aa r X i v : . [ ec on . T H ] S e p Random Non-Expected Utility: Non-Uniqueness
Yi-Hsuan Lin ∗ August 2020 † Abstract
In random expected utility (Gul and Pesendorfer, 2006), the distributionof preferences is uniquely recoverable from random choice. This paper showsthrough two examples that such uniqueness fails in general if risk preferencesare random but do not conform to expected utility theory. In the first, non-uniqueness obtains even if all preferences are confined to the betweenness class(Dekel, 1986) and are suitably monotone. The second example illustrates randomchoice behavior consistent with random expected utility that is also consistentwith random non-expected utility. On the other hand, we find that if risk pref-erences conform to weighted utility theory (Chew, 1983) and are monotone infirst-order stochastic dominance, random choice again uniquely identifies the dis-tribution of preferences. Finally, we argue that, depending on the domain of riskpreferences, uniqueness may be restored if joint distributions of choice across alimited number of feasible sets are available.
Keywords: random choice, random utility/preference, non-expected utility,identification
A classic model of random choice is random utility theory. It can be interpretedas follows. In a heterogeneous population, suppose each individual maximizes her ∗ Institute of Economics, Academia Sinica; [email protected]. I started this projectduring my PhD studies at Boston University. I am deeply grateful to Larry Epstein for his advice andencouragement. I also thank Jay Lu for his helpful comments and for raising the question addressedin Section 3.2. † First draft: October 2018.
Please click here for the latest version reference. Given any feasible set D of alternatives, as different individual mightmake different choice, the choice behavior of this population is summarized by adistribution over D , which is determined by the distribution of preferences in thepopulation. In the context of choice under risk, a special case of the model israndom expected utility (REU). All individuals in the population are expectedutility maximizers, but their risk attitudes are not identical. Under REU, thedistribution of preferences is uniquely recoverable from random choice (Gul andPesendorfer, 2006). In other words, when an analyst observes only the choicefrequencies for this population then, under the assumption that each individual’spreference conforms to expected utility theory, she is able to identify a uniquedistribution of preferences consistent with the observed behavior.The focus on expected utility preferences is natural as a first step, but is notcompletely satisfactory in light of its well-known descriptive failures, such as theAllais paradox. One might suspect that the observed random choice of lotteriescould be rationalized by random non-expected utility but not by REU. On theother hand, in an abstract choice setting, it is well-known that two distinctrandom preferences can rationalize the same random choice behavior. Henceif risk preferences are unrestricted at all, random choice of lotteries does notidentify the underlying distribution of preferences uniquely.Nonetheless, even if a modeler wants to deviate from expected utility the-ory, she does not need to embrace all kinds of risk preferences. Many classesof non-expected utility preferences have certain structures for tractability andare consistent with many stylized behaviors violating expected utility theory.Therefore, a natural question is if the uniqueness result of REU remains truewhen we make a small deviation from expected utility? Moreover, if a classof non-expected utility does not yield unique identification, how can we restoreuniqueness by enriching the observable behavior? This paper aims to respond tothese questions.Firstly, we show by example that random choice may not identify a uniquedistribution of preferences when risk preferences are random and restricted butdo not conform to expected utility theory. In particular, we have three find- Random utility theory can also model the stochastic behavior of a single agent. It hypothesizesthat the agent’s preference is random according to a fixed distribution. Facing a feasible set ofalternatives, she first perceives the realized preference and then makes a rational choice. Hence, fromex-ante point of view, her choice appears random. See Barber´a and Pattanaik (1986) for an example of random choice that can be rationalized bymore than one random utility. ngs for non-uniqueness: (i) non-uniqueness obtains even if risk preferences areconfined to the betweenness (implicit expected utility) class, developed by Dekel(1986) and Chew (1983, 1989), and are all monotonic with respect to first-orderstochastic dominance; (ii) non-uniqueness obtains even if risk preferences areconfined to the weighted utility class (Chew, 1983); (iii) random choice may berationalized by both REU and random non-expected utility. Thus, non-uniqueidentification seems to be a generic problem for random non-expected utilitymodels.The third finding above demonstrates a subtle difference between the ex-pected utility hypothesis at the individual and population levels. Even if theobserved choice frequencies from a population can be rationalized by random ex-pected utility, it is still possible that no individual in the population is an expectedutility maximizer.
Therefore, while violating any axiom of REU implies that notall individuals are expected utility maximizers, consistency with REU does notvalidate the expected utility hypothesis at the individual level either.We then deepen our analysis under weighted utility theory and provide somepositive results on restoring unique identification. While random weighted util-ity is not uniquely identified from random choice in general, we may regainuniqueness by requiring all preferences to be monotonic with respect to first-order stochastic dominance. We provide a formal argument for the case of threeprizes. In some choice settings, for example, when the prizes are monetary, suchmonotonicity is normative appealing. Therefore, if unique identification is de-sirable for a modeler, random weighted utility with monotonicity in stochasticdominance may be a good choice for random non-expected utility models.Without such monotonicity, we can still uniquely identify random weightedutility if we suitably enrich the observable behavior. In the case of three prizes, weshow that the distribution of weighted utility preferences is uniquely recoverablefrom joint distribution of choice across three binary menus. Note that underthe classic notion of random choice, only the distribution of choice from eachfeasible set is observable.
The joint distribution of choice across any two sets isnot.
Under REU, random choice implicitly reveals all joint choice probabilitiesand so pins down a unique distribution of preferences. Once we deviate fromexpected utility, random choice no longer discloses such information. Thus, non-uniqueness obtains in general. If we can observe joint choice frequencies acrosschoice sets, we will be able to improve identification. Our finding suggests thatwhen risk preferences are sufficiently restricted, joint choice distributions across small number of menus suffice for unique identification.To the best of our knowledge, this is the first study to address the identi-fication of random non-expected utility from a decision-theoretic perspective. The paper does not aim to dive deep into any particular model of random non-expected utility. Instead, it is mainly illustrative. We argue that some classesof non-expected utility preferences do not admit unique identification of randomutility, and some do. We suggest that allowing joint choice probabilities as theobservable can help to regain uniqueness. To convey these points easily, we con-duct all the analyses in a three-prize setting so that a risk preference can bevisualized on a two-dimensional plane. While risk preferences in this paper allbelong to the betweenness class or a subclass, we acknowledge the existence ofmany other types of non-expected utility widely used in economics but not con-sidered here yet. Future work may focus on a particular class of risk preferencesand perform a more detailed analysis.The rest of this paper is organized as follows. Section 2 introduces randomimplicit expected utility (RIEU). Section 3 provides two examples of two distinctRIEUs that induce the same random choice. Hence the distribution of riskpreferences is not uniquely identified. Section 4 points out a class of non-expectedutility that yields unique identification of random utility. It also discusses thereason for general non-uniqueness and demonstrates that uniqueness may berestored if choice data are suitably enriched. In the appendix, we point out threebehavioral properties of RIEU.
We first review the general random utility model. There is a universal space ofchoice alternatives X . A choice set, or called a menu, is a finite subset of X . Let D denote the collection of all menus. Lu (2020) studies a random utility model where each individual has an ambiguity-averse pref-erence over Anscombe-Aumann acts. Thus, the population’s choice of acts is not consistent withrandom subjective expected utility. Nonetheless, he considers the set of all lotteries over acts as thechoice domain and assumes that each individual is still an expected-utility maximizer when evaluatinglotteries. Hence his identification result essentially follows from the uniqueness of REU. hoice from a menu is modeled as a random set. In particular, for any A, D ∈ D , we denote by ρ D ( A ) the probability that A is the set of all optimalalternatives in D . To ensure feasibility, we require that if ρ D ( A ) > A is anonempty subset of D . Let Π be the set of all simple probability measures overthe class of all finite subsets of X . The observable choice behavior is summarizedby a random choice correspondence. Definition 1.
A random choice correspondence (RCC) is a function ρ : D → Π with ρ D ( { A ∈ D : ∅ 6 = A ⊂ D } ) = 1 for all D ∈ D . A preference relation % is a complete and transitive binary relation over X .For each menu D ∈ D , let M ( D, % ) denote the set of all optimal options in D according to % ; that is, M ( D, % ) := { x ∈ D : x % y ∀ y ∈ D } . Fix a set of preference relations Ω. Let N ( D, A ) denote the set of preferencesin Ω under which A is the set of all optimal options in D ; that is, N ( D, A ) := { % ∈ Ω : A = M ( D, % ) } ⊂ Ω . Let C := { N ( D, A ) :
A, D ∈ D} , (1)and let F ( C ) denote the smallest field that contains every element of C .A random utility, or called a random preference, is a finitely additive prob-ability measure µ on (Ω , F ( C )). Say that ρ is rationalized by µ if ρ D ( A ) equalsthe probability under µ that A is the set of all optimal alternatives in D . Definition 2.
Random choice correspondence ρ is rationalized by random utility µ if, for all D, A ∈ D , ρ D ( A ) = µ ( N ( D, A )) . We consider multi-valued random choice to avoid dealing with ties in choice. Literature oftenstudies single-valued random choice. Namely, the choice out of A is summarized by a distributionover A , but not over the class of subsets of A . With single-valued random choice, a random utilitymodel must assume that ties occur with zero probability, or impose a tie-breaking rule. Otherwise,the connection between the observable and the model would be loose. .2 Random utility models for choice under risk andidentification problem This paper exclusively focuses on random choice under risk. There is a finite setof prizes denoted W = { w , w , · · · , w N +1 } for N ≥
1. The objects of choice arelotteries over W . Let ∆ := { p ∈ R N + : P Nn =1 p n ≤ } be the set of all lotteries.For each p ∈ ∆, its n th coordinate p n is the probability of winning the prize w n ,for all n = 1 , · · · , N , and p N +1 := 1 − P Nn =1 p n is the probability of winning theprize w N +1 . A lottery assigning probability one to the prize w ∈ W is denotedby w .Let X = ∆, and then we can define a random risk preference µ as before. Theidentification problem concerns if random choice of lotteries ρ identifies a uniquedistribution of risk preferences µ . That is, if ρ is rationalizable by both µ and µ ′ , is it necessary that µ = µ ′ ? The answer to this question crucially depends onthe domain of preferences Ω.In order to facilitate the analysis, we assume that there are only three prizes( N = 2) throughout the paper. Thus a lottery is identified as a point in aprobability simplex in R , and a risk preference can be described by its indiffer-ence map. For an expected-utility preference, the indifference sets are all linearand parallel to each other. A well-known generalization of expected utility isbetweenness preference, which relaxes the parallelism of indifference curves. Definition 3 (Dekel, 1986) . A binary relation % over ∆ is called a betweennesspreference if it satisfies:1. % is complete and transitive.2. There exist best and worst elements in ∆ which are the sure prizes in W .3. If p ≻ q ≻ r , then there exists α ∈ (0 , such that αp + (1 − α ) q ∼ r .4. (Betweenness)If p ≻ q , then p ≻ αp + (1 − α ) q ≻ q for all α ∈ (0 , .If p ∼ q , then p ∼ αp + (1 − α ) q ∼ q for all α ∈ (0 , . A notable feature of such preference is that for any p ∈ ∆, the indifferenceset { q ∈ ∆ : q ∼ p } is the intersection of a hyperplane and ∆. Moreover,this hyperplane divides ∆ into the upper and the lower contour sets of p (i.e. { q ∈ ∆ : q ≻ p } and { q ∈ ∆ : q ≺ p } ). However, two indifference sets need not beparallel (i.e. their corresponding hyperplanes can intersect outside the simplex). I ( p ) I ( p ) w w ¯ w w w ¯ wp p Figure 1
A stylized betweenness preference is depicted in (a). The indifference set containing p ,denoted I ( p ), is the intersection of a straight line and the simplex. Two indifference setscan be non-parallel. A stylized expected-utility preference is depicted in (b), where allindifference sets are parallel. (Arrows indicate the direction of increasing preference.) If we strengthen Betweenness to Independence: ∀ α ∈ (0 , , ∀ p, q, r, ∈ ∆ , p % q ⇔ αp + (1 − α ) r % αq + (1 − α ) r, then % becomes an expected-utility preference whose indifference sets are allparallel to each other. See Figure 1 as examples of betweenness preference andexpected-utility preference in the Marschak-Machina triangle.Each betweenness preference has an implicit expected utility representationwhich we state in Appendix A. Thus, if Ω is the set of all betweenness preferences,we call µ random implicit expected utility (RIEU).All random preferences considered in this paper are RIEUs. We can furtherrestrict Ω by considering special cases of betweenness preferences. We will alsoconsider monotonicity in first-order stochastic dominance. It requires an exoge-nous and fixed ranking of prizes. And then for every possible preference % , p % q if p is obtained from q by shifting probability mass from a worse prize to a betterprize (i.e. p first-order stochastic dominates q ). In general, an RCC may be rationalized by more than one RIEU. We showthis through two examples. In particular, the first example shows that non-uniqueness obtains even if risk preferences are all monotonic with respect tofirst-order stochastic dominance. The second example illustrates random choice onsistent with random expected utility that can also be rationalized by morethan one random weighted utility, a special case of RIEU. We define two RIEUs µ and µ ′ as follows. Under µ , the realized preferenceis either % or % with equal probability. Figure 2 depicts their indifferencemaps. Under µ ′ , the realized preference is either % ′ or % ′ with equal probability.Figure 3 depicts their indifference maps.Note that these four preferences have identical indifference set of w . Abovethat set, % and % ′ have the same indifference map, and so do % and % ′ . Belowthat set, % and % ′ have the same indifference map, and so do % and % ′ .Below we provide numerical representations of % i and % ′ i , for i ∈ { , } . Inparticular, % and % both follow weighted utility theory (Chew, 1983), whichis a special case of betweenness preference. Weighted utility preferences arerepresented by the function V ( p ) = P N +1 n =1 p n g ( w n ) u ( w n ) P N +1 n =1 p n g ( w n ) , where u ( · ) and g ( · ) are real-valued functions defined on W , and g is non-zeroand nonnegative (or nonpositive).We have assumed N = 2. Let u ( · ) be such that u ( w ) = 0, u ( w ) = 1, and u ( w ) = . Let g ( · ) be such that g ( w ) = g ( w ) = 1 and g ( w ) = , andlet g ( · ) be such that g ( w ) = g ( w ) = 1 and g ( w ) = 2. Let V i ( · ) be theweighted utility function defined by u and g i for i ∈ { , } ; that is, V ( p ) = p + [(1 − p − p ) × × ] p + p + [(1 − p − p ) × ] and V ( p ) = p + [(1 − p − p ) × × ] p + p + [(1 − p − p ) × . Define utility function V ′ i ( · ) for i ∈ { , } such that V ′ ( p ) = ( V ( p ) if V ( p ) ≥ ,V ( p ) otherwise,and V ′ ( p ) = ( V ( p ) if V ( p ) ≥ ,V ( p ) otherwise.For i ∈ { , } , V i represents % i , and V ′ i represents % ′ i . Obviously, these four w w w x x q ′ p ′ qp w w w x x q ′ p ′ qp Figure 2 (a): The preference % is represented by the weighted utility function V . The worstlottery is w and the best lottery is w . All indifference curves intersect at x = ( − , − ).(b): The preference % is represented by the weighted utility function V . The worstlottery is w and the best lottery is w . All indifference curves intersect at x = (1 , (a) (b) w w w x x q ′ p ′ qp w w w x x q ′ p ′ qp Figure 3 (a): The preference % ′ is represented by the utility function V ′ . The worst lottery is w and the best lottery is w . For lotteries which are better than w , their indifference curvesintersect at x = ( − , − ). For lotteries which are worse than w , their indifference curvesintersect at x = (1 , % ′ is represented by the utility function V ′ . The worst lottery is w and the best lottery is w . For lotteries which are better than w , their indifference curvesintersect at x = (1 , w , their indifference curvesintersect at x = ( − , − ). references all are betweenness preferences. They also agree on the ranking ofprizes: w is the worst prize, and w is the best prize. Moreover, if we definefirst-order stochastic dominance based on this ranking of prizes, then under allthese preferences, p is better than q whenever p dominates q . RCC ρ is rationalized by µ if and only if it is rationalized by µ ′ . To see this,suppose that lottery p is such that V ( p ) ≥ . Then V ( p ) ≥ . For any otherlottery q , p % q ⇔ p % ′ q and p % q ⇔ p % ′ q . Similarly, if V ( p ) < ,then p % q ⇔ p % ′ q and p % q ⇔ p % ′ q . Since µ ( % i ) = µ ′ ( % ′ j ) for all i, j ∈ { , } , the probability that p is better than q is the same under µ and µ ′ .It is straightforward to extend the argument to show that, for any menu D and A ⊂ D , the probability that A is the set of all optimal lotteries in D is the sameunder µ and µ ′ .Therefore, RCC induced by µ can be rationalized by at least two RIEUs. Weconclude that, under RIEU, random choice may not identify a unique distributionof preferences. It is true even if we impose monotonicity in stochastic dominance. Proposition 1.
There exists a random choice correspondence rationalizable bymore than one random implicit expected utility.
We will review in Section 4 that a random choice correspondence cannot berationalized by two different random expected utilities (REU). However, it mightbe rationalizable by REU and also by random non-expected utility. The followingexample illustrates. In particular, it shows that random expected utility andrandom weighted utility (RWU) can induce identical random choice.When there are only three prizes, each weighted utility preference (Chew,1983) is characterized by (i) a point outside the Marschak-Machina triangle atwhich all indifference curves intersect, and (ii) the direction of increasing prefer-ence (either clockwise or counterclockwise). In fact, % ′ and % ′ both follow semi-weighted utility (Chew, 1989). A key geometric feature ofsuch preference is that, there are at most two points outside the Marschak-Machina triangle at whichtwo indifference curves can intersect. Our example illustrates that non-uniqueness obtains even if wefurther confine risk preferences to the class of semi-weighted utility. In Figures 2 and 3, preferences increase when we shift a lottery toward northwest. Therefore,they are monotonic with respect to first-order stochastic dominance. Here, we exclude expected utility from the class of weighted utility instead of viewing it as aspecial case. That is, we mean “strict” weighted utility. onstruct RWU ν as follows (see Figure 4). Fix a circle surrounding thetriangle. Let all possible intersection points of indifference curves be distributeduniformly on the circle. Moreover, conditional on any intersection point, let thedirection of increasing preference be clockwise or counterclockwise with equalprobability.Figure 5 illustrates the implication of ν for random choice. Suppose that ρ is rationalized by ν . Take any p, q, r ∈ ∆. Taking p as the vertex, let α be theangle (in degrees) formed by these lotteries ( ∡ qpr = α ◦ ). It can be shown that ρ { p,q,r } ( p ) = (cid:0) − α (cid:1) .Let ν be a uniform distribution over expected utility preferences. Then,under ν , the probability that p is optimal in { p, q, r } is also (cid:0) − α (cid:1) . In fact,for any p ∈ ∆, ν and ν induce the identical distribution over the class of lowercontour sets { p ′ ∈ ∆ : p % p ′ } . Therefore, ν and ν induce the same randomchoice.Note that when constructing ν , the circle surrounding the triangle is chosenarbitrarily. By choosing a different circle, we can construct another RWU thatrationalizes the same ρ . Thus, non-uniqueness obtains even if all preferences areconfined to the (strict) weighted utility class. Proposition 2.
There exists a random choice correspondence rationalizable bymore than one random weighted utility. There exists a random choice correspon-dence rationalizable by both random expected utility and random weighted utility.
Suppose that an analyst aims to test expected utility theory at the individuallevel, but only has choice data at the group level. Then she might consider testingREU theory instead. However, while the consistency between observed randomchoice and REU is necessary for all individuals to be expected-utility maximizers,it is not sufficient. Our example illustrates that, even if the observed behaviorsatisfies all the axioms of REU, it could be that no individual in the populationis an expected-utility maximizer.This observation connects to classical demand theory. It is known that, evenif each consumer behaves irrationally, aggregate demand could satisfy the weakaxiom of revealed preference, even be consistent with maximization of a singlepreference (Becker, 1962; Grandmont, 1992). Our result here is an analogy ifwe view expected utility theory as a rational choice behavior under risk. Theexample demonstrates that if individuals violate the independence axiom in di-verse directions, then their irrational behaviors “cancel out”. Hence the observedpopulation choice could be fully rational. w w w x w w w x Figure 4 (a): The preference follows weighted utility theory. All indifference curves intersect at x ,and the preference increases counterclockwise.(b): The preference follows weighted utility theory. All indifference curves intersect at x ,and the preference increases clockwise. pq rD AF C BE α ◦ Figure 5
Under ν , the realized preference ranks p optimal in D ≡ { p, q, r } if and only if either (i)the indifference curves intersect at some point on arc \ ABC and the preference increasesclockwise, or (ii) the indifference curves intersect at some point on arc \ DEF and thepreference increases counterclockwise. Because all possible intersection points distributeuniformly on the circle, the realized point lies on \ ABC ∪ \ DEF with probability 1 − α .Because the direction of increasing preference is clockwise or counterclockwise with equalprobability, ρ D ( p ) = (cid:0) − α (cid:1) if ρ is rationalized by ν . r ¯ a ˜ m a = a ˜ m + (1 − a ) ˜ m ˜ m ˜ m − aa Figure 6
The best and worst prizes, ¯ w and w , are identified with the points (0 ,
1) and (0 , − w is identified with ( − , x . The slopes of the lines ←→ x ¯ w and ←→ xw are ˜ m and ˜ m respectively.The lottery p a equals a ¯ w + (1 − a ) w , and the slope of its indifference curve equals a ˜ m +(1 − a ) ˜ m . This section provides some positive results on the unique identification of ran-dom non-expected utility. We find one class of non-EU preferences under whichrandom choice identifies underlying distribution of preferences uniquely. Thenwe discuss the reason for general non-uniqueness and suggest how to enrich theobservable to improve identification.
Suppose that W = { w, w, ¯ w } , where ¯ w is the best prize, and w the worst. Asshown in Figure 6, identify ¯ w and w with the points (0 ,
1) and (0 , −
1) in R respectively, and let the point ( − ,
0) denote w . If a weighted utility preference s monotone with respect to first-order stochastic dominance, the intersectionpoint of the indifference curves must lie in the green or blue area in the figure.If it lies in the green area, the preference increases counterclockwise. If it lies inthe blue area, the preference increases clockwise.We can depict the possible locations of the intersection point in the followingway. Draw a line passing w with slope m , and another line passing ¯ w withslope m . The intersection of these two lines lies in the green or blue areaif and only if ( m , m ) ∈ [ − , . Therefore, a random weighted utility withmonotonicity in first-order stochastic dominance can be identified with a randomvector ( ˜ m , ˜ m ) ∈ R whose support is [ − , .For any a ∈ (0 , p a = a ¯ w + (1 − a ) w . The slope of the indifference curveof p a is equal to a ˜ m + (1 − a ) ˜ m (see Figure 6). The probability that p a is chosenover another lottery q is equal to the probability that the slope of the indifferencecurve of p a is less than the slope of ←→ p a q (i.e. q lies below the indifference curveof p a ). Therefore, we can recover the distribution of a ˜ m + (1 − a ) ˜ m for any a ∈ (0 ,
1) from random choice. This allows us to pin down all the joint momentsof ˜ m and ˜ m .To find out the moments, fix any natural number n . Take any n + 1 distinct b , · · · , b n ∈ (0 , k ∈ { , · · · , n } , we can compute E [( ˜ m + b k ˜ m ) n ].Observe that E [( ˜ m + b k ˜ m ) n ] = E [ ˜ m n ] + · · · + (cid:18) nj (cid:19) b jk E [ ˜ m n − j ˜ m j ] + · · · + b nk E [ ˜ m n ] . Hence we have a system of linear equations for the unknown moments E [ ˜ m n − j ˜ m j ],0 ≤ j ≤ n : · · · (cid:0) nj (cid:1) b j · · · b n ... . . . ...... · · · (cid:0) nj (cid:1) b jk · · · b nk ... . . . ...1 · · · (cid:0) nj (cid:1) b jn · · · b nn × E [ ˜ m n ]... E [ ˜ m n − j ˜ m j ]... E [ ˜ m n ] = E [( ˜ m + b ˜ m ) n ]... E [( ˜ m + b k ˜ m ) n ]... E [( ˜ m + b n ˜ m ) n ] . The ( n + 1)-by-( n + 1) matrix on the left-hand side is equivalent to a squareVandermonde matrix. It is invertible as long as all b , · · · , b n are distinct. Thusthis system of linear equations has a unique solution. Consequently, randomchoice pins down E [ ˜ m i ˜ m j ] for any nonnegative integers i and j .Now the identification problem translates into a classic moment problem: ifall the joint moments are given, is there a unique joint distribution that generates hese moments? The answer is yes in our case because the support of ( ˜ m , ˜ m )is compact. A distribution with a compact support is uniquely determined byits moments. Note that without first-order stochastic dominance, the supportwould not be bounded. So the argument is not valid for general random weightedutility, and as shown in Section 3, uniqueness fails. Our result here showsthat stochastic dominance may help restoring unique identification when riskpreferences do not conform to expected utility theory.
Proposition 3.
Suppose that there are three prizes, and the best and worstones are prespecified. Then a random choice correspondence is rationalized byat most one random weighted utility under which all possible risk preferences aremonotonic with respect to first-order stochastic dominance.
This result is valid for local random choice data. The above identificationstrategy works if we observe random choice from menus { p a , q } for all a in anarbitrary open subset of [0 ,
1] and all q in a neighborhood of p a . Note that theidentification of REU has a similar property. Fixing any p , random choice frommenu { p, q, r } for all q, r in a neighborhood of p is sufficient to pin down REU. The lack of unique identification implies that random choice does not fully cap-ture the implications of a random utility model. Sometimes we can add moreassumptions into a model to improve identification. However, a modeler mayfind those assumptions unappealing, and the actual choice data may not be con-sistent with a more constrained model. Another way to restore uniqueness, asdemonstrated in this section, is to enrich the observable.Under the classic notion of random choice, the distribution of choice out ofeach menu is observable. Although there are infinitely many menus in our setting,random choice is by no means rich choice data in general. To see this, we firstreview how the identification of REU works. In contrast to Proposition 1, whenall possible preferences conform to expected utility theory then random choiceidentifies a unique distribution of preferences. It can be proved by the Stone-Weierstrass theorem, which implies that any continuous functionon a compact subset of a multi-dimensional Euclidean space can be uniformly approximated by poly-nomials. In general, two different distributions can generate identical moments. The moment problem hasbeen studied for decades. See Kleiber and Stoyanov (2013) for a recent study. heorem 1 (Gul and Pesendorfer, 2006) . A random choice correspondence isrationalized by at most one random expected utility.
We suggest the reason for this difference between REU and RIEU. The keyconcerns the joint distribution of choice across two or more menus. Such infor-mation is implicitly revealed through random choice under REU, but not underRIEU. A brief proof of Theorem 1 demonstrates this point.
Sketch of Proof.
We need some preliminaries. First, define REU formally. LetΩ E denote the set of all expected utility preferences over ∆. Let N E ( D, A ) := { % ∈ Ω E : A = M ( D, % ) } . Then let C E := { N E ( D, A ) :
A, D ∈ D} , and let F ( C E ) denote the smallest field that contains every element of C E . Arandom expected utility (REU) is a finitely additive probability measure µ E on(Ω E , F ( C E )).A class A of subsets of a set X is called a semiring if (i) ∅ ∈ A ; (ii) if A, B ∈ A then A ∩ B ∈ A ; (iii) if A, B ∈ A then A \ B = ∪ mk =1 C k for some mutuallydisjoint sets C , · · · , C m ∈ A . Say that A is a semifield if it is a semiring and X ∈ A . A set function is a mapping ν : A → R . The set function ν is finitelyadditive if ν ( ∪ mi =1 A i ) = P mi =1 ν ( A i ) for any finite collection of mutually disjointsets { A , · · · , A m } ⊂ A such that ∪ mi =1 A i ∈ A .The argument for the unique identification of REU is the following. GivenRCC ρ , define a set function µ ∗ E : C E → [0 ,
1] such that µ ∗ E ( N E ( D, A )) = ρ D ( A )for all N E ( D, A ) ∈ C E . Suppose that ρ is rationalized by REU. Then µ ∗ E is afinitely additive set function. Since C E is a semifield, the extension theorem (Raoand Rao, 1983, Theorem 3.5.1) applies, and the set function µ ∗ E can be extendedto a finitely additive measure µ E on F ( C E ). Moreover, such extension is unique.Hence µ E is the unique REU that rationalizes ρ .The key to the uniqueness of REU is that C E is a semifield. This ensures thatthe extension of µ ∗ E is unique. To identify RIEU, define C by (1) and then define aset function µ ∗ : C → [0 ,
1] such that, for all N ( D, A ) ∈ C , µ ∗ ( N ( D, A )) = ρ D ( A ).RIEU µ rationalizes ρ if and only if it is an extension of µ ∗ . However, C is not asemifield. Hence the extension of µ ∗ may not be unique. If, on (Ω , F ( C )), thereexist two different probability measures µ and µ ′ that both agree with µ ∗ on C ,then both of them are RIEUs and rationalize ρ . he class C fails to be a semifield because it is not closed under finite intersec-tions. That is, even if N ( D, A ) and N ( D ′ , A ′ ) are both in C , N ( D, A ) ∩ N ( D ′ , A ′ )may be not. This reflects the fact that, under RIEU, random choice does notreveal the joint distribution of choice across two menus . That is, ρ does not givethe joint probability that A is the set of all optimal lotteries in D and A ′ is theset of all optimal lotteries in D ′ .For example, consider RIEUs µ and µ ′ defined in Section 3.1. Let p = (0 , ), q = ( , ), p ′ = ( ,
0) and q ′ = ( , ). They are depicted in Figures 2 and 3.Note that p ≻ ( ≻ ′ ) q and p ′ ≺ ( ≻ ′ ) q ′ , and that p ≺ ( ≺ ′ ) q and p ′ ≻ ( ≺ ′ ) q ′ .Therefore, under µ , the probability that p is chosen over q and p ′ is chosen over q ′ is 0. However, under µ ′ , this probability equals . Thus, µ and µ ′ disagree onthe joint distribution of choice across { p, q } and { p ′ , q ′ } . On the other hand, the class C E is closed under finite intersections because,for any λ ∈ (0 , N E ( D, A ) ∩ N E ( D ′ , A ′ ) = N E ( λD + (1 − λ ) D ′ , λA + (1 − λ ) A ′ ) ∈ C E . This follows from the independence axiom of expected utility. Specifically, forany expected utility preference % E , { p, q } ⊂ D , { p ′ , q ′ } ⊂ D ′ , and λ ∈ (0 , p % E q ⇔ λp + (1 − λ ) p ′ % E λq + (1 − λ ) p ′ ; p ′ % E q ′ ⇔ λp + (1 − λ ) p ′ % E λp + (1 − λ ) q ′ . These imply that λp + (1 − λ ) p ′ is optimal in λD + (1 − λ ) D ′ if and only if p and p ′ are optimal in D and D ′ respectively. In words, under REU, the distributionof choice from the mixture of D and D ′ reveals the joint distribution of choiceacross D and D ′ . Once we deviate from REU, we may lose such informationfrom random choice.For a given random utility model, a natural question is what is the minimalrequirement on choice data for unique identification. Or at least, one would like Similarly, in our second example, v and v could be distinguished if we were given joint dis-tributions of choice across two menus. Consider menus { p, q } and { p ′ , q ′ } , where p ′ = p + r and q ′ = q + r for some r . Under ν , the probability of p being chosen from { p, q } and q ′ being chosenfrom { p ′ , q ′ } is positive. Under ν , that probability is zero. In other words, if ν is the actual dis-tribution of preferences, then with positive probability, the joint choice will violate the independenceaxiom . Define mixtures of menus D and D ′ by λD + (1 − λ ) D ′ := { λp + (1 − λ ) p ′ : p ∈ D, p ′ ∈ D ′ } forany λ ∈ (0 , o know how to enrich data to narrow down the set of rationalizing random pref-erences. The above discussion suggests that collecting data on joint frequenciesof choice across different menus may be helpful.Regardless of the domain of preferences, if the joint distribution of choiceacross any number of menus is observable, then the distribution of preferences isuniquely recoverable. It is because the class {∩ ki =1 N ( D i , A i ) : A i , D i ∈ D ∀ ≤ i ≤ k ; k ≥ } is closed under finite intersection. Such class is called a π -system. If two prob-ability measures agree on a π -system that generates the field where they aredefined, then they are identical. However, it seems far-fetched to assume that all joint distributions of choiceare observable. An advantage of REU is that no joint choice distribution isneeded to identify the distribution of preferences. If we deviate from REU butstill consider a restrictive domain of preferences, then joint distributions of choiceacross a limited number of menus might be sufficient. Our last result for randomweighted utility demonstrates this point.What are the minimal observations to completely determine a weighted utilitypreference on the Marschak-Machina triangle? If we know p ∼ q , p ′ ∼ q ′ , and p ′′ ≻ q ′′ , then we can infer the entire indifference map. This is because thelines ←→ pq and ←→ p ′ q ′ determine the intersection point of all the indifference curves,and then p ′′ ≻ q ′′ determines the direction of the increasing preference. A similarresult holds for random weighted utility: Joint distributions of choice across threebinary menus pin down a unique distribution of weighted utility preferences. Proposition 4.
Suppose that there are three prizes. Suppose that the joint distri-bution of choice across any three binary menus is observable. Then such behavioris rationalizable by at most one random weighted utility.Proof.
Now Ω denotes the set of all weighted utility preferences. The class C isstill defined by (1), and F ( C ) is the smallest field generated by C .As shown in Appendix C, if two random weighted utilities µ and µ ′ agree onthe class {∩ i =1 N ( { p i , q i } , A i ) : A i ⊂ { p i , q i } ⊂ ∆ ∀ i = 1 , · · · , } , then they also agree on the class E := {∩ ki =1 N ( { p i , q i } , A i ) : A i ⊂ { p i , q i } ⊂ ∆ ∀ i = 1 , · · · , k ; k ≥ } . This follows from Dynkin’s π - λ theorem. he class E is a π -system. Moreover, the smallest field generated by E is also F ( C ). Therefore, µ and µ ′ agree on F ( C ).The key is that the joint distribution of choice across any four or more binarymenus can be deduced from the observable. Figure 7 illustrates. There arefour pairs of lotteries, { ( p i , q i ) } i =1 . If p i ≻ q i , then % rotates at some point (theintersection of all its indifference curves) in one side of the line ←→ p i q i clockwise (thedirection of increasing preference), or in another side counterclockwise. Thus, inFigure 7(a), if p i ≻ q i for all i , then % rotates at some point in the blue areaclockwise, or in the green area counterclockwise.In Figure 7(b), we plot three additional lotteries r , s , and t . The lines ←→ rs and ←→ rt divide the blue and green areas into five regions. Note that, for instance,if r ≻ s , t ≻ r , and p ≻ q , then % rotates at some point in the region R clockwise, or in R counterclockwise. One can verify that p i ≻ q i ∀ i = 1 , · · · , ⇐⇒ ( p ≻ q ∧ p ≻ q ∧ s ≻ r ) ∨ ( r % s ∧ t % r ∧ p ≻ q ) (2) ∨ ( p ≻ q ∧ r ≻ t ∧ p ≻ q ) . The right-hand side is a disjunction of three mutually exclusive statements. Let P D ,D ,D denote the joint distribution of choice across menus D , D , and D .Then (2) implies thatProbability that p i is chosen from { p i , q i } for all i = 1 , · · · , P { p ,q } , { p ,q } , { r,s } ( p , p , s ) + P { r,s } , { r,t } , { p ,q } ( r ∨ { r, s } , t ∨ { r, t } , p )+ P { p ,q } , { r,t } , { p ,q } ( p , r, p ) . Therefore, although the joint probability that p i is chosen over q i for all i =1 , · · · , This paper addresses the identification of random utility in a three-prize setting ofchoice under risk. When risk preferences conform to the expected utility theory,the distribution of preferences is uniquely recoverable from random choice of q p q p q p q p q q p q p p q p rs t R R R R R Figure 7 (a): Suppose that % is a weighted utility preference. Then p i ≻ q i for all i = 1 , · · · , % rotates at some point in the blue area clockwise or rotates at some point inthe green area counterclockwise.(b): Suppose that % is a weighted utility preference. Then p ≻ q , p ≻ q , and s ≻ r ifand only if % rotates at some point in the area R ; r ≻ s , t ≻ r , and p ≻ q if and onlyif % rotates at some point in the area R or R ; p ≻ q , r ≻ t , and p ≻ q if and only if % rotates at some point in the area R or R . lotteries. But such uniqueness fails in general if risk preferences deviate fromexpected utility. We show by example that even if preferences are confinedto a specific class of non-expected utility, such as the betweenness class or asubclass, non-uniqueness can obtain. On the other hand, unique identificationis not hopeless at all. We find that if risk preferences conform to the weightedutility theory and are monotone in first-order stochastic dominance, then thedistribution of preferences is again uniquely recoverable from random choice. Ingeneral, collecting data on joint choice distributions across different menus canimprove identification. If preferences are suitably restricted, random joint choiceacross a small number of menus will suffice for unique identification.The paper is illustrative but does not dive into any particular model of ran-dom non-expected utility. Many classes of non-expected utility are not con-sidered in this paper, such as rank-dependent expected utility (Quiggin, 1982),disappointment-averse utility (Gul, 1991), or cautious expected utility (Cerreia-Vioglio et al., 2015). A random utility modeler may choose a domain of pref-erences based on what describes each individual’s behavior better in her belief.But she may also rather select a model with a unique identification. This paperpoints out a generic difficulty a modeler may face and suggests some possiblesolutions. Nonetheless, a comprehensive study of random non-expected utility is eyond the scope of this paper and is left for future research. Appendices
A Representation of Betweenness Preference
A betweenness preference has the following representation.
Proposition A.1 (Implicit Expected Utility Representation) . A preference over ∆ is a betweenness preference if and only if there exists u ( · , · ) : W × [0 , → R ,continuous in the second argument, such that p % q ⇔ V ( p ) ≥ V ( q ) , where V ( p ) is defined implicitly as the unique v ∈ [0 , that solves n +1 X i =1 u ( w i , v ) p i = vu ( ¯ w, v ) + (1 − v ) u ( w, v ) . (3) Furthermore, u ( w, v ) is unique up to positive affine transformations which arecontinuous in v . A particular transformation exists setting u ( w, v ) = 0 and u ( ¯ w, v ) = 1 for all v ∈ [0 , .Proof. See Dekel (1986, Proposition A.1).
B Behavioral Properties of RIEU
For any
D, D ′ ∈ D and λ, λ ′ ≥
0, let λD + λ ′ D ′ := { λx + λ ′ y : x ∈ D, y ∈ D ′ } .Note that λD + λ ′ D ′ is also a menu.For any convex set C , a convex set F ⊂ C is called a face of C if for all x, y ∈ C and λ ∈ (0 , λx + (1 − λ ) y ∈ F ⇒ { x, y } ⊂ F. For any set A , let chA denote the convex hull of A . For any D, A ∈ D , we saythat A is a face of D if chA is a face of chD and chA ∩ D = A . Axiom B.1.
Monotonicity: ρ D ( A ) ≤ ρ D \ B ( A \ B ) for all D, A, B ∈ D with A \ B = ∅ . Monotonicity captures the intuition that the probability of lottery p beingoptimal does not decrease as some lotteries are removed from the menu. Thisproperty holds under any random utility model. xiom B.2. Extremeness: ρ D ( A ) > implies that A is a face of D , for all A, D ∈ D . Extremeness states that lotteries p and q are both optimal whenever a mixtureof them is optimal. This property reflects the fact that each indifference set of abetweenness preference is linear. Axiom B.3.
Stochastic Betweenness: ρ λD +(1 − λ ) p ( λA + (1 − λ ) p ) = ρ D ( A ) forall p ∈ A ⊂ D ∈ D , λ ∈ (0 , . Stochastic Betweenness states that the probability of lottery p being optimalremains unchanged when each other lottery q in the menu is replaced by a mixtureof p and q . This property captures the betweenness axiom, which requires thata mixture of two lotteries should lie in between them in preference. Proposition B.1. If ρ is rationalized by RIEU, then ρ satisfies Monotonicity,Extremeness, and Stochastic Betweenness.Proof. Suppose that ρ is rationalized by RIEU µ .Monotonicity: If A = M ( D, % ), then A \ B = M ( D \ B, % ). Thus N ( D, A ) ⊂ N ( D \ B, A \ B ). It follows that ρ D ( A ) = µ ( N ( D, A )) ≤ µ ( N ( D \ B, A \ B )) = ρ D \ B ( A \ B ). Hence ρ satisfies Monotonicity.Extremeness: Suppose that A = M ( D, % ), where % is a betweenness prefer-ence. Betweenness property implies that p ∼ p ′ for all p, p ′ ∈ chA , and p ≻ p ′′ for all p ∈ chA and p ′′ ∈ chD \ chA . Thus chA = M ( chD, % ). Suppose that p ∈ chA and p = λq + (1 − λ ) r where λ ∈ (0 ,
1) and { q, r } ⊂ chD . Since p isoptimal, p % q and p % r . If p ≻ q or p ≻ r , then by betweenness property, p ≻ λq + (1 − λ ) r = p , a contradiction. Hence p ∼ q ∼ r , implying that q and r are both in chA . Thus chA is a face of chD . Since A = M ( D, % ), chA ∩ D = A .Thus A is a face of D and so ρ satisfies Extremeness.Stochastic Betweenness: Suppose that p ∈ A = M ( D, % ), where % is abetweenness preference. Then for all q ∈ A and r ∈ D \ A , p ∼ q and p ≻ r . Bybetweenness property, p ∼ λq + (1 − λ ) p and p ≻ λr + (1 − λ ) p for all λ ∈ (0 , N ( D, A ) ⊂ N ( λD + (1 − λ ) p, λA + (1 − λ ) p ). The argument can bereversed to conclude that N ( D, A ) ⊃ N ( λD + (1 − λ ) p, λA + (1 − λ ) p ). Thus ρ λD +(1 − λ ) p ( λA + (1 − λ ) p ) = ρ D ( A ). So ρ satisfies Stochastic Betweenness. Proof of Proposition 4
We want to show that if two random weighted utilities µ and µ ′ agree on theclass E := {∩ i =1 N ( { p i , q i } , A i ) : A i ⊂ { p i , q i } ⊂ ∆ ∀ i = 1 , · · · , } , then they also agree on the class E := {∩ ki =1 N ( { p i , q i } , A i ) : A i ⊂ { p i , q i } ⊂ ∆ ∀ i = 1 , · · · , k ; k ≥ } . This would be true if any E ∈ E can be expressed as E = ∪ Mm =1 E m , where E m ∈ E for all m , and E m ∩ E m ′ = ∅ for all m = m ′ .Consider four pairs of lotteries, ( p i , q i ) for i = 1 , · · · ,
4. Let L i denote theline ←→ p i q i . Each L i divides R into two half spaces, H + i and H − i . Without lossof generality, assume that p i % q i if % rotates at some point in H + i clockwise orrotates at some point in H − i counterclockwise.We want to show that the set { % : p i % q i ∀ i = 1 · · · , } can be decomposedinto finitely many mutually disjoint subsets. Moreover, each subset takes theform { % : r j ≻ s j ∨ r j ∼ s j ∀ j = 1 · · · , } . By induction, we can extend theclaim to the case of more than four pairs of lotteries.It suffices to show the following: There exist r jk , s jk for j ∈ { , , } and k ∈ { , · · · , K } such that ∩ i =1 { % : p i % q i } = ∪ Kk =1 (cid:0) ∩ j =1 { % : r jk % s jk } (cid:1) , and, for all k = k ′ , (cid:0) ∩ j =1 { % : r jk % s jk } (cid:1) ∩ (cid:0) ∩ j =1 { % : r jk ′ % s jk ′ } (cid:1) ⊂ { % : r ∼ s } for some r, s ∈ ∆.If L i = L j for some i = j , then the claim is trivial. Now assume that L i = L j for all i = j . Without loss of generality, assume that ∩ i =1 H + i = ∅ . Case 1: ∩ i =1 H + i has only 1 one-dimensional face. That is, it is a half space.Without loss of generality, assume that H +1 ⊂ · · · ⊂ H +4 . Then H − ⊃ · · · ⊃ H − . Suppose that p % q and p % q . Then % either rotates at some pointin H +1 clockwise or rotates at some point in H − counterclockwise. Thus p % q and p % q . We have ∩ i =1 { % : p i % q i } = { % : p % q } ∩ { % : p % q } . Case 2: ∩ i =1 H + i has two one-dimensional faces. Without loss of generality,assume that L ∦ L and ∩ i =1 H + i = H +1 ∩ H +2 . ase 2-1: L k L . Then H − ∩ H − = ∅ . If p % q and p % q , then % rotates at some point in H +1 ∩ H +2 clockwise. Thus p % q and p % q . Wehave ∩ i =1 { % : p i % q i } = { % : p % q } ∩ { % : p % q } . Case 2-2: L ∦ L and L k L . Without loss of generality, assume H +3 ⊂ H +4 .Then ∩ i =1 H + i = H +1 ∩ H +2 ∩ H +4 and ∩ i =1 H − i = H − ∩ H − ∩ H − . Thus ∩ i =1 { % : p i % q i } = { % : p % q } ∩ { % : p % q } ∩ { % : p % q } . Case 2-3: L ∦ L , L ∦ L , and L ∩ L ⊂ ∆. If L ∩ L int ( H − ∩ H − ),then either L or L is redundant. That is, without loss of generality, ∩ i =1 H + i = H +1 ∩ H +2 ∩ H +3 and ∩ i =1 H − i = H − ∩ H − ∩ H − . Thus ∩ i =1 { % : p i % q i } = { % : p % q } ∩ { % : p % q } ∩ { % : p % q } . Now, consider L ∩ L ⊂ int ( H − ∩ H − ). Let r ∈ ∆ denote the intersection of L and L . Pick s ∈ ∆ such that ←→ rs passes L ∩ L . Note that ←→ rs divides H +1 ∩ H +2 into two fans. Let H +5 and H − denote the half spaces generated by ←→ rs such that H +1 ∩ H +2 = ( H +1 ∩ H +5 ) ∪ ( H − ∩ H +2 ). Without loss of generality, assume that r % s if % rotates at some point in H +5 clockwise and that H − ∩ H − ⊂ H − ∩ H − .Then, (cid:0) ∩ i =1 H + i (cid:1) ∪ (cid:0) ∩ i =1 H − i (cid:1) = (cid:2) ( H +1 ∩ H +3 ∩ H +5 ) ∪ ( H − ∩ H − ∩ H − ) (cid:3) ∪ (cid:2) ( H +2 ∩ H +4 ∩ H − ) ∪ ( H − ∩ H − ∩ H +5 ) (cid:3) . This implies that ∩ i =1 { p i % q i } = ( { p % q } ∩ { p % q } ∩ { r % s } ) ∪ ( { p % q } ∩ { p % q } ∩ { r - s } )Case 2-4: L ∦ L , L ∦ L , and L ∩ L int ∆. Without loss of generality,assume that { p , q } ⊂ H +1 . As in the previous case, we shall consider L ∩ L ⊂ int ( H − ∩ H − ).Pick a lottery r that is a strict mixture of p and q . Pick s ∈ ∆ such that ←→ rs k L . Let H +5 and H − denote the half spaces generated by ←→ rs such that H +5 ⊂ H +1 . Without loss of generality, assume that r % s if % rotates at somepoint in H +5 clockwise.Pick s ∈ ∆ such that ←→ rs passes L ∩ L . Let H +6 and H − denote the halfspaces generated by ←→ rs such that H +5 ∩ H +2 = ( H +5 ∩ H +6 ) ∪ ( H − ∩ H +2 ). Withoutloss of generality, assume that r % s if % rotates at some point in H − clockwiseand that H +6 ∩ H − ⊂ H +6 ∩ H − . hen (cid:0) ∩ i =1 H + i (cid:1) ∪ (cid:0) ∩ i =1 H − i (cid:1) = (cid:2) ( H +1 ∩ H +2 ∩ H − ) ∪ ( H − ∩ H − ∩ H +5 ) (cid:3) ∪ (cid:2) ( H +2 ∩ H +4 ∩ H − ) ∪ ( H − ∩ H − ∩ H +6 ) (cid:3) ∪ (cid:2) ( H +1 ∩ H +3 ∩ H +5 ∩ H +6 ) ∪ ( H − ∩ H − ∩ H − ∩ H − ) (cid:3) . This implies that ∩ i =1 { p i % q i } = ( { p % q } ∩ { p % q } ∩ { r - s } ) ∪ ( { p % q } ∩ { p % q } ∩ { r % s } ) ∪ ( { p ≻ q } ∩ { p ∩ q } ∩ { r % s } ∩ { r - s } )Note that { p ≻ q } ∩ { p ∩ q } ∩ { r % s } ∩ { r - s } can be decomposed as inCase 2-3. Case 3: ∩ i =1 H + i has three one-dimensional faces. Without loss of generality,assume that ∩ i =1 H + i = H +1 ∩ H +2 ∩ H +3 , L ∦ L , and L ∦ L .Case 3-1: L k L or ∩ i =1 H + i is bounded. Note that ∩ i =1 H − i = ∅ . Thus ∩ i =1 { p i % q i } = { p % q } ∩ { p % q } ∩ { p % q } . Case 3-2: L ∦ L , L ∩ L ⊂ H − ∩ H − . Note that H − ∩ H − ⊂ H − . Thus, (cid:0) ∩ i =1 H + i (cid:1) ∪ (cid:0) ∩ i =1 H − i (cid:1) = (cid:0) ∩ i =1 H + i (cid:1) ∪ (cid:0) ∩ i =1 H − i (cid:1) . Therefore, ∩ i =1 { p i % q i } = { p % q } ∩ { p % q } ∩ { p % q } . Case 3-3: L ∦ L , L ∩ L ⊂ H − ∩ intH +4 , and [( L ∩ L ) ∪ ( L ∩ L )] ∩ ∆ = ∅ .Without loss of generality, assume that L ∩ L ≡ { r } ⊂ int ∆. Pick s ∈ ∆such that ←→ rs passes the intersection of L and L . Let H +5 and H − denote thehalf spaces generated by ←→ rs such that ∩ i =1 H + i = ( H +1 ∩ H +2 ∩ H +5 ) ∪ ( H − ∩ H +3 ).Without loss of generality, assume that r % s if % rotates at some point in H − clockwise. Then (cid:0) ∩ i =1 H + i (cid:1) ∪ (cid:0) ∩ i =1 H − i (cid:1) = (cid:2)(cid:0) H +1 ∩ H +2 ∩ H +5 (cid:1) ∪ (cid:0) H − ∩ H − ∩ H − (cid:1)(cid:3) ∪ (cid:2)(cid:0) H +3 ∩ H +4 ∩ H − (cid:1) ∪ (cid:0) H − ∩ H − ∩ H +5 (cid:1)(cid:3) . This implies that ∩ i =1 { p i % q i } = ( { p % q } ∩ { p % q } ∩ { s % r } ) ∪ ( { p % q } ∩ { p % q } ∩ { r % s } ) . ase 3-4: L ∦ L , L ∩ L ⊂ H − ∩ intH +4 , and [( L ∩ L ) ∪ ( L ∩ L )] ∩ ∆ = ∅ .Without loss of generality, assume that { p , q } ⊂ H − . Let L denote theline passing L ∩ L and parallel to L . Note that L must intersect ∆ at twopoints, say r and s . One may pick r as a mixture of p and p , s as a mixtureof p and q . Let H +5 and H − denote the half spaces generated by L such that ∩ i =1 H + i = ( H +1 ∩ H +2 ∩ H +5 ) ∪ ( H − ∩ H +3 ). Without loss of generality, assumethat r % s if % rotates at some point in H − clockwise. Then (cid:0) ∩ i =1 H + i (cid:1) ∪ (cid:0) ∩ i =1 H − i (cid:1) = (cid:2)(cid:0) H +1 ∩ H +2 ∩ H +5 (cid:1) ∪ (cid:0) H − ∩ H − ∩ H − (cid:1)(cid:3) ∪ (cid:2)(cid:0) H +1 ∩ H +3 ∩ H +4 ∩ H − (cid:1) ∪ (cid:0) H − ∩ H − ∩ H − ∩ H +5 (cid:1)(cid:3) . Therefore, ∩ i =1 { p i % q i } = ( { p % q } ∩ { p % q } ∩ { s % r } ) ∪ ( { p % q } ∩ { p % q } ∩ { p % q } ∩ { r % s } ) . Note that { p % q } ∩ { p % q } ∩ { p % q } ∩ { r % s } can be decomposed as inCase 2-4. Case 4: ∩ i =1 H + i has four one-dimensional faces. Without loss of generality,assume that L ∦ L ; L ∦ L and L ∩ L ⊂ H +1 ; L ∦ L and L ∩ L ⊂ H +2 .Case 4-1: ∩ i =1 H + i is bounded; that is, it is a convex polygon with four edges.Note that, for at least one diagonal, the line containing it intersects ∆ at morethan one point. Without loss of generality, pick r, s ∈ ∆ such that ←→ rs passes L ∩ L and L ∩ L . Let H +5 and H − denote the half spaces generated by ←→ rs such that L ∩ L ⊂ H +5 . Without loss of generality, assume that r % s if % rotates at some point in H +5 clockwise. Then (cid:0) ∩ i =1 H + i (cid:1) ∪ (cid:0) ∩ i =1 H − i (cid:1) = (cid:2)(cid:0) H +1 ∩ H +2 ∩ H − (cid:1) ∪ (cid:0) H − ∩ H − ∩ H +5 (cid:1)(cid:3) ∪ (cid:2)(cid:0) H +3 ∩ H +4 ∩ H +5 (cid:1) ∪ (cid:0) H − ∩ H − ∩ H − (cid:1)(cid:3) . This implies that ∩ i =1 { p i % q i } = ( { p % q } ∩ { p % q } ∩ { s % r } ) ∪ ( { p % q } ∩ { p % q } ∩ { r % s } ) . Case 4-2: L k L . Let L be the line that passes L ∩ L and is parallel to L and L . Note that L intersects ∆ at more than one point. One may pick r ∈ L as a mixture of p and p , s ∈ L as a mixture of p and q . Then followthe same argument as in Case 4-1.Case 4-3: L ∩ L ⊂ H − ∩ H − . Note that ∩ i =1 H − i = H − ∩ H − , i.e., ∩ i =1 H − i has only two one-dimensional faces. We go back to Case 2. eferences Barber´a, S. and Pattanaik, P. K. (1986). Falmagne and the rationalizability ofstochastic choices in terms of random orderings.
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