E [ ˜ Z · U ] ≥ E [( Q X ( U ) + Q Y ( U )) · U ] since by construction, Q X ( U ) = d X and Q Y ( U ) = d Y , and the desired equality follows.Conversely, we now prove that a preference relation % satisfying Axioms 1 ′ , 2 ′ and 3 ′ is represented by a functional γ defined for all prospect X by γ ( X ) = E [ Q X ( U ) · φ ( U )] for a function φ such that φ ( U ) ∈ ( R − ) d almost surely. ByAxiom 1 ′ , there exists a functional γ representing % and there is a point Z ∈ L d , where γ is Fr´echet differentiable withnon-zero gradient D . Let Q Z be the generalized quantile of Z relative to µ . There exists a U ∈ L d with distribution µ such that Z = Q Z ( U ) almost surely. Let X and Y be two prospects in L dd with µ -quantile functions Q X and Q Y respectively. By the definition of µ -comonotonicity, Q X ( U ), Q Y ( U ) and Z = Q Z ( U ) are µ -comonotonic. By Axiom 2 ′ , γ is law invariant, so that γ ( X ) ≥ γ ( Y ) is equivalent to γ ( Q X ( U )) ≥ γ ( Q Y ( U )). Hence, by Axiom 3 ′ , γ ( X ) ≥ γ ( Y )implies that for any 0 < ǫ ≤
1, we have γ ( ǫQ X ( U )+(1 − ǫ ) Z ) ≥ γ ( ǫQ Y ( U )+(1 − ǫ ) Z ). Hence, γ ( Z + ǫ ( Q X ( U ) − Z )) ≥ γ ( Z + ǫ ( Q Y ( U ) − Z )) and, therefore, γ ( Z ) + E [ D · ǫ ( Q X ( U ) − Z )] ≥ γ ( Z ) + E [ D · ǫ ( Q Y ( U ) − Z )] − o ( ǫ ), or, finally, E [ D · Q X ( U )] ≥ E [ D · Q Y ( U )].Suppose now that X and Y are two prospects such that E [ D · Q X ( U )] = E [ D · Q Y ( U )]. We shall show that γ ( Q X ( U )) = γ ( Q Y ( U )) and, hence, that γ ( X ) = γ ( Y ), thereby concluding that the functional X E [ D · Q X ( U )]represents % . Indeed, suppose that E [ D · Q X ( U )] = E [ D · Q Y ( U )]. We will show shortly that there exists a function φ such that E [ D · ∇ φ ( U )] > E [ D · ( Q X ( U ) + ǫ ∇ φ ( U ))] > E [ D · Q Y ( U )] and E [ D · ( Q X ( U ) − ǫ ∇ φ ( U ))] < E [ D · Q Y ( U )] for any ǫ >
0. Using the result above yields γ ( Q X ( U ) + ǫ ∇ φ ( U )) ≥ γ ( Q Y ( U )) and γ ( Q X ( U ) − ǫ ∇ φ ( U )) ≤ γ ( Q Y ( U )). Hence, γ ( Q X ( U )) = γ ( Q Y ( U )) by the continuity of γ . Let us now show that E [ D · ∇ φ ( U )] = 0 for all gradient functions ∇ φ yields a contradiction. Calling V Z the convex function such that Z = Q Z ( U ) = ∇ V Z ( U ) almost surely, D is the Fr´echet derivative of γ at Z = ∇ V Z ( U ). Hence, E [ D · ∇ φ ( U )] = 0implies that γ ( ∇ ( V Z ( U ) + ǫφ ( U ))) = γ ( ∇ V Z ( U )) + o ( ǫ ). This is true for all gradient functions ∇ φ and, in particular,for φ = ( V ǫ − V Z ) /ǫ , where V ǫ is such that Z ǫ = ∇ V ǫ ( U ) converges to Z in L . We then have γ ( Z ǫ ) − γ ( Z ) = o ( ǫ ) and,hence, D = 0, which contradicts Axiom 1 ′ . We have shown that % is represented by the functional X E [ D · Q X ( U )].As µ is absolutely continuous with respect to Lebesgue measure, D can be written as φ ( U ) for some function φ whichtakes values in ( R − ) d by Axiom 2 ′ . (cid:3) A.4.
Proof of Theorem 2.
That (a) implies (b) follows from Proposition 4. We now show that (b) implies (c).Axiom 4 implies that γ ( Q X + α ( ˜ X − Q X )) ≥ γ ( X ) for all α ∈ (0 ,
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