on the representation of incomplete preferences under...

On the Representation of Incomplete Preferences underUncertainty with Indecisiveness in Tastes and Beliefs

Gil Riella�

March 27, 2014

Abstract

Recently, there has been some interest on models of incomplete preferences un-der uncertainty that allow for incompleteness due the multiplicity of tastes and be-liefs. In particular, Galaabaatar and Karni (2013) work with a strict partial order andpresent axiomatizations of the Multi-prior Expected Multi-utility and the Single-priorExpected Multi-utility representations. In this paper we characterize both models us-ing a preorder as the primitive. In the case of the Multi-prior Expected Multi-utilityrepresentation, like all the previous axiomatizations of this model in the literature, ourcharacterization works under the restriction of a �nite prize space. In our axiomati-zation of the Single-prior Expected Multi-utility representation the space of prizes isa compact metric space. Later in the paper we present two applications of our char-acterization of the Single-prior Expected Multi-utility representation and discuss thenecessity of an axiomatization of the Multi-prior Expected Multi-utility model whenthe prize space is not �nite. In particular, we explain how the two applications wedevelop in this paper could be generalized to that model if we had such an axiomati-zation.

1 Introduction

Recently, there has been some interest on models of incomplete preferences under uncertaintythat allow for the multiplicity of tastes and beliefs. (See García del Amo and Ríos Insua(2002), Nau (2006) and Seidenfeld, Schervish, and Kadane (1995), for example.) In par-ticular, Nau (2006) and Galaabaatar and Karni (2013) present axiomatizations of a modelwhere the agent is represented by a set of pairs of priors and state independent utilityfunctions�the Multi-prior Expected Multi-utility representation. (See Seidenfeld et al. (1995)for the axiomatization of a relation that can be represented by a set of pairs of priors andalmost state independent utilities.) Also, axiomatizations of a model that allows only for

�Department of Economics, Universidade de Brasília. Email: [email protected].

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incompleteness due to the multiplicity of tastes�the so called Single-prior Expected Multi-utility representation�were presented by Ok, Ortoleva, and Riella (2012) and Galaabaatarand Karni (2013).

The primitive in Galaabaatar and Karni (2013) is a strict partial order and they workunder the restriction of a �nite prize space. In this paper, we present an axiomatization ofthe Multi-prior Expected Multi-utility representation using a preorder as the primitive ofthe model. We keep the assumption of a �nite prize space and show that three di¤erentproperties can be used to obtain such a model: the �rst is a weaker and simpler versionof Nau�s Strong State-independence postulate. The second is the natural adaptation ofGalaabaatar and Karni�s Dominance axiom to the case of a preorder. The third one is aproperty we call Mixture Monotonicity.

Galaabaatar and Karni (2013) also show that when we add a property that can beinterpreted as a postulate of completeness of beliefs to their axiomatization of the Multi-prior Expected Multi-utility representation, we obtain a characterization of the Single-priorExpected Multi-utility model. We show that this is also the case when we use a preorder asthe primitive. Moreover, we show that in the case of the Single-prior Expected Multi-utilityrepresentation the restriction of a �nite prize space is not necessary. That is, we show thatif we add a completeness of beliefs axiom to our axiomatization of the Multi-prior ExpectedMulti-utility representation, we obtain the Single-prior Expected Multi-utility model even ifwe allow for a general compact metric prize space, .

Although the fact that we can obtain an axiomatization of the Single-prior ExpectedMulti-utility representation even when the prize space is an arbitrary compact metric spaceis mainly a technical contribution, this is likely to be important for some applications of themodel. We illustrate this point with two examples. We �rst provide a result in the spiritof the main result in Gilboa, Maccheroni, Marinacci, and Schmeidler (2010) for the Single-prior Expected Multi-utility model. Later, we generalize the main result in Cerreia-Vioglio,Dillenberger, and Ortoleva (2014) to the case of multiple tastes under uncertainty. Bothresults rely heavily on the fact that we have an axiomatization of the Single-prior ExpectedMulti-utility representation when the prize space is metric and compact.

We end the paper with a discussion section. First, we make a comment about thenecessity of an axiomatization of the Multi-prior Expected Multi-utility representation for anon �nite prize space. In particular, we explain how the two applications developed in thepaper could easily be generalized to the Multi-prior Expected Multi-utility model if we hadan axiomatization of that model for a compact metric prize space. Second, we present a newaxiomatization of the Single-prior Expected Multi-utility representation that weakens theassumption that there exist best and worst constant acts, which was used in our previousaxiomatization of the model.

In the next section we introduce the setup of the paper and recall Ok et al. (2012)�s ax-iomatization of the Additively Separable Expected Multi-utility representation. In Section 3,we present our axiomatization of the Multi-prior Expected Multi-utility model. In Section 4,we study the Single-prior Expected Multi-utility model. We �rst recall Ok et al. (2012)�s ax-iomatization of this model and, after that, we use this result to show that, as in Galaabaatarand Karni (2013), if we add a completeness of beliefs axiom to the axioms that characterize

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the Multi-prior Expected Multi-utility representation, we obtain an axiomatization of theSingle-prior Expected Multi-utility model. We note that this is true even when we generalizethe prize space to be a compact metric space. In Section 5, we present the two applicationsof the result in Section 4 we discussed above. We end the paper with a discussion in Section6. Detailed proofs of the results in the main text appear in the appendix.

2 The Additively Separable ExpectedMulti-utility Rep-resentation

In this section we revisit a result about the representation of incomplete preferences underuncertainty that has appeared recently in the literature. We recall the axiomatization ofthe Additively Separable Expected Multi-utility representation that appeared in Ok et al.(2012). (The same result was axiomatized by Nau (2006), under the restriction of a �niteprize space. See also García del Amo and Ríos Insua (2002) for a related result when boththe prize space as well as the state space are subsets of euclidean spaces.)

2.1 Setup

Let X be a compact metric prize space. Occasionally, we will assume that X is a �niteset. We will denote the elements of X by x; y; z, etc.. We will write �(X) to representthe space of Borel probability measures on X. We metrize �(X) in such a way that metricconvergence coincides with weak convergence of Borel probability measures. The elements of�(X) are called lotteries and denoted by p; q; r, etc.. The degenerate lottery that pays prizex 2 X with probability one is denoted by �x. The linear space of all continuous real mapson X is denoted by C(X). Throughout the exposition we metrize C(X) by the supnorm.The expectation of any function u 2 C(X) with respect to a probability measure p 2 �(X)is denoted by Ep(u). That is,

Ep(u) :=ZX

udp:

Let S be a �nite state space. We denote the space of probability measures on S by �(S).An act is a function that maps the state space S into the space �(X) of lotteries. We denotethe space of all acts by F . That is, F := �(X)S. The elements of F are denoted by f; g; h,etc.. We follow the usual abuse of notation and write simply p to represent the constantact that returns the lottery p in every state of nature. Given that, we often write f(s) torepresent the constant act that returns the lottery f(s) in every state of nature. For anysubset T of S, and any two acts f and g, we write fTg to represent the act h such thath(s) = f(s) for every s 2 T and h(s) = g(s) for every s 2 S n T . Our primitive is a preorder(i.e., a re�exive and transitive binary relation) %� F � F . (As usual, the asymmetric partof this preorder is denoted by �, and its symmetric part by �.)

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2.2 Representation Theorem

The following are two standard postulates imposed on %.

Axiom 1 (Independence). For any acts f; g and h in F , and � 2 (0; 1),

f % g implies �f + (1� �)h % �g + (1� �)h.

Axiom 2 (Continuity). % is a closed subset of F � F .1

Ok et al. (2012) prove the following result.

Theorem 1 (Ok et al. (2012), Theorem 0). A preorder %� F � F satis�es Independenceand Continuity if, and only if, there exists a nonempty subset U of C(X�S) such that f % gi¤ X

s2SEf(s)(U(:; s)) �

Xs2S

Eg(s)(U(:; s)) for every U 2 U ,

for any acts f and g in F .

The same result is proved in Nau (2006) under the restriction of a �nite prize space X.García del Amo and Ríos Insua (2002) prove a related result when X and S are subsets ofEuclidean spaces.

3 The Multi-prior Expected Multi-utility Representa-tion

In this section we present an axiomatization of the Multi-prior Expected Multi-utility model.The Multi-prior Expected Multi-utility model was axiomatized �rst by Nau (2006) and laterby Galaabaatar and Karni (2013). (See also Seidenfeld et al. (1995) for a related result.)Both papers work under the restriction of a �nite prize space and the existence of best andworst constant acts. The axiomatization in Nau (2006) works with a preorder as the primitiveof the model and relies on a postulate called Strong State-independence. This postulate isa stronger version of the property Mixture Separability, discussed below. Galaabaatar andKarni (2013) use a strict partial order as the primitive of the model and axiomatize the Multi-prior Expected Multi-utility model with a version of the postulate One Side Monotonicity,discussed below. Like Nau (2006), we work with a preorder as our primitive, and we discussthree di¤erent postulates that can be used in order to obtain an axiomatization of the Multi-prior Expected Multi-utility model.

In this section, we impose the additional restriction that X be a �nite set. We imposealso the following restriction on %.

1That is, for any convergent sequences (fm) and (gm) in F , with fm % gm for each m, we have lim fm %lim gm. We note that in the results of this paper that make use of a �nite prize space X this property canbe replaced by the weaker requirement that the sets f� : �f + (1 � �)g % hg and f� : h % �f + (1 � �)ggare closed in [0; 1], for any f; g and h in F .

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Axiom 3 (Best and Worst). There exist x0 and x1 in X such that, for any x 2 X;

�x1 % �x % �x0,

and �x1 � �x0.

We will investigate the consequences of the following postulates when imposed on %.

Axiom 4 (Mixture Separability). For any p; q; r 2 �(X), f; g 2 F and � 2 [0; 1], if f % gand �p+ (1� �)f % �q + (1� �)g, then

�(pTr) + (1� �)f % �(qTr) + (1� �)g;

for any T � S:

Axiom 5 (One Side Monotonicity). For any two acts f and g, f(s) % g for every s 2 Simplies that f % g:

Axiom 6 (Mixture Monotonicity). If f; g; h and j in F , and � 2 [0; 1] are such that �h(s)+(1� �)f % �j(s) + (1� �)g for every s 2 S, then �h+ (1� �)f % �j + (1� �)g.

The Mixture Separability axiom is a weaker and simpler version of Nau (2006)�s StrongState-independence postulate.2 One Side Monotonicity is Galaabaatar and Karni (2013)�sDominance postulate adapted to preorders. In the presence of Independence and Continuityeach one of these properties implies the standard Monotonicity axiom, presented below.

Axiom 7 (Monotonicity). For any two acts f and g, f(s) % g(s) for all s 2 S implies thatf % g:

We can now state the following lemma, which is valid even when X is an arbitrarycompact metric space.

Lemma 1. Suppose % satis�es Independence and Continuity. Then, % satis�es Mixture Sep-arability, or One Side Monotonicity, or Mixture Monotonicity only if it satis�es Monotonic-ity.

The following is the main result of this section.

Theorem 2. The following statements are equivalent.2In our notation Nau�s Strong State-independence axiom can be written as:

Strong State-independence. For any p; q; r 2 �(X), f; g 2 F , T; T � S, a; b 2 (0; 1] and � 2 [0; 1], iff % g, (�x1T�x0) % a�x1 + (1� a)�x0 , b�x1 + (1� b)�x0 % �x1 T �x0 and �pTr+ (1� �)f % �qTr+ (1� �)g,then �pT r+ (1� �)f % �qT r+ (1� �)g, for � = 1, if � = 1 and otherwise for all � such that �

1�� 1��

ab :

Our Mixture Separability Axiom is implied by the postulate above when T = S and a = b = 1.

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1. There exists a nonempty subsetM of �(S)�C(X) with u(x1) = 1 and u(x0) = 0 forevery (�; u) 2M such that f % g i¤X

s2S�(s)Ef(s)(u) �

Xs2S

�(s)Eg(s)(u) for every (�; u) 2M,

for any acts f and g in F ;

2. % satis�es Independence, Continuity, Best and Worst, and Mixture Separability;

3. % satis�es Independence, Continuity, Best and Worst, and One Side Monotonicity;

4. % satis�es Independence, Continuity, Best and Worst, and Mixture Monotonicity.

Although two di¤erent axiomatizations of the Multi-prior Expected Multi-utility modelhave already appeared in the literature, we believe the result above is of interest for tworeasons. First, as we have pointed out before, Galaabaatar and Karni (2013) use a strictpartial order as the primitive of the model, so that Theorem 2, although not entirely sur-prising, con�rms that a similar result can be obtained under the more common option ofhaving a preorder as the primitive. Second, Galaabaatar and Karni (2013)�s proof is basedon a constructive argument and uses convex analysis only indirectly. In contrast, our proofis more standard and it is heavily based on convex analysis. It is possible that a reader fa-miliar with the literature about incomplete preferences under risk and uncertainty will �ndour argument easier to follow.

Intuition for the proof of Theorem 2. Checking that 1 implies 2, 3 and 4 is straightforward.We give the intuition why 2, 3 and 4 imply 1. Suppose that % satis�es all the axioms in 2,3or 4. By Theorem 1, we know that % admits an Additively Separable Expected Multi-utilityRepresentation U . Without loss of generality, we can assume that U is closed and convexand we can normalize all functions in U so that it is as if x0 and x1 have state-independentutilities of 0 and 1, respectively.3 Suppose now that U has an extreme point U� that cannotbe written as a prior utility pair (�; u). We �rst use a separation argument to construct actsf and g such that X

s2SEf(s) (U� (:; s)) =

Xs2S

Eg(s) (U� (:; s))

and Xs2S

Ef(s) (U (:; s)) >Xs2S

Eg(s) (U (:; s)) ;

for every U 2 Un fU�g. Without loss of generality, we may assume that f is a constant act.Since U� cannot be written as a prior utility pair, we can �nd lotteries p; q and T � S suchthat X

s2SEp (U� (:; s)) >

Xs2S

Eq (U� (:; s)) ;

3Formally, the set U can be normalized so that U(x0; s) = 0, for all s 2 S andP

s2S U(x1; s) = 1, for allU 2 U .

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but Xs2T

Ep (U� (:; s)) <Xs2T

Eq (U� (:; s)) :

Using a continuity argument, we can show that for � 2 (0; 1), � small enough,Xs2S

E�p+(1��)f(s)U (:; s) >Xs2S

E�q+(1��)g(s) (U (:; s))

for every U 2 U . That is,

�p+ (1� �)f � �q + (1� �)g:

But observe thatXs2S

E�(pTq)+(1��)f(s)U� (:; s) <Xs2S

E�q+(1��)g(s) (U� (:; s)) :

That is, it is not true that

�(pTq) + (1� �)f % �q + (1� �)g:

This contradicts the Mixture Separability and the Mixture Monotonicity axioms. Becausewe can choose f to be constant, this is also a contradiction to One Side Monotonicity. k

4 The Single-prior Expected Multi-utility Representa-tion

Galaabaatar and Karni (2013) show that when we add an axiom that can be interpreted asa completeness of beliefs property to their version of Theorem 2, we obtain a representationwith a unique prior and a set of utilities�the so called Single-prior Expected Multi-utilitymodel. In this section, we show that this is also true when we use a preorder as primitive.Moreover, in the case of the Single-prior Expected Multi-utility representation, we can provethat this characterization works even when we assume a compact metric prize space. Thatis, we can drop the assumption of a �nite prize space. From now on, we get back to theassumption that X is a compact metric space. We begin by recalling a result from Ok et al.(2012).

For each probability measure � on S and any act f 2 F , let f� be the constant act thatreturns the lottery

Ps2S �(s)f(s) in every state of nature. Ok et al. (2012) introduced the

following postulate.

Axiom 8 (Reduction). For any act f 2 F , there exists �f 2 �(S) such that f�f � f .

Ok et al. (2012) prove the following result.

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Theorem 3 (Ok et al. (2012), Theorem 1). A preorder %� F � F satis�es Independence,Continuity, and Reduction if, and only if, there exists a nonempty subset U of C(X) and aprior � 2 �(S) such that f % g i¤X


Xs2S

�(s)Eg(s)(u) for every u 2 U ,


Although the theorem above characterizes the Single-prior Expected Multi-utility repre-sentation in quite a general setup, the Reduction axiom makes use of an existential quanti�erand, therefore, it is not testable, in principle. Galaabaatar and Karni (2013) showed thatif we add a postulate that could be interpreted as completeness of beliefs to their charac-terization of the Multi-prior Expected Multi-utility representation, then we obtain anothercharacterization of the Single-prior Expected Multi-utility model. As it was the case be-fore, Galaabaatar and Karni (2013) worked with a strict partial order as their primitive andimposed that the prize space was �nite and contained best and worst elements. We nowshow that if we add a completeness of beliefs postulate to our axiomatization of the Multi-prior Expected Multi-utility representation, we also get a characterization of the Single-priorExpected Multi-utility model. Moreover, we also show that we do not need the restrictionthat the prize space be �nite for that. Instead, our characterization is valid for an arbitrarycompact metric prize space.

Consider the following postulate.

Axiom 9 (Complete Beliefs). For any p; q 2 �(X), � 2 [0; 1] and T � S, if p % q, then

pTq % �p+ (1� �) q or �p+ (1� �) q % pTq.

We can now state the following result.

Theorem 4. A preorder %� F � F satis�es Independence, Continuity, Best And Worst,One Side Monotonicity and Complete Beliefs if, and only if, there exists a nonempty subsetU of C(X) with u(x1) = 1 and u(x0) = 0 for every u 2 U , and a prior � 2 �(S) such thatf % g i¤ X


Xs2S



As we have discussed before, the result above di¤ers from the related result in Galaabaatarand Karni (2013) in two aspects. First, our primitive is a preorder, not a strict partial order,as in their case. Second, the result is obtained under the more general assumption that Xis a compact metric space, not necessarily a �nite set. As we will see in the next section,although this is a technical contribution, it is something essential for some applications ofthe model.

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Intuition for the proof of Theorem 4. Checking that the necessity part of the proof works isstraightforward. For the su¢ ciency part, suppose �rst that X is �nite. In this case, Theorem2 applies and we know that % admits a Multi-prior Expected Multi-utility RepresentationMsuch that, for every (�; u) 2M, we have u(x1) = 1 and u(x0) = 0. If there exist (�1; u1) and(�2; u2) inM with �1 6= �2, then we can easily �nd T � S and � 2 [0; 1] such that �x1T�x0is not comparable to ��x1 + (1 � �)�x0, which contradicts Complete Beliefs. We concludethat % admits a Single-prior Expected Multi-utility Representation. Now suppose that X isa compact metric space, not necessarily �nite, and �x any f 2 F . Let Y := f(S)[f�x1 ; �x0gand look at the restriction of % to acts g 2 F such that g(S) � conv(Y ).4 This restriction�ts the �nite prize space case we investigated above and, consequently, admits a Single-prior Expected Multi-utility representation. By Theorem 3, this implies that there exists� 2 �(S) such that f � f�. Since f was chosen arbitrarily in our analysis, we conclude that% satis�es the Reduction axiom. Now we can apply Theorem 3 once more to conclude that% admits a Single-prior Expected Multi-utility representation. k

Remark 1. From Theorem 2, we know that when X is �nite and % satis�es the otherpostulates in the statement of Theorem 4 One Side Monotonicity is equivalent to MixtureSeparability and Mixture Monotonicity. Since One Side Monotonicity only plays a role inthe proof of Theorem 4 when X is �nite, this implies that we obtain the same result if wereplace it by Mixture Separability or Mixture Monotonicity.

Remark 2. It is clear from the argument above that we could restrict the lotteries p and qin the statement of the Complete Beliefs axiom to be such that supp(p)[ supp(q) � fx0; x1g.This observation will be useful for the proof of Theorem 6 below.

5 Applications

In this section, we present two applications of Theorem 4. Both results utilize a compactmetric prize space and cannot be obtained otherwise.

5.1 Objective and Subjective Rationality in a Single-prior Multi-ple Tastes Model

Gilboa et al. (2010) prove an interesting bridge between a pair of relations where one of themadmits a Multi-prior Expected Single-utility representation à la Bewley (2002) and the otheradmits a representation à la Gilboa and Schmeidler (1989). Formally, they characterizewhen the two relations share the same utility over lotteries and the same set of priors.We can perform a similar exercise starting from the Single-prior Expected Multi-utilityrepresentation in a world with monetary lotteries.

In this section, we will specialize X to be a closed and bounded interval [a; b] � R. Allother de�nitions remain as in the previous sections. We will work with a pair of preorders< and <. We will introduce the following new postulates for a preorder %:

4Notation: For any subset Y of a vector space, we write conv(Y ) to represent the convex hull of Y .

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Axiom 10 (Certainty Dominance). For any x; y 2 X, �x % �y i¤ x � y.5

Axiom 11 (Certainty Continuity). For any f 2 F , the sets fx 2 X : �x % fg and fx 2 X :f % �xg are closed.

We will also work with the following postulates linking < and <.

Axiom 12 (Consistency). For every pair of acts f and g in F , f < g implies f<g.

Axiom 13 (Caution). For every f 2 F and x 2 X, if it is not true that f < �x, then �x<f .

Axiom 14 (Default to Certainty). For every f 2 F and x 2 X, if it is not true that f < �x,then �x�f .

Now, given a prior � on S, a strictly increasing function u 2 C(X), and an act f 2 F ,de�ne x�;uf to be the certainty equivalent of f with respect to � and u. That is, de�ne x�;ufby

x�;uf := u�1

Xs2S

�(s)Ef(s)(u)

!.

We can now state the following result:

Theorem 5. The following statements are equivalent.

1. There exists a nonempty set of strictly increasing functions U � C(X) and a prior� 2 �(S) such that, for any acts f and g in F , f < g i¤X


Xs2S

�(s)Eg(s)(u) for every u 2 U ;

and f<g i¤infu2U

x�;uf � infu2U

x�;ug ;

2. < satis�es Independence, Continuity, One Side Monotonicity, Complete Beliefs andCertainty Dominance, < is complete and satis�es Certainty Continuity, and together< and < satisfy Consistency and Default to Certainty;

3. < satis�es Independence, Continuity, One Side Monotonicity, Complete Beliefs andCertainty Dominance, < is complete and satis�es Certainty Continuity and CertaintyDominance, and together < and < satisfy Consistency and Caution.

Remark 3. An observation similar to Remark 2 applies here. It is enough to impose theComplete Beliefs axiom only to lotteries p and q such that supp(p) [ supp(q) � fa; bg. Thisobservation will be used in the proof of Theorem 6 below.

5Our �rst version of the main result in this section used a more technical axiom based on First OrderStochastic Dominance. We owe Cerreia-Vioglio et al. (2014) for the observation that Certainty Dominanceis enough to do the job.

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5.2 The Minimum Certainty Equivalent Representation

In this section we generalize the main result in Cerreia-Vioglio et al. (2014) to the case of asingle prior and a set of utilities. Again, we specialize X to be a closed and bounded interval[a; b] � R. Let %� F �F be a complete preorder. We will consider the following additionalpostulates on %:

Axiom 15 (Negative Certainty Independence). For all f; g 2 F , x 2 X and � 2 [0; 1],f % �x implies that �f + (1� �)g % ��x + (1� �)g.

Axiom 16 (Best and Worst Mixture Consistency). For every h; j 2 F , �; 2 (0; 1] andf; g 2 F such that supp(f(s)) [ supp(g(s)) � fa; bg for every s 2 S, if �f + (1 � �)h ��g + (1� �)h, then f + (1� )j % g + (1� )j.

Axiom 17 (One Side Mixture Monotonicity). If f , g and h in F , and � 2 [0; 1] are suchthat �h(s) + (1� �)f % g for every s 2 S, then �h+ (1� �)f % g.

We can now state the following result:

Theorem 6. A complete preorder %� F � F satis�es Continuity, Certainty Dominance,Negative Certainty Independence, Best and Worst Mixture Consistency, and Mixture OneSide Monotonicity if, and only if, there exists a nonempty set of strictly increasing functionsU � C(X) and a prior � 2 �(S) such that, for any acts f and g in F , f % g i¤

infu2U

x�;uf � infu2U

x�;ug ,

and the function V : F ! R de�ned by V (f) := infu2U x�;uf , for every f 2 F is continuous.

6 Discussion

6.1 Multi-prior Expected Multi-utility with a Compact MetricPrize Space

The analysis in this paper leaves an important question unanswered. The question concernsthe axiomatization of the Multi-prior Expected Multi-utility model when the prize spaceis not �nite. The arguments used in the proof of Theorem 2 in this paper and in thecharacterizations of the Multi-prior Expected Multi-utility model in the other papers arevery �nite dimensional and it is not clear how to generalize them to the case of a compactmetric prize space, for example. Perhaps an argument similar to the one used in the proofof Theorem 4 might work, but we were not able to provide one.

We must emphasize that this is not only a technical question. The results in Section 5show that such a generalization is essential for some applications of the model. For example,a version of Theorem 5 can easily be obtained for the Multi-prior Expected Multi-utilitymodel, except for the fact that we do not have an axiomatization of this model when the

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prize space is not �nite. Formally, under the same setup as in Section 5, we can show thefollowing result:

Proposition 1. The following statements are equivalent.

1. There exists a nonempty subsetM of �(S)�C(X) with u being strictly increasing forevery (�; u) 2M such that, for any acts f and g in F , f < g i¤X


Xs2S

�(s)Eg(s)(u) for every (�; u) 2M,

f<g i¤inf

(�;u)2Mx�;uf � inf

(�;u)2Mx�;ug ;

2. < admits a Multi-prior Expected Multi-utility representation and satis�es CertaintyDominance, < is complete and satis�es Certainty Continuity, and together < and <satisfy Consistency and Default to Certainty;

3. < admits a Multi-prior Expected Multi-utility representation and satis�es CertaintyDominance, < is complete and satis�es Certainty Continuity and Certainty Domi-nance, and together < and < satisfy Consistency and Caution.

The proof of this result is very similar to the proof of Theorem 5 and, therefore, it isomitted. Theorem 6 would also be easily generalizable to the case of multiple beliefs andtastes if we had an axiomatization of the Multi-prior Expected Multi-utility model with acompact metric prize space. All we need in order to be able to use the same argument inthe proof of Theorem 6 is that the relation <� F �F de�ned by f < g i¤ �f + (1� �)h %�g+(1��)h for every h 2 F and every � 2 (0; 1] admit a Multi-prior Expected Multi-utilityrepresentation.

6.2 The Single-prior Expected Multi-utility Representation with-out Best and Worst

The Best and Worst postulate, specially when used together with Independence, reduces thedegree of incompleteness of the relation % considerably. On the other hand, the axiomatiza-tion of the Single-prior Expected Multi-utility representation in Ok et al. (2012) makes useof an existential quanti�er and it is, in principle, not testable. We now obtain an axioma-tization of the Single-prior Expected Multi-utility representation that replaces the existenceof a best and a worst act by the existence of two constant acts that are strictly comparable.

Consider the following postulates.

Axiom 18 (Constant Nontriviallity). There exists lotteries p; q 2 �(X) such that p � q.

Axiom 19 (Reduction Invariance). If p; q 2 �(X), T � S and � 2 [0; 1] are such that p � qand pTq � �p+ (1� �)q, then pT q � �p+ (1� �)q, for every p; q 2 �(X).

12

We can now state the following result.

Theorem 7. A preorder %� F �F satis�es Independence, Continuity, Monotonicity, Con-stant Nontriviality, Complete Beliefs and Reduction Invariance if, and only if, there exists anonempty subset U of C(X) and a prior � 2 �(S) such that f % g i¤X


Xs2S


for any acts f and g in F , and Ep(u) � Eq(u) for every u 2 U , with strict inequality forsome u 2 U , for some pair of lotteries p and q in �(X).

The main advantages of the theorem above over Theorem 4 are that it weakens OneSide Monotonicity to Monotonicity and Best and Worst to Constant Nontriviality. Thedownside is that it imposes the additional Reduction Invariance postulate, which relies on acondition based on an indi¤erence. Therefore, although it is in principle testable, in practiceit may be di¢ cult to test this property. We note that, except for Continuity and ConstantNontriviality, the other axioms in the statement of Theorem 7 are testable.

A Proofs

A.1 Proof of Lemma 1

It is clear that Mixture Monotonicity is stronger than Monotonicity. Suppose now that %satis�es Mixture Separability, Independence and Continuity. By Theorem 1, we know that %has an additively separable expected multi-utility representation. We can prove the followingclaim:

Claim 1. For any p; q 2 �(X), if p % q, then pfsgf % qfsgf for any f 2 F and s 2 S.

Proof of Claim. Suppose p % q and �x s 2 S and f 2 F . By Independence 12p + 1

2p %

12q + 1

2p. Since p % p, Mixture Separability implies that 1

2(pfsgp) + 1

2p % 1

2(qfsgp) + 1

2p. By

Independence, p % qfsgp. It is easy to see that the fact that % has an additively separableexpected multi-utility representation now implies that pfsgf % qfsgf . k

Now suppose f and g are acts such that f(s) % g(s) for all s 2 S. By the claim above,f = ffs1gf % gfs1gf % gfs1; s2gf % ::: % g:Now suppose that % satis�es One Side Monotonicity, Independence and Continuity. Sup-

pose f and g are acts such that f(s) % g(s) for all s 2 S. De�ne the act ~g by

~g(s�) :=1

jSj � 1Xs 6=s�

g(s).

Note that 1jSjg +

jSj�1jSj ~g is the constant act

1jSjPg(s). By Independence, we have that

1jSjf(s) +

jSj�1jSj ~g(s) %

1jSjg(s) +

jSj�1jSj ~g(s) for every s 2 S. Since

1jSjg +

jSj�1jSj ~g is a constant

13

act, One Side Monotonicity implies that 1jSjf +

jSj�1jSj ~g %

1jSjg +

jSj�1jSj ~g. Another application

of Independence gives us that f % g.6 �

A.2 Proof of Theorem 2

[Necessity] Suppose that % has a Multi-prior Expected Multi-utility representationM suchthat, for any (�; u) 2 M, u (x1) = 1 � u (x) � 0 = u (x0) for every x 2 X. It is clearthat such a representation satis�es Best and Worst, and, by Theorem 1, it also satis�esContinuity and Independence. Now suppose the acts f; g; h; j and � 2 (0; 1] are such that�h(s) + (1 � �)f % �j(s) + (1 � �)g for all s 2 S. Fix a generic pair (�; u) 2 M. Theassumption above implies thatX

s2S� (s)E�h(s�)+(1��)f(s)(u) �

Xs2S

� (s)E�j(s�)+(1��)g(s)(u);

for every s� 2 S. But this now implies thatXs2S

�(s)E�h(s)+(1��)f(s)(u) �Xs2S

� (s)E�j(s)+(1��)g(s)(u).

Since the above is true for all (�; u) 2M, we conclude that �h+ (1� �)f % �j + (1� �)g.That is, % satis�es Mixture Monotonicity. A similar reasoning shows that % also satis�esMixture Separability and One Side Monotonicity.

[Su¢ ciency] By Theorem 1, % has an additively separable expected multi-utility rep-resentation U . Given Lemma 1, we can normalize every U 2 U so that x0 and x1 havestate-independent utilities of 0 and 1, respectively. That is, we can normalize each U 2 U sothat U(x0; s) = 0, for every s 2 S and

Ps2S U(x1; s) = 1.

7 We can also assume, without lossof generality, that U is closed and convex. Let�s agree to say that a given function U 2 Uis state independent if, for every p; q 2 �(X),

Ps2S Ep(U(:; s)) �

Ps2S Eq(U(:; s)) implies

that Ep(U(:; s�)) � Eq(U(:; s�)) for every s� 2 S. Now, for each U 2 U , de�ne a probabilitymeasure �U on S by �U(s) := U(x1; s), for every s 2 S. De�ne also a function uU : X ! Rby uU(x) :=

Ps2S U(x; s) for every x 2 X. We need the following claim.

Claim 1. A function U 2 U is state independent if, and only if, U(x; s) = �U(s)uU(x) forevery x 2 X and s 2 S.

6Here we are using the fact that it is clear from the additively separable representation of % that bothdirections of the Independence axiom are true. That is, it is also true that, for any acts f , g and h in F ,and any � 2 (0; 1], �f + (1� �)h % �g + (1� �)h implies f % g.

7By Lemma 1, for every s 2 S and x 2 X, �x1fsg�x0 % �xfsg�x0 % �x0 , which implies that U(x1; s) �U(x; s) � U(x0; s) for every s 2 S and every U 2 U . Also, we can ignore the functions U 2 U such thatU(:; s) is constant for every s 2 S, since they do not matter for additively separable expected multi-utilityrepresentations. Now, for each U 2 U , de�ne ~U by ~U(:; s�) := U(:;s�)�U(x0;s�)P

s2S U(x1;s)�U(x0;s) , for every s

� 2 S. Noticethat ~U(x0; s) = 0 for every s 2 S, and

Ps2S

~U(x1; s) = 1. Moreover, it is clear that for every pair of acts fand g in F we have

Ps2S Ef(s)(U(:; s)) �

Ps2S Eg(s)(U(:; s)) i¤

Ps2S Ef(s)( ~U(:; s)) �

Ps2S Eg(s)( ~U(:; s)).

14

Proof of Claim. It is clear that if U(x; s) = �U(s)uU(x) for every x 2 X and s 2 S, thenU is state independent. Suppose now that U is state independent. That is, suppose that,for every s 2 S and lotteries p and q in �(X), Ep(uU) � Eq(uU) implies that Ep(U(:; s)) �Eq(U(:; s)). Given the uniqueness properties of expected-utility representations and that Uis normalized so that x1 and x0 have state independent utilities of one and zero, respectively,this can happen only if, for each s 2 S, either U(:; s) is constant and equal to zero, orU(:; s) is a positive a¢ ne transformation of uU . In the �rst case, we have �U(s) = 0,which implies that �U(s)uU(x) = 0 = U(x; s) for every x 2 X. In the second case, wehave U(x; s) = �su

U(x) + �s, for some �s 2 R++ and �s 2 R, for every x 2 X. SinceU(x0; s) = u

U(x0) = 0, we must have � = 0. Now, from U(x1; s) = �U(s) and uU(x1) = 1

we get that � = �U(s). k

We will now show that every extreme point U of U is state independent. For that,suppose that U� is an extreme point of U and U� is state-dependent. By the Straszewicz�sTheorem, we may assume, without loss of generality, that U� is in fact an exposed point ofU .8 So, there exists a �nite signed measure � on X � S and � 2 R such thatX

(x;s)2X�S

�(x; s)U�(x; s) = � >X

(x;s)2X�S

�(x; s)U(x; s);

for every U 2 U n fU�g. Consider now a measure � such that � (fx1; sg) = �� for everys 2 S, and � is identically null in (X � S) n (fx1g � S). De�ne �0 := �+ � and observe thatX

(x;s)2X�S

�0(x; s)U�(x; s) =X

(x;s)2X�S

�(x; s)U�(x; s) +X

(x;s)2X�S

�(x; s)U�(x; s)

= � � �= 0:

A similar reasoning shows thatP

(x;s)2X�S �0(x; s)U(x; s) < 0 for every U 2 Un fU�g. Fi-

nally, consider a measure ~� such that ~� (f(x0; s)g) = ��0 (X � fsg) for every s 2 S, and ~�is identically null elsewhere. De�ne �00 by �00 := �0 + ~�. Observe that, for any U 2 U ;X

(x;s)2X�S

�00(x; s)U(x; s) =X

(x;s)2X�S

�0(x; s)U(x; s) +X

(x;s)2X�S

~�(x; s)U(x; s)

=X

(x;s)2X�S

�0(x; s)U(x; s):

Also, �00 (X � fsg) = 0 for every s 2 S. By the Jordan decomposition we can �nd posi-tive measures �+ and �� such that �00 = �+ � ��. Moreover, �+ can be chosen so that�+ (X � fsg) > 0 for every s 2 S.9 Let s� be such that �+ (X � fs�g) � �+ (X � fsg) forevery s 2 S. Without loss of generality, we can assume that �+ (X � fs�g) = 1. Now, for

8An exposed point x of a convex set C is an extreme point of C that has a supporting hyperplane whoseintersection with C is only x. The mentioned theorem says that in Rn the set of exposed points of a closedand convex set C is a dense subset of the set of extreme points of C:

9Just apply the Jordan decomposition state by state.

15

each s 2 S, de�ne a probability measure +s on X by

+s (x) =�+ (f(x; s)g)�+ (X � fsg) ,

for every x 2 X. De�ne �s analogously. Now, for every s 2 S, de�ne probability measures'+s ; '

�s on X by

'+s = �s +s + (1� �s) �s

and'�s = �s

�s + (1� �s) +s

where

�s =�+ (X � fsg) + 1

2:

We note that, for each x 2 X and s 2 S,

'+s (x)� '�s (x) = �00 (f(x; s)g) .

But then, if we de�ne acts f and g by

g(s) = '+s and f(s) = '�s , for every s 2 S,

we have that Xs2S

Eg(s)(U(:; s))�Xs2S

Ef(s)(U(:; s)) =X

(x;s)2X�S

�00(x; s)U(x; s);

for every U 2 U . That is, we have just found two acts f and g such thatXs2S

Eg(s)(U(:; s))�Xs2S

Ef(s)(U(:; s)) < 0;

for every U 2 U n U� andXs2S

Eg(s)(U�(:; s))�Xs2S

Ef(s)(U�(:; s)) = 0.

Without loss of generality, we can assume that g is a constant act.10 Since U� is state-dependent, we can �nd p; q 2 �(X) and T � S such thatX

s2SEp(U�(:; s)) >

Xs2S

Eq(U�(:; s))

and Xs2T

Ep(U�(:; s)) <Xs2T

Eq(U�(:; s)):11

10For every act f 2 F , there exists another act ~f and � 2 (0; 1) such that �f + (1� �) ~f is constant. Seethe proof of Lemma 1 for the details.

16

By continuity, the two inequalities above are still true in some open neighborhood N (U�)of U�. For any act f , de�ne

U (f) :=Xs2S

Ef(s)(U(:; s)).

Since U n N (U�) is a compact set and

U (f)� U (g) > 0;

for every U 2 U n N (U�), we know that there exists � 2 (0; 1) such that

U (f)� U (g) > �

1� �;

for every U 2 U n N (U�). This implies that, for any U 2 U n N (U�),

U (�p+ (1� �)f) > U (�q + (1� �)g) .12

Also, for any U 2 U \ N (U�) ;

U (�p+ (1� �)f)� U (�q + (1� �)g) > (1� �) (U (f)� U (g))� 0:

This two facts imply that�p+ (1� �)f � �q + (1� �)g:

But observe that

U�(�(pTq)+(1� �)f)� U�(�q+(1� �)g) < (1� �) (U� (f)� U� (g))= 0:

This implies that it is not true that

�(pTq) + (1� �)f % �q + (1� �)g;

which contradicts Mixture Separability and Mixture Monotonicity. Since g is a constant act,this also contradicts One Side Monotonicity. �


[Necessity] Suppose that % has a Single-prior Expected Multi-utility representation (�;U)such that, for any u 2 U , u(x1) = 1 � u (x) � 0 = u (x0) for every x 2 X. It is clear that11Since U� is state-dependent, by de�nition, there exist ~p and ~q in �(X) such that

Ps2S E~p(U�(:; s)) �P

s2S E~q(U�(:; s)), but E~p(U�(:; s�)) < E~q(U�(:; s�)). Now, de�ne p := ��x1+(1��)~p and q := ��x0+(1��)~qfor � small enough so that it is still true that Ep(U�(:; s�)) < Eq(U�(:; s�)). Finally, de�ne T := fs�g andnote that

Ps2S Ep(U�(:; s)) >

Ps2S Eq(U�(:; s)).

12Recall that, because of our normalization, for any lottery p 2 �(X) and any U 2 U , 0 � U(p) � 1.

17

such a representation satis�es Best and Worst, and, by Theorem 1, it also satis�es Continuityand Independence. It is also easily checked that it satis�es Complete Beliefs. Finally, theargument that shows that a Multi-prior Expected Multi-utility representation satis�es OneSide Monotonicity does not rely on the �niteness of the prize space X, so we can repeat thatargument in order to show that % satis�es One Side Monotonicity.[Su¢ ciency] Fix any act f 2 F . De�ne Y := f(S) [ f�x0 ; �x1g. Let �(Y ) be the space

of probability measures on Y and let FY := �(Y )S. That is, FY is the space of acts whenthe state space is S and the prize space is Y . For each act � 2 FY , de�ne the act f � 2 Fby f �(s) :=

Ps2S �(s) (f(s)) f(s) + �(s)(�x0)�x0 + �(s)(�x1)�x1.

13 Now de�ne the relation%Y� FY �FY by � %Y � i¤ f � % f � . It can be checked that %Y inherits all the propertiesof %. That is, %Y is a preorder that satis�es Independence, Continuity, Best And Worst,One Side Monotonicity and Complete Beliefs. We need the following claim.

Claim 1. The relation %Y admits a Single-prior Expected Multi-utility representation.

Proof of Claim. By Theorem 2, there exists a Multi-prior Expected Multi-utility represen-tation M of %Y such that u(�x1) = 1 � u(y) � 0 = u(�x0) for every y 2 Y and every(�; u) 2M. Now suppose there exist (�1; u1) and (�2; u2) inM with �1 6= �2. In this case itis easy to construct an act � 2 FY such that �(S) = f�x0 ; �x1g and � is not %Y -comparableto ��x0 + (1� �)��x1 for some � 2 [0; 1].

14 We conclude that �1 = �2 for every (�1; u1) and(�2; u2) inM. k

The claim above shows that %Y admits a Single-prior Expected Multi-utility represen-tation which, by Theorem 3, implies that %Y satis�es the Reduction axiom. Now �x anyact f 2 F and de�ne FY as above. Let � 2 FY be such that �(s) = �f(s) for every s 2 S.That is, for each s 2 S, �(s) is the degenerate lottery that assigns probability one to theprize f(s) 2 Y . Since %Y satis�es the Reduction axiom, there exists � 2 �(S) such that� �Y

Ps2S �(s)�(s). By the de�nition of %Y , this implies that f �

Ps2S �(s)f(s). Since

f was completely arbitrary in this analysis, we conclude that % satis�es the Reduction ax-iom and, consequently, it admits a Single-prior Expected Multi-utility representation (�;U).That all functions u 2 U can be chosen so that u(x1) = 1 � u(x) � 0 = u(x0) for everyx 2 X comes from a simple normalization. �


It is easily checked that 1 implies 2 and 3. We �rst show that 2 implies 1. Suppose that 2is satis�ed. Let a; b 2 R be such that X = [a; b]. We �rst need the following claim:

Claim 1. For any act f 2 F , �b < f < �a.13Notation: For � 2 FY , s 2 S and y 2 Y , we write �(s)(y) to represent the probability that the lottery

�(s) assigns to the prize y.14Pick a state s� 2 S such that �1(s�) > �2(s�) and let � 2 R be such that �2(s�) < � < �1(s�). Notice

that, for � := ��x1fs�g��x0 we have

Ps2S �1(s)E�(s)(u1) > E��x1+(1��)��x0 (u1), but E��x1+(1��)��x0 (u2) >P

s2S �2(s)E�(s)(u2). That is, � and ��x1 + (1� �)��x0 are not comparable.

18

Proof of Claim. A standard inductive argument based on Independence and Transitivityshows that �b < p < �a for any lottery p with �nite support. Since X is a compact metricspace, the set of �nite support probability measures on X is dense in the set of all probabilitymeasures on X, so Continuity now implies that �b < p < �a for every p 2 �(X). The claimnow comes from the fact that < satis�es Monotonicity. (See lemma 1.) k

The claim above shows that < satis�es the Best and Worst axiom and, consequently, <satis�es all the postulates in the statement of Theorem 4. Therefore, that theorem guaranteesthat there exists a non-empty set U � C(X), with u(b) = 1 and u(a) = 0, for every u 2 U ,and a prior � 2 �(S) such that (�;U) is a Single-prior Expected Multi-utility representationof <. Certainty Dominance immediately implies that all functions in U are non-decreasing.Moreover, for every x; y 2 X with x > y, there must exist some u 2 U with u(x) > u(y).We need the following claim.

Claim 2. There exists a strictly increasing function u� 2 C(X) such that, for any acts fand g in F , f < g implies X

s2S�(s)Ef(s)(u�) �

Xs2S

�(s)Eg(s)(u�):

Proof of Claim. Let QX := Q\ (a; b). That is, QX is the set of rational numbers in the openinterval (a; b). Enumerate QX so that we can write QX = fx1; x2; :::g. Let u�1 2 U be suchthat u�1 (x1) > u�1 (a) and u

+1 2 U be such that u+1 (b) > u+1 (x1). Now, for i = 2; 3; :::; let

u�i 2 U be such that

u�i (xi) > u�i (maxfx 2 fa; b; x1; :::; xi�1g : x < xig) ;

and let u+i 2 U be such that

u+i (xi) < u+i (minfx 2 fa; b; x1; :::; xi�1g : x > xig) :

Now de�ne u� : X ! [0; 1] by u� := 12

P1i=1

12iu�i +

12

P1i=1

12iu+i . It is easily checked that

u� 2 C(X). Now suppose that x; y 2 X are such that x > y. Let z; w 2 QX be such thatx > z > w > y. By construction, it is clear that u�(x) � u�(z) > u�(w) � u�(y). That is,u�(x) > u�(y) and we conclude that u� is strictly increasing. Finally, suppose that f and gin F are such that f < g. This implies thatX


Xs2S

�(s)Eg(s)(u) for every u2 U .

19

But note that, for any act h 2 F ,

Xs2S

�(s)Eh(s)(u�) =Xs2S

�(s)Eh(s)

1

2

1Xi=1

1

2iu�i +

1

2

1Xi=1

1

2iu+i

!

=1

2

Xs2S

�(s)

1Xi=1

1

2iEh(s)(u�i ) +

1Xi=1

1

2iEh(s)(u+i )

!

=1

2

1Xi=1

1

2i

Xs2S

�(s)Eh(s)(u�i )

!+

1Xi=1

1

2i

Xs2S

�(s)Eh(s)(u+i )

!!:

Now it is clear that Xs2S

�(s)Ef(s)(u�) �Xs2S

�(s)Eg(s)(u�),

which concludes the proof of the claim. k

Let u� be as in the claim above. De�ne the set of strictly increasing functions U� � C(X)by U� := [�2(0;1)f�u� + (1� �)Ug. Note that, for any pair of acts f and g in F , f < g i¤X


Xs2S

�(s)Eg(s)(u) for every u2 U�.

That is, (�;U�) is a Single-prior Expected Multi-utility representation of <.For each act f , let xf be de�ned by

xf := max�x 2 X : f<�x

:

Notice that, Consistency and the fact that < satis�es Certainty Continuity guarantee thatxf is well-de�ned for every f 2 F . It is also clear that, for any two acts f and g, f<g i¤xf � xg. We will obtain the desired representation if we can show that, for each act f ,

xf = infu2U�

x�;uf :

For that, �rst note that, since f<�xf , Default to Certainty implies that f < �xf . Conse-quently, for every u 2 U�, we must haveX

s2S�(s)Ef(s)(u) � u (xf ) ;

which implies that x�;uf � xf . That is, infu2U� x�;uf � xf . Suppose that there exists x 2 Xsuch that infu2U� x

�;uf � x > xf . This implies that f < �x and, by Consistency, we get f<�x.

But the de�nition of xf implies that �x�f which gives us a contradiction. We learn thatinfu2U x

�;uf = xf and, consequently, 1 is satis�ed.

Let�s now show that 3 implies 2. Suppose that 3 is satis�ed. All we have to do is toshow that < and < satisfy Default to Certainty. We note that Claim 1 is still true. Now, �x

20

f 2 F and x 2 X, and suppose it is not true that f < �x. By Claim 1, this can happen onlyif x > a. Since < satis�es Continuity, we know that there exists " > 0 such that x � " � aand it is not true that f < �x�". By Caution, this implies that �x�"<f . But now the factthat < satis�es Certainty Dominance implies that �x��x�"<f . That is, < and < satisfyDefault to Certainty. �


[Necessity] It is easily checked that the representation satis�es Continuity, Certainty Dom-inance and Negative Certainty Independence. To see that it satis�es One Side MixtureMonotonicity, pick f , g and h in F , � 2 [0; 1] and suppose that �h(s) + (1 � �)f % g forevery s 2 S. This implies that, for every u 2 U ,

�Eh(s�)(u) + (1� �)Xs2S

�(s)Ef(s)(u) � u�infu2U

x�;ug

�,

for every s� 2 S. But this implies thatXs2S

�(s)E�h(s)+(1��)f(s)(u) � u�infu2U

x�;ug

�;

for every u 2 U , and, consequently, infu2U x�;u�h+(1��)f � infu2U x�;ug .

Now pick f and g in F such that supp(f(s)) [ supp(g(s)) � fa; bg for every s 2 S.Fix �; 2 (0; 1] and h; j 2 F . We can assume, without loss of generality, that everyu 2 U is such that u(a) = 0 and u(b) = 1. This implies that, for every u and u in U ,P

s2S �(s)Ef(s)(u) =P

s2S �(s)Ef(s)(u) andP

s2S �(s)Eg(s)(u) =P

s2S �(s)Eg(s)(u). So,either

Ps2S �(s)Ef(s)(u) �

Ps2S �(s)Eg(s)(u) for every u 2 U or

Ps2S �(s)Eg(s)(u) �P

s2S �(s)Ef(s)(u) for every u 2 U . Without loss of generality, let�s assume that wehave

Ps2S �(s)Ef(s)(u) �

Ps2S �(s)Eg(s)(u) for every u 2 U . This now implies thatP

s2S �(s)E�f(s)+(1��)h(s)(u) �P

s2S �(s)E�g(s)+(1��)h(s)(u) andP

s2S �(s)E f(s)+(1� )j(s)(u) �Ps2S �(s)E g(s)+(1� )j(s)(u) for every u 2 U . It is easy to see that this now implies that

�f +(1��)h % �g+(1��)h and f +(1� )j % g+(1� )j. This shows that % satis�esBest and Worst Mixture Consistency.

[Su¢ ciency] Suppose that % satis�es all the postulates in the statement of the theorem.De�ne the relation <� F � F by f < g i¤ �f + (1� �)h % �g + (1� �)h for every h 2 Fand every � 2 [0; 1]. It is easily checked that < is a preorder that satis�es Independenceand Continuity. Negative Certainty Independence together with the fact that % satis�esCertainty Dominance imply that < satis�es Certainty Dominance. We need the followingclaim:

Claim 1. < satis�es One Side Monotonicity.

Proof of Claim. Suppose f and g in F are such that f(s) < g for every s 2 S. This impliesthat, for every s 2 S, every h 2 F , and every � 2 [0; 1], �f(s)+(1��)h % �g+(1��)h. Since

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% satis�es One Side Mixture Monotonicity, this implies that �f + (1� �)h % �g + (1� �)hfor every h 2 F and every � 2 [0; 1]. That is, f < g and, consequently, < satis�es One SideMonotonicity. k

We now need the following claim.

Claim 2. For any T � S and � 2 [0; 1], �bT�a < ��b+(1��)�a or ��b+(1��)�a < �bT�a.

Proof of Claim. This is a straightforward consequence of Best and Worst Mixture Consis-tency. k

By Negative Certainty Independence, we know that < and % satisfy Default to Certainty.Since % is continuous, it satis�es Certainty Continuity. By Theorem 5 and Remark 3, thereexist a nonempty set of strictly increasing functions U � C(X) and a prior � 2 �(S) suchthat, for any acts f and g in F , f < g i¤X


Xs2S


and f % g i¤infu2U

x�;uf � infu2U

x�;ug .

It remains to show that the function V : F ! R de�ned by V (f) := infu2U x�;uf , for everyf 2 F , is continuous. For that, recall, from the proof of Theorem 5, that infu2U x

�;uf =

max fx 2 X : f % �xg =: xf , for every f 2 F . Since % is complete, continuous and satis�esCertainty Dominance, it is clear that xf is the unique element of X such that f � �xf . Nowsuppose that fm ! f . We will be done if we can show that xmf ! xf . Since fxmf g � X andX is compact, it is enough to show that every convergent subsequence of (xmf ) converges toxf . Suppose, thus, that x

mkf ! y. Since �xmkf � fmk for every k, Continuity of % implies

that �y � f and, consequently, y = xf . We conclude that V is continuous. �


[Necessity] Suppose that % has a representation as stated in the theorem. By Theorem 1,% satis�es Independence and Continuity. It is straightforward to show that % satis�es theother postulates in the statement of the theorem.

[Su¢ ciency] Suppose that % satis�es all the axioms in the statement of the theorem. We�rst need the following claim:

Claim 1. For any T � S and lotteries p and q in �(X), there exists � 2 [0; 1] such thatpTq � �p+ (1� �) q:

Proof of Claim. Fix T � S and lotteries p and q in �(X). By Constant Nontrivial-ity, there exist lotteries p and q in �(X) with p � q. Since p � q, by Monotonicity,we know that p % pT q and pT q % q. By Continuity, we know that the sets U% :=

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f� 2 [0; 1] : �p+ (1� �) q % pT qg and L% := f� 2 [0; 1] : pT q % �p+ (1� �) qg are closedand, by our previous observation, they are both nonempty. Moreover, by Complete Beliefs,U% [ L% = [0; 1]. Since [0; 1] is a connected set, we must have U% \ L% 6= ;, which impliesthat there exists � 2 [0; 1] such that pT q � �p + (1 � �)q. The claim now comes fromReduction Invariance. k

Since % satis�es Continuity and Independence, we know, by Theorem 1, that it has anadditively separable expected multi-utility representation U . Without loss of generality, wecan assume that for every U 2 U there exists s 2 S such that U(:; s) is not constant.15 Wewill now show that every U 2 U is state independent. To see that, suppose that there existsU 2 U and lotteries p and q in �(X) such that

Ps2S Ep(U(:; s)) �

Ps2S Eq(U(:; s)), but

Ep(U(:; s�)) < Eq(U(:; s�)) for some s� 2 S. It is clear that this can happen only if thereexists s 2 S with Ep(U(:; s)) > Eq(U(:; s)). Let T := fs 2 S : Ep(U(:; s)) � Eq(U(:; s))g.It is clear that

Ps2T Ep(U(:; s)) +

Ps2SnT Eq(U(:; s)) >

Ps2S E�p+(1��)q(U(:; s)) for every

� 2 [0; 1], which implies that for no � 2 [0; 1] we have �p + (1 � �)q % pTq. Since thiscontradicts the claim above, we conclude that all utilities in U are state independent. Nowwe can use a standard normalization argument to show that, for every utility U 2 U , thereexist a unique prior �U over S and a non-constant function uU : X ! R, unique up topositive a¢ ne transformations, such that, for any pair of acts f and g,

Ps2S Ef(s)(U(:; s)) �P

s2S Eg(s)(U(:; s)) i¤P

s2S �U(s)Ef(s)(uU) �

Ps2S �

U(s)Eg(s)(uU).16 To complete the proofof the theorem, we now have to show that �U = �V for every U; V 2 U . For that, �x anyT � S and pick lotteries p and q in �(X) such that Ep(uU) > Eq(uU). By the claim above,there exists � 2 [0; 1] such that pTq � �p + (1 � �)q. It is clear that this can happen onlyif �U(T ) = �. Now pick any lotteries p and q in �(X) such that Ep(uV ) > Eq(uV ). ByReduction Invariance, we have that pT q � �p + (1 � �)q. Again, this can happen only if�V (T ) = � = �U(T ). Since T was chosen arbitrarily, we conclude that �U = �V . �

References

Bewley, T. F. (2002). Knightian uncertainty theory: part i. Decisions in Economics andFinance 25 (2), 79�110.

Cerreia-Vioglio, S., D. Dillenberger, and P. Ortoleva (2014). Cautious expected utility andthe certainty e¤ect. Manuscript , Columbia University.

Galaabaatar, T. and E. Karni (2013). Subjective expected utility with incomplete prefer-ences. Econometrica 81 (1), 255�284.

García del Amo, A. and D. Ríos Insua (2002). A note on an open problem in the foundationof statistics. Rev. R. Acad. Cien. Serie A. Mat. 96 (1), 55�61.

15Notice that if U(:; s) is constant for every s 2 S, then U is completely irrelevant for the representationand, consequently, can be ignored.16See step 2 in section 5 of Ok et al. (2012), for example.

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Gilboa, I., F. Maccheroni, M. Marinacci, and D. Schmeidler (2010). Objective and subjectiverationality in a multiple prior model. Econometrica 78 (2), 755�770.

Gilboa, I. and D. Schmeidler (1989). Maxmim expected utility with non-unique prior. Journalof Mathematical Economics 18 (2), 141�153.

Nau, R. (2006). The shape of incomplete preferences. Annals of Statistics 34 (5), 2430�2448.

Ok, E. A., P. Ortoleva, and G. Riella (2012). Incomplete preferences under uncertainty:Indecisiveness in beliefs versus tastes. Econometrica 80 (4), 1791�1808.

Seidenfeld, T., M. J. Schervish, and J. B. Kadane (1995). A representation of partiallyordered preferences. Annals of Statistics 23 (6), 2168�2217.

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on the representation of incomplete preferences under...

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