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Research Laboratory on Political Behavior, Institutions and Public Policy−LAPCIPP

Master’s Program in Public Economics−MESP

On the Representation of Incomplete Preferences under Uncertainty with

Indecisiveness in Tastes

Gil Riella

University of Brasilia

Economics and Politics Working Paper 01/2013 June 12, 2013

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Economics and Politics Research Group CERME-CIEF-LAPCIPP-MESP Working Paper Series

ISBN: !

On the Representation of Incomplete Preferencesunder Uncertainty with Indecisiveness in Tastes

Gil Riella

March 21, 2013

Abstract

Recently, there has been some interest on models of incomplete preferences underuncertainty that allow for incompleteness of tastes. In particular, Galaabaatar andKarni (2013) and Ok, Ortoleva, and Riella (2012) presented di§erent axiomatizationsof the so called Single-prior Expected Multi-utility model. In this paper we prove twocharacterizations of the Single-prior Expected Multi-utility model. First, we show thatthe restriction of a Önite prize space in Galaabaatar and Karni (2013)ís result is notnecessary. That is, their characterization is still true when the prize space is a compactmetric space. Second, we present an axiomatization of the Single-prior Expected Multi-utility model that replaces the existence of best and worst constant acts by the weakerassumption that there exists a constant act that is strictly preferred to some otherconstant act.

1 Introduction

Recently, there has been some interest on models of incomplete preferences under uncer-tainty that allow for incompleteness of tastes. (See GarcÌa del Amo and RÌos Insua (2002),Nau (2006) and Seidenfeld, Schervish, and Kadane (1995), for example.) In particular,Galaabaatar and Karni (2013) and Ok et al. (2012) presented di§erent axiomatizations ofthe so called Single-prior Expected Multi-utility model.

The axiomatization in Galaabaatar and Karni (2013) works under the restriction of aÖnite prize space and impose the existence of best and worst acts, which turn out to beconstant given the other axioms. The axiomatization in Ok et al. (2012) works on the moregeneral environment of a compact metric prize space and does not impose the existence ofbest nor worst acts. However, the main axiom in Ok et al. (2012) makes use of an existentialquantiÖer and, consequently, is non-testable, in principle.

Department of Economics, Universidade de BrasÌlia. Email: riella@unb.br.

1

In this paper we prove two characterizations of the Single-prior Expected Multi-utilitymodel. First, we show that the restriction of a Önite prize space in Galaabaatar and Karni(2013)ís result is not necessary. That is, their characterization is still true even when theprize space is a compact metric space.1 As we have said above, the result in Galaabaatar andKarni (2013) imposes that there exist best and worst constant acts. Our second result is anaxiomatization of the Single-prior Expected Multi-utility model that replaces the existence ofbest and worst constant acts by the weaker assumption that there exists a constant act thatis strictly preferred to some other constant act. Except for this postulate and Continuity,all other postulates in the axiomatic system are testable.

In the next section we discuss some representations of Incomplete Preferences under Un-certainty that allow for incompleteness of tastes and beliefs simultaneously. In particular,we present Ok et al. (2012)ís result about Additively Separable Expected Multi-utility repre-sentations and we present the adaptation of Galaabaatar and Karni (2013)ís result aboutthe Multi-prior Expected Multi-utility representation to the case of a preorder as primitive.In Section 3, we study the Single-prior Expected Multi-utility model. We Örst recall Oket al. (2012)ís axiomatization of this model in Section 3.1. After that, in Section 3.2, weuse this result to show that Galaabaatar and Karni (2013)ís characterization of the Single-prior Expected Multi-utility model is true even when we generalize the prize space to be acompact metric space. Finally, we give a new axiomatization of this model that weakensthe assumption that there exist best and worst constant acts to the assumption that thereexist one constant act that is strictly preferred to some otheróSection 3.3. In Section 4we present two applications of the results in Section 3. In Section 4.1 we present a resultin the style of Gilboa, Maccheroni, Marinacci, and Schmeidler (2010)ís main result for theSingle-prior Expected Multi-utility model. In Section 4.2 we generalize the main result inCerreia-Vioglio, Dillenberger, and Ortoleva (2013) to the case of multiple tastes under un-certainty. We conclude in Section 5, where we discuss some open problems in the literatureof incomplete preferences under uncertainty and argue that the results in the current paperare more than technical curiosities. Detailed proofs of the results in the main text appear inthe appendix.

2 Multi-utility Representations of Incomplete Prefer-ences under Uncertainty

In this section we revisit some results about the representation of incomplete preferencesunder uncertainty that have appeared recently in the literature. We recall the axiomatizationof the 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 Önite

1Some qualiÖcation is necessary here. The primitive in Galaabaatar and Karni (2013) is a strict partialorder, while ours is a preorder. Formally, what we show is that we can obtain an axiomatization of theSingle-prior Expected Multi-utility model when the prize space is a compact metric space, using postulatesthat are adaptations of Galaabaatar and Karni (2013)ís postulates to the case of a preorder. We conjecturethis is also true in their setup, though.

2

prize space. See also GarcÌa del Amo and RÌos Insua (2002) for a related result when boththe prize space as well as the state space are subsets of euclidean spaces.)

After the introduction of the Additively Separable Expected Multi-utility representa-tion, we present an axiomatization of the Multi-prior Expected Multi-utility model. TheMulti-prior Expected Multi-utility model was axiomatized Örst by Nau (2006) and later byGalaabaatar and Karni (2013). (See also Seidenfeld et al. (1995) for a related result.) Bothpapers work under the restriction of a Önite prize space and the existence of best and worstconstant acts. The axiomatization we present follows Galaabaatar and Karni (2013), butour primitive is a preorder, di§erent from theirs which is a strict partial order.

2.1 Setup

Let X be a compact metric prize space. Occasionally we will assume that X is a Önite set.We will denote the elements of X by x; y; z, etc.. We will write (X) to represent the spaceof Borel probability measures onX. We metrize(X) in such a way that metric convergencecoincides with weak convergence of Borel probability measures. The elements of (X) arecalled lotteries and denoted by p; q; r, etc.. The linear space of all continuous real maps onX is denoted by C(X). Throughout the exposition we metrize C(X) by the 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) :=Z

X

udp:

Let S be a Önite state space. An act is a function that maps the state space S into thespace (X) of lotteries. We denote the space of all acts by F . That is, F := (X)S. Theelements of F are denoted by f; g; h, etc.. We follow the usual abuse of notation and writesimply p to represent the constant act that returns the lottery p in every state of nature.Given that, we often write f(s) to represent the constant act that returns the lottery f(s) inevery state of nature. Similarly, we write x to represent the lottery that assigns probabilityone to the prize x 2 X. For any subset T of S, and any two acts f and g, we write fTg torepresent the act h such that h(s) = f(s) for every s 2 T and h(s) = g(s) for every s 2 S nT .Our primitive is a preorder (i.e., a reáexive and transitive binary relation) % F F . (Asusual, the asymmetric part of this preorder is denoted by , and its symmetric part by .)

2.2 The Additively Separable Expected Multi-utility Representa-tion

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.

3

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

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

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

s2S

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

s2S

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 Önite prize space X.GarcÌa del Amo and RÌos Insua (2002) prove a related result when X and S are subsets ofEuclidean spaces.

2.3 The Multi-prior Expected Multi-utility Representation

In this section we work with the additional restriction that X is Önite. We impose also thefollowing restriction on %.

Axiom 3 (Best and Worst). There exists 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.

2That 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 Önite 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 .

4

TheMixture Separability axiom is a weaker and somewhat simpler version of Nau (2006)ísStrong State-independence postulate.3 One Side Monotonicity is Galaabaatar and Karni(2013)ís Dominance postulate adapted to preorders. In the presence of Independence andContinuity each one of these properties implies the standard Monotonicity axiom, presentedbelow.

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.

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

ity.

We can now state the main result of this section.

Theorem 2. The following statements are equivalent.

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) X

s2S

(s)Eg(s)(u) for every (; u) 2M,

for any acts f and g in F ;

2. % satisÖes Independence, Continuity, Best and Worst, and Mixture Separability;

3. % satisÖes Independence, Continuity, Best and Worst, and One Sided Monotonicity;

4. % satisÖes Independence, Continuity, Best and Worst, and Mixture Monotonicity.

We call the representation in statement 1 above the Multi-prior Expected Multi-utilitymodel. Nau (2006) axiomatized the Multi-prior Expected Multi-utility model using a moretechnical version of Mixture Separability.4 Galaabaatar and Karni (2013) axiomatized thesame model using the One Sided Monotonicity axiom, but their primitive was a strict partialorder.

Although we use Theorem 2 only as an intermediate step to prove the results in Section3, the result is of independent interest for two reasons. First, as we have pointed out before,

3In our notation Nauís Strong State-independence axiom can be written as:

Strong State-independence. For any p; q; r 2 (X), f; g 2 F , T; T̂ S, a; b 2 (0; 1] and 2 [0; 1], iff % g, (x1Tx0) % ax1 + (1 a)x0, bx1 + (1 b)x0 % x1T̂ x0 and pTr + (1 )f % qTr + (1 )g, thenpT̂ r + (1 )f % qT̂ 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 = 1.4See footnote 3 above.

5

Galaabaatar and Karni (2013) use a strict partial order as primitive, so Theorem 2, althoughnot surprising, conÖrms that their result is still true when we use the more common optionof having a preorder as the primitive. Second, Galaabaatar and Karni (2013)ís 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 readerfamiliar with the literature about incomplete preferences under risk and uncertainty willÖnd our 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 % satisÖes 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.5 Suppose now that U has an extreme point U that cannotbe written as a prior utility pair (; u). We Örst use a separation argument to construct actsf and g such that X

s2S

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

s2S

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

and X

s2S

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

s2S

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

for all U 2 Un fUg. 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 Önd lotteries p; q and T S suchthat X

s2S

Ep (U (:; s)) >X

s2S

Eq (U (:; s)) ;

but X

s2T

Ep (U (:; s)) <X

s2T

Eq (U (:; s)) :

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

s2S

Ep+(1)f(s)U (:; s) >X

s2S

Eq+(1)g(s) (U (:; s))

for all U 2 U . That is,p+ (1 )f q + (1 )g:

But observe thatX

s2S

E(pTq)+(1)f(s)U (:; s) <X

s2S

Eq+(1)g(s) (U (:; s)) :

5Formally, 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 .

6

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 g to be constant, this is also a contradiction to One Side Monotonicity. k

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

In this section we use the characterization of the Single-prior Expected Multi-utility repre-sentation introduced by Ok et al. (2012) to derive two new results about the same model. WeÖrst show that the characterization given by Galaabaatar and Karni (2013) for a Önite prizespace X is still true when X is a compact metric space. As in Galaabaatar and Karni (2013),this Örst result imposes that there exist best and worst constant acts. We then present acharacterization of the Single-prior Expected Multi-utility representation that replaces thisassumption by the weaker requirement that there exist two constant acts that are strictlycomparable. From now on we get back to the assumption that X is a compact metric space.

3.1 The Single-prior Expected Multi-utility Representation withthe Reduction Axiom

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 ff f .

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

Theorem 3 (Ok et al. (2012), Theorem 1). A preorder % F F satisÖes 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

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u) for every u 2 U ,

for any acts f and g in F .

7

3.2 The Single-prior Expected Multi-utility Representation withComplete Beliefs Axiom

Consider the following postulate introduced by Galaabaatar and Karni (2013).6

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 satisÖes Independence, Continuity, Best And Worst,One Side Monotonicity and Complete Beliefs if, and only if, there exists a nonempty subset

U 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

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u) for every u 2 U ,

for any acts f and g in F .

The result above di§ers from the related result in Galaabaatar and Karni (2013) in twoaspects. First, our primitive is a preorder, not a strict partial order, as in their case. Secondthe result is obtained under the more general assumption that X is a compact metric space,not necessarily a Önite set.

Intuition for the proof of Theorem 4. Checking that the necessity part of the proof works isstraightforward. For the su¢ciency part, suppose Örst that X is Önite. In this case, Theorem2 applies and we know that % admits a Multi-prior Expected Multi-utility RepresentationM such 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 Önd T S and 2 [0; 1] such that x1Tx0is not comparable to x1+(1)x0, which contradicts Complete Beliefs. We conclude that% admits a Single-prior Expected Multi-utility Representation. Now suppose that X is acompact metric space, not necessarily Önite, and Öx any f 2 F . Let Y := f(S) [ fx1; x0gand look to the restriction of % to acts g 2 F such that g(S) conv(Y ).7 This restrictionÖts the Önite 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% satisÖes 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 Önite and % satisÖes the otherpostulates in the statement of Theorem 4 One Side Monotonicity is equivalent to Mixture

Separability and Mixture Monotonicity. Since One Side Monotonicity only plays a role in

6To be precise, because the primitive in Galaabaatar and Karni (2013) is a strict partial order, theirstatement of this property is slightly di§erent, but the idea is the same.

7Notation: For any subset Y of a vector space, we write conv(Y ) to represent the convex hull of Y .

8

the proof of Theorem 4 when X is Önite, 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 Theorem 7 below.

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

The Best and Worst postulate that appears in the previous section, specially when usedtogether with Independence, reduces the degree of incompleteness of the relation % con-siderably. On the other hand, the axiomatization of the Single-prior Expected Multi-utilityrepresentation in Ok et al. (2012) makes use of an existential quantiÖer and it is, in principle,not testable. The goal of the present section is to obtain an axiomatization of the Single-prior Expected Multi-utility representation that does not impose the existence of a best anda worst act, and that does not make use of existential quantiÖers in its main axioms.

We will make use of the following postulates.

Axiom 10 (Constant Nontriviallity). There exists lotteries p; q 2 (X) such that p q.

Axiom 11 (Reduction Invariance). If p; q 2 (X), T S and 2 [0; 1] are such that p qand pTq p+ (1 )q, then p̂T q̂ p̂+ (1 )q̂, for every p̂; q̂ 2 (X).

We can now state the following result.

Theorem 5. A preorder % F F satisÖes Independence, Continuity, Monotonicity, Con-stant Nontriviality, Complete Beliefs and Reduction Invariance if, and only if, there exists a

nonempty subset U of C(X) and a prior 2 (S) such that f % g i§X

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u) for every u 2 U ,

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 OneSided Monotonicity to Monotonicity and Best and Worst to Constant Nontriviality. Thedownside is that it imposes the additional Reduction Invariance postulate. We note that,except for Continuity and Constant Nontriviality, the axioms in the statement of Theorem5 are testable.

Intuition for the proof of Theorem 5. By Theorem 1, we know that % admits an AdditivelySeparable Expected Multi-utility representation U . Now pick any two lotteries p and q in(X) such that p q and Öx T S. By Monotonicity, p % pTq % q. By Complete

9

Beliefs, for every 2 [0; 1], either p + (1 )q % pTq or pTq % p + (1 )q. That is,f 2 [0; 1] : p + (1 )q % pTqg [ f 2 [0; 1] : pTq % p + (1 )qg = [0; 1], whileboth these sets are non-empty and closed. Because [0; 1] is a connected set, we must havepTq p + (1 )q for some 2 [0; 1]. We have just proved that for every T S thereexists 2 [0; 1] such that pTq p + (1 )q. By Reduction Invariance, this implies thatfor every T S and every pair of lotteries p̂ and q̂ in (X), there exists 2 [0; 1] such thatp̂T q̂ p̂ + (1 )q̂. Now we can basically replicate the proof of Theorem 2 in Ok et al.(2012) to show that % admits a Single-prior Expected Multi-utility representation. k

4 Applications

4.1 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 themadmit a Multi-prior Expected Single-utility representation ‡ la Bewley (2002) and the otheradmits a representation ‡ 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] in R. Allother deÖnitions remain as in the previous sections. We will work with a pair of preorders< and

We can now state the following result:

Theorem 6. 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

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u) for every u 2 U ,

f

5 Conclusion

We proved two representation theorems about the Single-prior Expected Multi-utility model.First, we showed that the adaptation of Galaabaatar and Karni (2013)ís axioms to preordersare equivalent to the Single-prior Expected Multi-utility representation even when the prizespace is a compact metric space. Second, we presented a new axiomatization of the modelwhere One Side monotonicity was weakened to the standard monotonicity axiom and Bestand Worst was weakened to the assumption that there existed a pair of constant acts strictlycomparable. We also had to impose a new axiom we called Belief Consistency. The mainadvantage of this new axiomatization over Galaabaatar and Karni (2013)ís one is that itallows for more incomparability, since it does not impose the existence of best and worstconstant acts. Regarding Ok et al. (2012)ís axiomatization of the same model, the advantageis that our main axioms do not use existential quantiÖers and, therefore, are in principletestable. The downside is that our new axiom, Belief Consistency, is based on indi§erences,so, although it is testable in principle, it might be hard to test it in practice.

We also presented two applications that illustrate why the generalization of Galaabaatarand Karni (2013)ís result to the case of a compact metric prize space is more than simply atechnical question. We Örst obtained a result in line with Gilboa et al. (2010)ís main resultlinking the Single-prior Expected Multi-utility representation to the Minimum CertaintyEquivalent representation, which is a generalization of Cerreia-Vioglio et al. (2013)ís CautiousExpected Utility representation. Second, we used this result to axiomatize the MinimumCertainty Equivalent representation, generalizing the main result in Cerreia-Vioglio et al.(2013) to the case of uncertainty when the individual has a single belief but multiple tastes.This two examples rely on our results and, in particular, require an axiomatization of theSingle-prior Expected Multi-utility model when the prize space is a compact metric space.

The analysis in this paper leaves an important question unanswered. The question con-cerns the axiomatization of the Multi-prior Expected Multi-utility model when the prizespace is not Önite. 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 Önite 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 do that.

We must emphasize that this is not only a technical question. The results in Section 4show that such a generalization is essential for some applications of the model. For example,a version of Theorem 6 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 theprize space is not Önite. Formally, we can show the following 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

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u) for every (; u) 2M,

12

f

Note that 1jSjf +jSj1jSj

~f is the constant act 1jSjPf(s). By Independence, we have that

1jSjf(s)+

jSj1jSj

~f(s) % 1jSjg(s)+jSj1jSj

~f(s) for every s 2 S. Since 1jSjf +jSj1jSj

~f is a constant act,

now One Side Monotonicity implies that 1jSjf +jSj1jSj

~f % 1jSjg +jSj1jSj

~f . Another applicationof Independence gives us that f % g.9

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 satisÖes Best and Worst, and, by Theorem 1, it also satisÖesContinuity and Independence. Now suppose the acts f; g; h; j and 2 (0; 1] are such thath(s) + (1 )f % j(s) + (1 )g for all s 2 S. Fix a generic pair (; u) 2 M. Theassumption above implies that

X

s2S

(s)Eh(s)+(1)f(s)(u) X

s2S

(s)Ej(s)+(1)g(s)(u);

for every s 2 S. But this now implies thatX

s2S

(s)Eh(s)+(1)f(s)(u) X

s2S

(s)Ej(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, % satisÖes Mixture Monotonicity. A similar reasoning shows that % also satisÖesMixture 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 Uso that U(x0; s) = 0, for every s 2 S and

Ps2S U(x1; s) = 1.

10 We can also assume, withoutloss of generality, that U is closed and convex. Letís 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 , deÖne a probabilitymeasure U on S by U(s) := U(x1; s), for every s 2 S. DeÖne 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) for9Here we are using the fact that it is clear from the additively separable representation of % that both

directions 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.10By Lemma 1, for every s 2 S and x 2 X, x1fsgx0 % xfsgx0 % 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 , deÖne ~U by ~U(:; s) := U(:;s

)U(x0;s)Ps2S U(x

1;s)U(x0;s) , for every s 2 S. Notice

that ~U(x0; s) = 0 for every s 2 S, andP

s2S~U(x1; s) = 1. Moreover, it is clear that for every pair of acts f

and g in F we haveP

s2S Ef(s)(U(:; s)) P

s2S Eg(s)(U(:; s)) i§P

s2S Ef(s)( ~U(:; s)) P

s2S Eg(s)( ~U(:; s)).

14

every x 2 X and s 2 S.

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 U(:; s) is constant and equal to zero, or if U(:; s) is a positive a¢netransformation of uU . In the Örst 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, we have U(x; s) = suU(x) + s, for somes 2 R++ and s 2 R, for every x 2 X. Since U(x0; s) = 0, we must have = 0. Now, fromU(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 StraszewiczísTheorem we may assume, without loss of generality, that U is in fact an exposed point ofU .11 So, there exists a Önite signed measure on X S and 2 R such that

X

(x;s)2XS

(x; s)U(x; s) = >X

(x;s)2XS

(x; s)U(x; s);

for all U 2 U n fUg. Consider now a measure ̂ such that ̂ (fx1; sg) = for every s 2 S,and ̂ is identically null in (X S) n (fx1g S). DeÖne 0 := + ̂ and observe that

X

(x;s)2XS

0(x; s)U(x; s) =X

(x;s)2XS

(x; s)U(x; s) +X

(x;s)2XS

̂(x; s)U(x; s)

= = 0:

A similar reasoning shows thatP

(x;s)2XS 0(x; s)U(x; s) < 0 for all U 2 Un fUg. Finally,

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

X

(x;s)2XS

00(x; s)U(x; s) =X

(x;s)2XS

0(x; s)U(x; s) +X

(x;s)2XS

~(x; s)U(x; s)

=X

(x;s)2XS

0(x; s)U(x; s):

Also, 00 (X fsg) = 0 for all s 2 S. By the Jordan decomposition we can Önd posi-tive measures + and such that 00 = + . Moreover, + can be chosen so that+ (X fsg) > 0 for every s 2 S. Let s be such that + (X fsg) + (X fsg) for11An exposed point x of a convex set C is an extreme point of C that has a supporting hyperplane whose

intersection 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:

15

every s 2 S. Without loss of generality, we can assume that + (X fsg) = 1. Now, foreach s 2 S, deÖne a probability measure +s on X by

+s (x) =+ (f(x; s)g)+ (X fsg)

,

for every x 2 X. DeÖne s analogously. Now, for every s 2 S, deÖne 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 deÖne acts f and g by

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

we have thatX

s2S

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

s2S

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

(x;s)2XS

00(x; s)U(x; s);

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

s2S

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

s2S

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

for every U 2 U n U andX

s2S

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

s2S

Ef(s)(U(:; s)) = 0.

Without loss of generality, we can assume that f is a constant act.12 Since U is state-dependent, we can Önd p; q 2 (X) and T S such that

X

s2S

Ep(U(:; s)) >X

s2S

Eq(U(:; s))

12For every act f , there exists another act ~f and 2 (0; 1) such that f + (1 ) ~f is constant. See theproof of Lemma 1 for the details.

16

and X

s2T

Ep(U(:; s)) <X

s2T

Eq(U(:; s)):13

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

U (f) :=X

s2S

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

Since U n N (U) is a compact set and

U (f) U (g) > 0;

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

U (f) U (g) >

1 ;

for all 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) .14

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. 13Since U is state-dependent, by deÖnition, there exist ~p and ~q in (X) such that

Ps2S E~p(U

(:; s)) Ps2S E~q(U

(:; s)), but E~p(U(:; s)) < E~q(U(:; s)). Now, deÖne 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, deÖne T := fsg andnote that

Ps2S Ep(U

(:; s)) >P

s2S Eq(U(:; s)).

14Recall that, because of our normalization, for any lottery p 2 (X) and any U 2 U , 0 U(p) 1.

17

A.3 Proof of Theorem 4

[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 thatsuch a representation satisÖes Best and Worst, and, by Theorem 1, it also satisÖes Continuityand Independence. It is also easily checked that it satisÖes Complete Beliefs. Finally, theargument that shows that a Multi-prior Expected Multi-utility representation satisÖes OneSide Monotonicity does not rely on the Öniteness of the prize space X, so we can repeat thatargument in order to show that % satisÖes One Side Monotonicity.[Su¢ciency] Fix any act f 2 F . Enumerate the states in S, so that S := fs1; :::; sjSjg,

and let ~Y := fy1; :::; yjSjg be any Önite set with jSj elements, all of them distinct fromx1 and x0. DeÖne Y := ~Y [ fx0; x1g. Let (Y ) be the space of probability measureson Y and let FY := (Y )S. That is, FY is the space of acts when the state space isS and the prize space is Y . For each act 2 FY , deÖne the act f 2 F by f (s) :=PjSj

i=1 (s)(yi)f(si) + (s)(x0)x0 + (s)(x1)x1.15 Now deÖne the relation %Y FY FY by

%Y i§ f % f . It can be checked that %Y inherits all the properties of %. That is, %YsatisÖes Independence, Continuity, Best And Worst, One Side Monotonicity and CompleteBeliefs. 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) 2 M. Now suppose there exist (1; u1) and (2; u2) inM with 1 6= 2. In this caseit is easy to construct an act 2 FY such that (S) = fx0; x1g and is not %Y -comparableto x0 + (1 )x1 for some 2 [0; 1].16 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 representa-tion which, by Theorem 3, implies that %Y satisÖes the Reduction axiom. Now Öx any actf 2 F and deÖne FY as above. Let 2 FY be such that (si) = yi for i = 1; :::jSj. Since%Y satisÖes the Reduction axiom, there exists 2 (S) such that Y

Ps2S (s)(s). By

the deÖnition of %Y , this implies that f P

s2S (s)f(s). Since f was complete arbitrary inthis analysis, we conclude that % satisÖes the Reduction axiom and, consequently, it admitsa Single-prior Expected Multi-utility representation (;U). That all functions u 2 U canbe chosen so that u(x1) = 1 u(x) 0 = u(x0) for every x 2 X comes from a simplenormalization. 15Notation: 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.16Pick 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 := x1fsgx0 we haveP

s2S 1(s)E(s)(u1) > Ex1+(1)x0(u1), but Ex1+(1)x0(u2) >Ps2S 2(s)E(s)(u2). That is, and x1 + (1 )x0 are not comparable.

18

A.4 Proof of Theorem 5

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

[Su¢ciency] Suppose that % satisÖes all the axioms in the statement of the theorem. WeÖrst 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 p̂ and q̂ in (X) with p̂ q̂. Since p̂ q̂, by Monotonicity,we know that p̂ % p̂T q̂ and p̂T q̂ % q̂. By Continuity, we know that the sets U% :=f 2 [0; 1] : p̂+ (1 ) q̂ % p̂T q̂g and L% := f 2 [0; 1] : p̂T q̂ % p̂+ (1 ) q̂g 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 p̂T q̂ p̂ + (1 )q̂. The claim now comes fromReduction Invariance. k

Since % satisÖes 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.17 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 ŝ 2 S with Ep(U(:; ŝ)) > Eq(U(:; ŝ)). 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 Ep+(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).18 To complete the proofof the theorem, we now have to show that U = V for every U; V 2 U . For that, Öx 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 p̂ and q̂ in (X) such that Ep̂(uV ) > Eq̂(uV ). ByReduction Invariance, we have that p̂T q̂ p̂ + (1 )q̂. Again, this can happen only ifV (T ) = = U(T ). Since T was chosen arbitrarily, we conclude that U = V . 17Notice that if U(:; s) is constant for every s 2 S, then U is completely irrelevant for the representation

and, consequently, can be ignored.18See step 2 in section 5 of Ok et al. (2012), for example.

19

A.5 Proof of Theorem 6

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

Claim 1. For any act f 2 F , b < f < a.

Proof of Claim. A standard inductive argument based on Independence and Transitive showsthat b < p < a for any lottery p with Önite support. Since X is a compact metric space, theset of Önite support probability measures is dense in the set of all probability measures, soContinuity now implies that b < p < a for every p 2 (X). The claim now comes from thefact that < satisÖes Monotonicity. (See lemma 1.) k

The claim above shows that < satisÖes the Best and Worst axiom and, consequently, <satisÖes 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 y, there must exist some u 2 U with u(x) > u(y).We need the following claim.

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

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u):

Proof of Claim. Let QX := (Q \X) n fa; bg and enumerate QX so that we can write QX =fx1; x2; :::g. Let u1 2 U be such that 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 ui 2 U be such that

ui (xi) > ui (maxfx 2 fa; b; x1; :::; xi1g : x < xig) ;

and let u+i 2 U be such that

u+i (xi) < u+i (minfx 2 fa; b; x1; :::; xi1g : x > xig) :

Now deÖne u : X ! [0; 1] by u := 12

P1i=1

12iui +

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 that

X

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u) for every u2 U .

20

But note that, for any act h 2 F ,

X

s2S

(s)Eh(s)(u) =X

s2S

(s)Eh(s)

1

2

1X

i=1

1

2iui +

1

2

1X

i=1

1

2iu+i

!

=1

2

X

s2S

(s)

1X

i=1

1

2iEh(s)(ui ) +

1X

i=1

1

2iEh(s)(u+i )

!

=1

2

1X

i=1

1

2i

X

s2S

(s)Eh(s)(ui )

!+

1X

i=1

1

2i

X

s2S

(s)Eh(s)(u+i )

!!:

Now it is clear that X

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u),

which concludes the proof of the claim. k

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

X

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u) for every u2 U.

That is, (;U) is a Single-prior Expected Multi-utility representation of

Öx f 2 F and x 2 X, and suppose it is not true that f < x. By Claim 1, this can happenonly if x > a. Since < satisÖes Continuity, we know that there exists " > 0 such that it isnot true that f < x " a. By Caution, this implies that x "

We now need the following claim.

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

Proof of Claim. This is a straightforward consequence of Best and Worst Independence. k

It is also clear that < and % satisfy Caution and that % satisÖes Certainty Continuityand Certainty Dominance. By Theorem 6, there exist a nonempty set of strictly increasingfunctions U C(X) and a prior 2 (S) such that, for any acts f and g in F , f < g i§

X

s2S

(s)Ef(s)(u) X

s2S

(s)Eg(s)(u) for every u 2 U ,

and f % g i§infu2U

x;uf infu2U

x;ug .

It remains to show that the function V : F ! R deÖned by V (f) := infu2U x;uf , for everyf 2 F , is continuous. For that, recall, from the proof of Theorem 6, that infu2U x;uf =max fx 2 X : f % xg =: xf , for every f 2 F . Now suppose that fm ! f . We will be doneif we can show that xmf ! xf . Since fxmf g X and X is compact, it is enough to show thatevery convergent subsequence of xmf converges to xf . Suppose, thus, that x

mkf ! y. Since

xmkf f for every k, Continuity of % implies that y f and, consequently, y = xf . Weconclude that V is continuous.

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Galaabaatar, T. and E. Karni (2013). Subjective expected utility with incomplete prefer-ences. Econometrica 81 (1), 255ñ284.

GarcÌa del Amo, A. and D. RÌos Insua (2002). A note on an open problem in the foundationof statistics. Rev. R. Acad. Cien. Serie A. Mat. 96 (1), 55ñ61.

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Gilboa, I. and D. Schmeidler (1989). Maxmim expected utility with non-unique prior. Journalof Mathematical Economics 18 (2), 141ñ153.

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