Expected balanced uncertain utility

Grant, Simon; Roorda, B.; Yang, Jingni Article Expected balanced uncertain utility Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Grant, Simon; Roorda, B.; Yang, Jingni (2025) : Expected balanced uncertain utility, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 20, Iss. 1, pp. 1-25, https://doi.org/10.3982/TE5404 This Version is available at: https://hdl.handle.net/10419/320279 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 20 (2025), 1–25 1555-7561/20250001 Expected balanced uncertain utility Simon Grant Research School of Economics, Australian National University Berend Roorda Faculty of Behavioural, Management and Social Sciences, University of Twente Jingni Yang School of Economics, University of Sydney We introduce and analyze expected balanced uncertain utility (EBUU) theory. Apriorandabalanced outcome-set utility characterize an EBUU decision maker. Conditional on a reference or “balancing value,” the latter assigns a utility to each outcome-set. The decision maker associates with each act, its envelope,theminimal measurable mapping from states to outcome-sets that contains the act. She then (implicitly) ranks an act according to the balancing value at which the expected balanced utility of its associated envelope is zero. As a consequence, her risk preferences need only exhibit betweenness allowing for behavior that can accommodate Allais-type paradoxes. Keywords. Uncertainty, ambiguity, betweenness. JEL classification. D80, D81. 1. Introduction In the tradition of the voluminous literature initiated by Ellsberg (1961), consider a decision maker (hereafer, DM) who possesses only partial information about the underlying stochastic process that determines the resolution of the uncertainty she faces. In particular, this means she is not comfortable quantifying with a precise probability the uncertainty she associates with each and every event. There does exist, however, a rich collection of events she deems measurable over which is defined her prior, a unique probability representing her beliefs over those events. An object of choice for our DM is an uncertain prospect or act that maps each state to an outcome. Using her prior, any act measurable with respect to her prior can be mapped to a corresponding probability distribution or lottery over outcomes. Thus the restriction of her preferences to measurable acts may be viewed as inducing a preference relation over lotteries, which we refer to as the DM’s risk preferences. Simon Grant: [email protected] Berend Roorda: [email protected] Jingni Yang: [email protected] We thank audiences at Royal Holloway U. of London, ANU, Erasmus U., SAET2021, and U. Alabama as well as Jürgen Eichberger, George Mailath, Ron Stauber, and Peter Wakker for comments and suggestions. We are also very grateful to the referees who pushed us to improve the exposition and tighten the analysis. ©2025 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5404 2Grant, Roorda, and Yang Theoretical Economics 20 (2025) In Gul and Pesendorfer’s (2014) expected uncertain utility model, the DM deems measurable precisely those events in which the event and its complement jointly satisfy a version of Savage’s (1954) state-separability postulate P2.1As a consequence, her risk preferences conform to expected utility. The Allais paradoxes, however, clearly illustrate that P2 is not only challenged when probabilities are unknown. This is certainly the case when it comes to descriptive modeling (see, e.g., Tversky and Shafir (1992)forempirical findings). Furthermore, even on normative grounds P2 has not gone unchallenged (see, e.g., Heukelom (2015) for an extensive historical account). Hence our goal is to characterize a class of DMs who may perceive ambiguity but whose risk preferences need not conform to expected utility theory. One approach taken by Grant, Rich, and Stecher (2022) that allows for risk-preferences that can accommodate Allais style violations of expected utility is to assume the set of measurable events are exogeneously specified and then axiomatize a family of preference relations in which the evaluation of an arbitrary act is characterized by a mapping to an equivalent measurable act along with a generalized notion of the certainty equivalent of a lottery.2 The approach taken here is to consider an alternative property an event must satisfy in order for it to be deemed measurable by the DM and then explore its implications for the corresponding risk-preferences. The one we propose is simple as it corresponds to Grant, Kajii, and Polak’s (2000) (weak) decomposability property (in its strict form). Letting denote the DM’s preferences over acts and writing fEgfor the act that agrees with the act fon the event Eand with the act gon its complement, an event Ris decomposable, if for any pair of acts fand g: [fRggand gRfg]=⇒ fg. Grant, Kajii, and Polak (2000) contend that by interpreting a statement like “fwould be preferred to gif the event Rwere known to obtain” as only entailing fRgg,decomposability may be interpreted as encapsulating the following reasoning: If the DM would prefer fto gknowing Robtains, and she would prefer fto gknowing Rdoes not obtain, then she should prefer fto geven though she currently does not know whether R will or will not obtain. That is, similar to the property employed by Gul and Pesendorfer to identify those events deemed measurable by the DM, decomposability provides a way to operationalize Savage’s (1954)extralogical Sure-Thing Principle (STP). With measurable events classified as those that satisfy decomposability, as Grant, Kajii, and Polak (2000) establish, the corresponding risk-preferences need only exhibit the betweenness property of Chew (1983) and Dekel (1986), and thus can accommodate Allais style violations of expected utility. 1Gul and Pesendorfer refer to any such event as ideal. 2Grant, Rich, and Stecher (2022) also consider endogenizing the set of events the DM views as measurable by utilizing Epstein and Zhang’s (2001) preference-based definition for classifying measurable events. Even then, however, in their representation theorem (Theorem 5, p. 14) the structure of the set of measurable events needed for the evaluation of arbitrary acts is assumed as part of the hypothesis of the theorem and not derived from the postulates they propose. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 20 (2025) Expected balanced uncertain utility 3 Our second point of departure concerns the handling of nonmeasurable acts. To model her valuation of such acts, we adapt Gul and Pesendorfer’s construction that assigns to each act a measurable envelope. We retain their notion of a measurable split of the state space induced by the preimage of the act. Each element of this split corresponds to an outcome-set (with finite cardinality), for which, given her limited information about the underlying stochastic process, renders her incapable of attributing any fraction of the probability her prior assigns to that element of the split to any strict subset of the corresponding outcome-set. The essential difference is that whereas Gul and Pesendorfer define an envelope as a mapping from states to intervals of outcomes, defined in terms of leastand most-preferred outcomes of the corresponding outcome-set, we retain the entire outcome-set. That is, the envelope of the act maps to an outcomeset precisely those states in the element of the measurable split induced by the act’s preimage that corresponds to that subset of outcomes. This in turn means the envelope of an act can be characterized as the minimal (with respect to set-inclusion) measurable mapping from states to outcome-sets that contains the act. Indeed from a perceptual perspective, we contend it makes sense to view the DM as incapable of distinguishing among acts that have a common envelope.3Theaxiomsweadoptguaranteetheexistence and uniqueness of envelopes. The interpretation of envelopes in terms of belief and plausibility functions, as described in Gul and Pesendorfer’s, becomes even more straightforward: the belief in a particular outcome-set obtaining is the total probability assigned to that outcome-set and all its subsets in the envelope, while its plausibility is the total probability of all subsets containing at least one element of that outcome-set. Moreover, the prior and the envelope of an act induce the outcome-set lottery in which for each outcome-set, the probability assigned to that outcome-set is given by the probability the prior assigns to the set of states that the envelope maps to that particular outcome-set. Imposing that the DM is indifferent among acts inducing the same outcome-set lottery, we introduce Expected Balanced Uncertain Utility (EBUU) preferences corresponding to the family of preferences that admit an implicit probability equivalent (utility) representation characterized by a pair μ,U,whereμis the DM’s prior defined over those events she deems measurable and a balanced outcome-set utility,U(Y,p),that specifies the utility of an outcome-set Yin a lottery that has a probability equivalent of p, by which we mean any lottery the DM views as equally valuable as a binary gamble that yields the best outcome with probability pand the worst outcome with the complementary probability 1−p. It exhibits a natural (outcome-set) monotonicity with respect to its first argument. Thus we obtain a clean separation of the ambiguity she perceives to be present given her knowledge about the random process governing the resolution of the uncertainty she faces from her attitude toward risk (i.e., measurable uncertainty). The former is characterized by those events that lie outside the domain of her prior while the restriction of her balanced outcome-set utility to singleton outcome-sets encodes the latter. 3In this regard, the measurable split induced by an act’s inverse image is reminiscent of Ghirardato’s (2001, p. 249) second scenario of an underspecified state space as one possible way to interpret his model in which preferences are defined over outcome-set acts. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4Grant, Roorda, and Yang Theoretical Economics 20 (2025) Finally, her attitude toward (general) uncertainty involving both risk and ambiguity is embodied in her (unrestricted) balanced outcome-set utility. This interpretation clarifies why we do not impose decomposability for nonmeasurable events. The principle relies on a sharp separation between outcomes of an act on an event and its complement, but this gets blurred when events are nonmeasurable, since outcomes on an event may contribute to the outcome-set outside that event. We refer to Gul and Pesendorfer (2014)(Section5)foranillustrationbywayof the Ellsberg paradox of this effect. Hence, their motivation not to impose P2 for nonmeasurable events in essence also applies to restricting decomposability to measurable events only. We develop the formal definition of EBUU preferences in Section 2with its axiomatic characterization appearing in Section 3. We provide three examples in Section 4.We conclude in Section 5. Proofs appear in the Appendix. 2. The model Our setting is one in which the purely subjective uncertainty the DM faces is described by a state space . The objects of choice are acts that for each state of nature ω∈, deliver an outcome xfrom a set X.Eachactfis simple, that is, its image f()is a finite subset of X. We denote the set of all acts by F. We identify any outcome x∈Xwith the (constant) act fin which f(ω)=xfor all ω. And with further (albeit fairly standard) abuse of notation, Xwill also refer to the set of constant acts. For any pair of events E,B⊆,B\Eshall denote the set of elements that are in B but not in E. For any pair of acts fand gin Fand any event E⊂,wewritefEgfor the act that agrees with fon Eand with gon \E. The DM is characterized by her preferences over acts, a binary relation on F,with asymmetric and symmetric parts denoted by and ∼, respectively. We begin our description of expected balanced uncertain utility preferences by first noting the DM possesses rich coherent beliefs. This entails the existence of a sufficiently rich collection of (risky) events, constituting a σ-algebra of subsets of , over which can be defined a countably-additive and convex-ranged probability measure μ(her “prior”) with which the DM precisely quantifies the uncertainty she associates with each risky event. We denote the domain of μby R. Countable-additivity requires the probability of the union of a countable collection of disjoint measurable events from Requals the infinite sum of the probabilities of these events. For μto be convex-ranged requires for any event Rin Rand any rin (0, 1)there exists a subset B⊂Rthat is in Rand for which μ(B)=rμ(R).LetFμ⊂Fdenote the set of acts that are measurable with respect to μ. From this point on, the term outcome-set will refer to any nonempty finite subset of Xwith generic elements denoted by Y,Z,Y, etc. As we alluded to in the Introduction, there is a natural way to use this prior to identify with each act its outcome-set envelope. Let Fμbe the set of measurable (with respect to μ) functions f:→{Y⊂ X:Y= ∅,|Y|<∞}. We refer to elements of Fμas outcome-set acts. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 20 (2025) Expected balanced uncertain utility 5 Definition 1 (Envelope of an Act). The outcome-set act f∈Fμis the envelope of fif: (i) f(ω)∈f(ω)for all ω∈,and (ii) for any outcome-set act g∈Fμ: f(ω)∈g(ω)for all ω∈=⇒ μω∈:f(ω)⊆g(ω)=1. To construct the envelope of an act, it is useful first to define the inner measure of μ, denoted by μ∗, that is derived from the prior by assigning to each event E⊂the weight μ∗(E)∈[0, 1]that is the solution to sup R∈R,R⊆E μ(R). Since μis countably additive, the supremum is attained. We shall refer to the measurable event [E]∗∈Ras the inner-sleeve of E,if[E]∗⊆Eand μ([E]∗)=μ∗(E).4 Following Gul and Pesendorfer, we associate with each act a measurable partition of the state space generated by the act’s preimage as follows. Definition 2 (Measurable Split). The measurable split (of the state space) associated with the act f:→Xand denoted by {RY f∈R:Y⊆f(),Y= ∅}is inductively defined as follows: 1. For each element x∈f(),setR{x} f:=[f−1(x)]∗. 2. For each Y⊆f()such that |Y|>1, set RY f:=f−1(Y)∗\ Z⊂Y,Z=∅ RZ f. We refer to RY fas the f-marginal inner-sleeve of the outcome-set Y. To see how the envelope of an act can be constructed using the measurable split generated by its inverse image, first consider a binary act xAy. The measurable split is the three element partition of the state-space R{x} xAy = [A]∗ ,R{y} xAy = [\A]∗ ,R{x,y} xAy = \([A]∗∪[\A]∗). The first (resp., second) element corresponds to the largest measurable subset in which the binary act xAyyields the outcome x(resp., y). For the third element, all the DM can discern is that the outcome will be either xor y. However, she is unable to attribute any fraction of the probability her prior assigns to this element of the split to either xor y 4Notice that the inner sleeve is unique up to a set of μ-measure 0. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6Grant, Roorda, and Yang Theoretical Economics 20 (2025) obtaining alone. Thus, it readily follows from Definition 1that the envelope of xAyis the outcome-set act f∈Fμfor which f(ω)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ {x}if ω∈[A]∗ {y}if ω∈[\A]∗ {x,y}otherwise This method readily extends to an arbitrary act fin F. Using the measurable split {RY f∈R:Y⊆f(),Y= ∅}, its (essentially unique) envelope is the outcome-set act f∈ Fμconstructed by setting f(ω):=Ywhenever ω∈RY f. An outcome-set lottery is a finite ranged function L:{Y⊂X:Y= ∅,|Y|<∞}→ [0, 1]satisfying Y⊂X,|Y|<∞L(Y)=1. We associate with the act fthe outcome-set lottery μ◦f−1. Recalling the approach of Dempster (1967)andShafer (1976), we shall interpret the outcome-set lottery μ◦f−1as encoding how the DM weights that part of the evidence supporting the belief that the act fleads to an outcome in a given set of outcomes obtaining that is not well specified enough to allow her to distribute any of it across any of the elements of that set or any of the other strict subsets of that set of outcomes. In other words, for each outcome-set Y⊆f()we shall interpret μ◦f−1(Y)(=μ(RY f)) as the weight assigned by the DM to evidence that directly supports the act fleading to an outcome in Yobtaining that cannot be further refined in terms of any of the strict subsets of Y. Analogous to Grant’s (1995, p. 163) rendition of Machina and Schmeidler’s (1992) concept of probabilistic sophistication, we require that no relevant preference information is lost by this association. Definition 3 (Coherent Beliefs). The prior μis a coherent belief for the preference relation , if for each pair of acts fand  f, with respective envelopes fandf,f∼ fwhenever μ◦f−1=μ◦f−1. One more element is needed, a balanced outcome-set utility that specifies the utility of each outcome-set in a lottery of a given value. Restricted to singletons, we impose the standard properties of utility in betweenness models, but to determine a useful concept of monotonicity for sets turns out to be a more delicate issue. It involves the choice of a dominance relation between sets of outcomes. To impose a complete dominance relation, as for singletons, would overly restrict the model class. However, it is natural to require the DM strictly (resp., weakly) prefer one outcome-set over another if the DM strictly (resp., weakly) prefers all the outcomes in the former to all the outcomes in the latter. To streamline the exposition, we assume that there exists a best outcome ¯ xand a worst outcome x(i.e., ¯ xxxfor all xin X), and impose a normalization that for any binary act ¯ xRxwith μ(R)=p, its expected balanced utility is zero. Definition 4 (Balanced Outcome-Set Utility). A balanced outcome-set utility is a function U:{Y⊂X:Y= {∅},|Y|<∞}×[0, 1]→Rthat: 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 20 (2025) Expected balanced uncertain utility 7 1. Exhibits outcome-set monotonicity, in the sense that for any pin (0, 1]and any pair of outcome-sets Yand Z,U(Y,p)>(≥)U(Z,p),wheneverforall(y,z)∈Y× Z:U({y},q)=U({z},p)=0=⇒ q>(≥)p;and, 2. Is normalized with respect to some maximal and minimal outcomes in X, denoted ¯ xand x, respectively, in the sense that: pU{¯ x},p+(1−p)U{x},p≡0. (1) It is deemed to be canonical if, in addition, U({¯ x},p)−U({x},p)≡1. For all the examples presented in Section 4, the balanced outcome-set utility we specify will be canonical. This amounts to setting U({¯ x},p):=1−pand U({x},p):= −p. That we always can choose Ucanonical, relies on the fact that rescaling (for a fixed p)U(·,p)by a strictly positive scalar λphas no effect in an EBUU representation, as it becomes evident from the next definition. Definition 5 (EBUU preferences). A preference is EBUU if there exists a prior μand a balanced outcome-set utility Usuch that admits a (probability equivalent) representation V:F→[0, 1]defined as the unique solution to  Y⊆f() UY,V(f)μf−1(Y)=0, where fis the envelope of f.(2) The left-hand side of equation (2) functions as a balance scale, in the sense that, for each probability pand any binary act ¯ xRxwith R∈Eand μ(R)=p:  Y⊆f() U(Y,p)μf−1(Y)≥0⇐⇒ f¯ xRx. 3. Characterization We begin our characterization of EBUU preferences by first specifying what property an event must satisfy that allows us to infer the DM deems it to be “risky,” and thus “measurable.” As we noted above in the Introduction,inGul and Pesendorfer’s (2014)expected uncertain utility theory, this requires both it and its complement satisfy a version of Savage’s (1954) postulate P2. Definition 6 (Ideal Events). An event E⊆is ideal if for any four acts f,g,hand hin F,[fEhgEhand hEfhEg]=⇒ [fEhgEhand h Efh Eg]. Instead, we propose the following property of decomposability. Definition 7 (Decomposable Events). An event R⊆is decomposable if for every pair of acts fand gin F,fRggand gRfg=⇒ fg. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 8Grant, Roorda, and Yang Theoretical Economics 20 (2025) Our axioms will ensure that decomposability is equivalent to the criterion with  replaced by ≺, and entails the nonstrict variants with and as well.5We shall refer to Ras the set of decomposable events. Notice that by construction the set R(like the set of ideal events) is closed under complements. Also, it readily follows that if is complete and transitive, then any ideal event is also decomposable.6The converse, however, need not hold. We refer to an act as decomposable if it is measurable with respect to R,thatis,an act gis decomposable if g−1({x})∈Rfor all x∈X. Hence, loosely speaking, a decomposable act coincides with its own envelope. Let G⊂Fdenote the set of decomposable acts. A subclass of decomposable events are those for which modifying any act on that event leaves it in the same indifference set. These are known as (Savage-)null events. Definition 8 (Null Events). An event N⊆is null if f∼gNffor all f,g∈F.LetNdenote the set of null events. Set R+:=R\N, the class of nonnull decomposable events. And for each non-null decomposable event R∈R+and each act f∈F,setf(R)+:={y∈f(R):f−1(y)∩R/∈N}. That is, f(R)+contains each element in the image of f(R)for which its preimage has a nonnull intersection with R. Analogous to the role played by ideal events in Gul and Pesendorfer (2014), we suppose an EBUU DM uses elements of R+to quantify the uncertainty of any event. So, it seems natural to view an event as maximally ambiguous if it and its complement contain no element of R+. Adopting the terminology of Gul and Pesendorfer, we will refer to such an event (as well as its complement) as diffuse. Definition 9(Diffuse Events). An event D⊆is diffuse if, for every nonnull decomposable event R∈R+,R∩D/∈N,andR∩(\D)/∈N.LetDdenote the set of diffuse events. We say an act his diffuse if its inverse image generates a diffuse partition of ,by which we mean h−1(x)∈Dfor all x∈h()+.LetH⊂Fdenote the set of diffuse acts.7 With these preliminaries in hand, we can now state the axioms. Our first is the standard ordering axiom. Axiom 1 (Ordering). The binary relation is complete and transitive. We next require the collection of events the DM deems unambiguous to be closed under conjunctions. That is, we require for any pair of decomposable events Rand  R that their intersection is also decomposable. 5Where is the relation derived from by setting ffif ff, and ≺is the asymmetric part of . 6If the event Eis ideal, then it follows from Gul and Pesendorfer’s (2014) Lemma B0 (p. 25) that it is also left ideal, that is, gEffimplies gfEgor equivalently, ¬(gfEg)implies ¬(gEff). Thus, it follows from the completeness of that fEggimplies fgEf.Henceif,inaddition,wehavegEfg,thenfg follows from the transitivity of , which in turn follows from the completeness and transitivity of . 7It will turn out that the envelope of any diffuse act h∈Hwill be the constant function h(ω)=h()+for all ω∈. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 20 (2025) Expected balanced uncertain utility 15 the standard (i.e., unadjusted) quasi-arithmetic mean. Moreover, if every Bernoulli utility in the set v(Y)is greater than (resp., less than or equal to) p, then the balanced utility is equal to the difference between the quasi-arithmetic mean of the Bernoulli utilities in the set v(Y)and p. However, if γ>1andpresides in the open interval (vY,¯ vY),then Mpv(Y)<φ −11 |W| v∈v(Y) φ(v). That is, the quasi-arithmetic mean is adjusted downward, reflecting the decisionmaker’s aversion to ambiguity about whether the outcome she will receive from the set Ywill prove disappointing because its Bernoulli utility is less than p,orwillbe a cause for elation because its Bernoulli utility is greater than p. The case for each example in which the balanced outcome-set utility is quasi-linear with respect to p(thus allowing for a straightforward rearrangement of the implicit representation to obtain an explicit representation) corresponds to β=0, α(p)≡0, and γ=0, respectively. And, for each of the three, standard definitions of aversion to ambiguity from the literature respectively correspond to u(v,¯ v)≤(v+¯ v)/2; α(p)≥1/2forall p;φis concave and γ≥0, respectively. 5. Concluding comments Like Gul and Pesendorfer’s (2014) expected uncertain utility, EBUU affords the outside observer the ability to infer those events the DM deems measurable, solely from her preferences. And just as it was the case for ideal events in expected uncertain utility, decomposable events may be viewed as ones for which Savage’s (1954)sure-thing principle applies. Unlike Gul and Pesendorfer, however, we have not “operationalized” this principle by imposing Savage’s postulate P2. Instead, following Grant, Kajii, and Polak (2000), we have interpreted statements like “fwould be preferred to gif the event R were known to obtain” as only entailing fRgg. As a consequence and following as an immediate corollary to Theorem 1, for a DM who exhibits rich coherent beliefs, her risk preferences (over outcome lotteries) need only satisfy the betweenness property of Chew (1983)andDekel (1986), rather than (full) independence. Moreover, the third example from Section 4provides us with a parsimoniously parameterized model that not only can accommodate Ellsberg-style choice patterns but allows the DM to exhibit the novel phenomenon of aversion to ambiguity about disappointment. Although arguably they have received a reasonable amount of attention in the risk literature, preferences that exhibit betweenness properties are almost completely absent in the ambiguity literature. However, as decomposability provides us with a natural way to operationalize the sure-thing principle, we contend EBUU theory provides us with a normatively attractive approach for modeling choice under uncertainty. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 16 Grant, Roorda, and Yang Theoretical Economics 20 (2025) Appendix Proof of Theorem 1 To set the stage for the proof, we first describe how every act can be expressed in terms of constants acts or (and) diffuse acts on its measurable split, as explained in Gul and Pesendorfer (2014). Without loss of generality, we take f()=f()+. Lemma 1. Fix a act f∈Fwith measurable split {RY f∈R:Y⊆f(),Y= ∅}.Foreach nonnull element RY finthemeasurablesplitoff,if|Y|>1, there exists a diffuse act hY∈Hsuch that hY RY f f=f,andifY={y},f=yRY ff. Proof. Given a nondecomposable f∈F, choose a nonnull RY fwith Y={y1,,yn}, n>1(thecasen=1 is obvious). There exists a sequence of disjoint events {Bn}such that Bi=f−1(yi)∩RY ffor all i. Lemma A2 of GP show that there exists a diffuse partition {D1,,Dn}∈Dof . Now define D∗ 1=D1∩\RY f∪B1, ······ D∗ n=Dn∩\RY f∪Bn. We next show that {D∗ 1,,D∗ n}is a diffuse partition of . Assume by way of contradiction that D∗ iis not a diffuse event for some i. Then there is R∈R+such that R∈D∗ i. Since Ris a σ-algebra, (\(RY f))\R∈R.Moreover,(\(RY f))\R∈Bi, which contradicts Bicontaining no decomposable event. Thus, D∗ iis a diffuse event. It is easy to check that D∗ is are all disjoint and their union is . Therefore, {D∗ 1,,D∗ n}is also a diffuse partition of .SethY=(D∗ 1:y1,,D∗ n:yn).ThenhY RY f f=f. Sufficiency of the axioms Step 1: Deriving the prior First, we show that the set of decomposable events constitutes a σ-algebra: That is, (1) it contains both the universal set and the empty set, ∅; (2) it is closed under complements; (3) it is closed under intersection; and (4) it is closed under countable unions. Lemma 2. The set of decomposable events Ris a σ-algebra. Proof. (1) From the definition of a decomposable event, it is immediate that ∅∈Rand ∈R.(2)IfR∈R, then also by definition we have \R∈R.(3)Axiom2ensures Ris closed under intersection. (4) Finally, let {Rn}be a set of (increasing) decomposable events with Rn⊂Rn+1. Assume by way of contradiction that R∞:=Rnis not a decomposable event. That is, there exist f,g∈Fsuch that fR∞ggand gR∞fgbut gf. But since each Rnis decomposable, this means for every neither gfRngor ggRnf(or both). Hence, we can find an infinite subsequence { Rn}of {Rn}with  Rn=R∞, and for which: 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 20 (2025) Expected balanced uncertain utility 17 (i) either gf Rng(x)foralln, (ii) or gg Rnf(x)foralln. If (i) (resp., (ii)) holds, axiom 5implies gfR∞g(resp., ggR∞f) contradicting fR∞gg (resp., gR∞fg). Thus we have established there must exist at least one nfor which fRnggand gRnfg, which since Rnisdecomposable,inturnimpliesfg—a contradiction. Thus, Rnis a decomposable event. Therefore, Ris a σ-algebra. The following auxiliary result is fundamental and used both here and in subsequent steps. Lemma 3. For all x∗x,allf,f∈F,andR∈R+,ifx∗RffxRf, then there is a R∈Rfor which (x∗Rx)Rf∼f. Proof.Ifeitherx∗Rf∼for f∼xRf,setR:=or set R:=∅. We now only need to find a decomposable event Rwhen x∗RffxRf. Since x∗Rff,Axiom6implies there is a decomposable event R1⊂Rsuch that (xR1x∗)Rff. Applying Axiom 6again implies there is a decomposable event R2⊂R\ R1such that (xR1∪R2x∗)Rff. By repeating the argument, this yields a series {Rn}⊂R+ of disjoint subsets of Rthat satisfies xk n=1Rnx∗Rfffor all k.(3) Let Adenote the collection of all such series, and define B:={∞ n=1Rn:{Rn}∈A}. Notice that B⊂R,andthat(x Rx∗)Rfffor each  R∈B,byAxiom5. We show that we can take R=R\Mwith Ma maximal element of Bif it exists, and invoke Zorn’s lemma to establish that otherwise we can take Mas the upper bound outside Bof a chain in B. Lemma (Zorn). Let Pbe a partially ordered set in which each chain Chas an upper bound. Then Phas at least one maximal element. This applies to Bas a partially ordered set, by set-inclusion. First, assume Bhas a maximal element M. We can exclude that (xMx∗)Rff, since otherwise the procedure above would determine ˜ R∈R+for which still ((xM∪˜ Rx∗)Eff, implying that also M∪ ˜ R∈B, as limit of the series (M∪˜ R,∅,)∈A, contradicting that Mis maximal of Bwith M⊂M∪˜ R.So,(x∗R\Mx)f∼f. Next, suppose Bhas no maximal element. By Zorn’s lemma, Bmust contain a chain Cwith upper bound ˜ R∈C˜ R/∈B.NowwecantakeMas this upper bound, we can again exclude that (xM∪˜ Rx∗)Rff. To conclude, in both cases, we can take Ras the decomposable event R\Msuch that (xMx∗)Rf∼f. Next, we establish Machina and Schmeidler’s (1992)AxiomP4∗holds on the restriction of to G(the set of decomposable acts). 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 18 Grant, Roorda, and Yang Theoretical Economics 20 (2025) Lemma 4. For any decomposable events R, R,T∈R,R∪ R⊆T, any four outcomes x∗x and y∗yin X, and any pair of acts f,f∈F: x∗ RxTfx∗  RxTf=⇒ y∗ RyTfy∗  RyTf. Proof.From(x∗ Rx)Tf(x∗  Rx)TfxTf, by applying Lemma 3we can find a decomposable event R⊆Rsuch that (x∗ Rx)Tf∼(x∗  Rx)Tf.ThusbyAxiom4it follows (y∗ Ry)Tf∼(y∗  Ry)Tf. And since R⊆Rit follows from Axiom 3that (y∗ Ry)Tf(y∗ Ry)Tf. The desired implication then follows from the transitivity of . Now we consider the restriction of to binary bets on decomposable events involving the best ¯ xand worst xoutcomes, denoted as F{x,¯ x}, and establish it admits an SEU representation. Our structural assumption on Xthat ¯ xximplies (Savage’s) P5.Axiom 1is P1.Axiom3directly implies P3.P4 is redundant, and Axiom 6is P6. Finally, in a setting with exactly two outcomes ¯ xx,P4∗(as derived above from Axiom 4)is equivalent to P2;cf.Machina and Schmeidler (1992, p. 764). Thus, there is a finitely additive, convex-ranged μon and a function v:{x,¯ x}→R such that V:F{x,¯ x}→R, defined by V(¯ xRx)=μ(R)v(¯ x)+(1−μ(R))v(x),represents on F{x,¯ x}. Without loss of generality, we can set v(x):=0andv(¯ x):=1. Hence V(¯ xRx)=μ(R).(4) Axiom 5implies that V(¯ xR1∪···∪Rnx)converges to V(¯ x∞ n=1Rnx)for any disjoint sequence of decomposable events {Rn},whichyields lim n→∞ n  n=1 μ(Ri)=μ∞  n=1 Rn, that is, μis countably additive. Step 2: Variants of decomposability To prepare for the construction and validation of the EBUU representation, we address some variants of decomposability conditions. Lemma 5. Let be a relation that satisfies Axiom 1and Axiom 3. For any decomposable event E∈Rand any pair of acts f,g∈F: 1. ffEgand fgEfimplies fg; and furthermore, if also satisfies Axioms 2and 6,then 2. ffEgand fgEfimplies fg;and, 3. fEggand gEfgimplies fg. Proof.FixE∈R. We show that statement 1 holds by the same techniques in Grant, Kajii, and Polak (2000). Assume by way of contradiction that there exist two acts f,g∈F, such that ffEg,fgEf,andgf. We consider two cases: 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 20 (2025) Expected balanced uncertain utility 19 (a) Suppose fEggEf.Thenwehave gffEggEf. Set  f:=fEgand  g:=gEf.Notice fE g=fand  gE f=g.Thus  gE f fE g f g. Since Eis decomposable,  gE f fE g fimplies that  g f, which contradicts  f g. (b) Now suppose, gEffEg.Thenwehave gfgEffEg. So again, set  f:=fEgand  g:=gEf, and again notice that  fE g=fand  gE f=g. Thus  gE f fE g g f. Since Eis decomposable,  gE f fE g gimplies that  f g, which contradicts  g f. Therefore, we have established that statement 1 holds. For statement 2, we first show that for all (arbitrary) acts f∈Fand decomposable acts g∈G, f∼fEg∼gEfimplies f∼g(5) Assume by way of contradiction that f∼fEg∼gEfand [fgor gf]: (a) f∼fEg∼gEfand fg. (i) In case, E⊂g−1(¯ x)or \E⊂g−1(¯ x).IfE⊂g−1(¯ x),then f∼fEg∼¯ xEfg=¯ xEg,thatis,fEg¯ xEg and Axiom 3is violated. Similarly, if \E⊂g−1(¯ x),then f∼fE¯ x∼gEfg=gE¯ x,thatis,gEfgE¯ x. Again, Axiom 3is violated. The same argument applies when g(E)or g(\E) only has outcomes indifferent to ¯ x. (ii) Otherwise, Axiom 6implies there is a partition {Ri}such that f¯ xRigfor all Ri. Since Eis nonnull, there exist Rjsuch that Ri∩Eis nonnull. Applying Axiom 6again, there is a partition {R i}such that f¯ xR i(¯ xRjg)for all R i. Then there is R jsuch that R j∩Ecis non-null. Let R=Rj∪R jand so f¯ xRg with both R∩Eand R∩Ecnonnull. Together with Axiom 3,fE(¯ xRg)fand (¯ xRg)Eff,andso ¯ xRgfand we reach a contraction. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 20 Grant, Roorda, and Yang Theoretical Economics 20 (2025) (b) f∼fEg∼gEfand gf. This case can be proved in the same way as what we have done in part (a) by applying Axiom 6on x. Next, we show statement 2 holds. Assume for contradiction that there are f,gsuch that any of the following holds: (i) f∼fEg∼gEfand gf. (ii) ffEg,f∼gEf,andgf. (iii) f∼fEg,fgEfand gf. We only need to show case (i) since the other two are in favor of the direction to get contradiction. Applying Lemma 3, there exist decomposable events R1 gand R2 gsuch that f∼fE(xR1 g¯ x)∼(xR2 g¯ x)Ef and so f∼xR1 g∪R2 g¯ xby Eq. (5). Similarly, there there exist decomposable events R1 fand R2 fsuch that f∼gE(xR1 f¯ x)∼(xR2 f¯ x)Eg(6) Now we let f∗=xR1 g∪R2 g¯ xand g∗=xR1 f∪R2 f¯ x, the above preferences are reduced to f∼f∗∼fEf∗∼f∗ Ef∼gEg∗∼g∗ Eg∼fEg∼gEf Notice that f∗ Ef=(f∗ Eg∗)E(gEf)and gEg∗=(gEf)E(f∗ Eg∗)and f∗ Eg∗∈Gand so f∗ Eg∗∼f∗ by expression (5). Similarly, g∗ Ef∗∼f∗,sof∗∼g∗.From(4), it follows that μ(R1 f)= μ(R1 g)and μ(R2 f)=μ(R2 g). Expression (6) becomes f∼gE(xR1 g¯ x)∼(xR2 g¯ x)Eg∼xR1 g∪R2 g¯ x and so g∼xR1 g∪R2 g¯ x∼f, which gives us the contradiction. Statement 3 can be proved in the same way as statement 2. Lemma 5implies the conditional independence of acts on an indifference class. Proposition 2. For any event R∈Rand any four acts f,f,f,f∗∈F: fRf∼fRf ∼f∗ Rf=⇒ f∗ Rf ∼f∗ Rf∼fRf∼fRf. Proof. By the nonstrict criterion in Lemma 5,fRffRf and fRff∗ Rfimplies fRff∗ Rf. Similarly, fRffRf and fRff∗ Rfimplies fRff∗ Rf.Thus,f∗ Rf∼ fRfimplies f∗ Rf ∼f∗ Rf∼fRf∼fRf. Step 3: Constructing the balanced outcome-set utility The preliminary results above (particularly, Lemmas 3and 5) enable us to define the balanced outcome-set utility U(·,·), as follows. Set U({¯ x},p):=1−pand U({x},p):=−p,forallp∈[0, 1]. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 20 (2025) Expected balanced uncertain utility 21 Fix an outcome set Yand p∈[0, 1]. We employ a shorthand notation ¯ xpxfor an act of the form ¯ xRxwith μ(R)=p.WhenYis not a singleton, choose a diffuse act hY∈H,forwhichitsenvelopehYis the constant function hY(ω)=Yfor all ω∈. When Y={y},sethY:=y. (a) For the case hY¯ xpx, determine a decomposable event Rsuch that hY Rx∼¯ xpx. Such an Rexists, since the balance probability of hY Rwith R∈Ronly depends on μ(R), again by Axiom 4, and we take any R∈Rwith μ(R)=q,forqthe maximum probability of Rsuch that hY R¯ xpx.Noticethatqdoes not depend on the choice of hY,byAxiom3(i). In order for to admit an EBUU representation requires qU(Y,p)+(1−q)U{x},p =pU{¯ x},p+(1−p)U{x},p=0. Solving for U(Y,p)yields U(Y,p):=1−q q×p. (b) Otherwise, determine a decomposable event Rsuch that hY R¯ x∼¯ xpx,andsetq:= μ(R). Again, qdoes not depend on the choice of Rand hY.Inorderforto admit a EBUU representation requires qU(Y,p)+(1−q)U{¯ x},p=0, which yields U(Y,p):=−1−q q×(1−p). By construction, this function satisfies the two properties required for a balanced outcome-set utility. Step 4: Establishing the EBUU representation It remains to verify that μand U, as specified above, constitute a proper EBUU representation that represents the given ordering on F. Fix an arbitrary act fin Fthat has associated with it the measurable split {RY f:Y⊆ f()+}, and the diffuse acts hY ffor which (hY f)RY ff=f. There exists a p∈[0, 1],suchthatf∼¯ xpx(again by Lemma 3, taking R=). For each Y⊆f()+, we can find a decomposable subevent ¯ RY f⊆RY ffor which [¯ x¯ RY fx]RY ff∼ f.Also,thereisA⊂\RY fin Rsuch that f∼fRY f[¯ xAx], and Proposition 2guarantees that also [¯ x¯ RY fx]RY f[¯ xAx]∼f. To analyze these expressions, notice that the value of an act of the form (hY Rx)E¯ xfor measurable events R⊂E, only depends on μ(R)and μ(E), again by Axiom 4. Since we can split Rand Ein ksubsets Ri,Eiwith equal probabilities, respectively μ(R)/k and μ(E)/k,wehave(hY Rix)Ei¯ x∼(¯ xAix)Ei¯ xfor events Ai⊂Ei,withalsoμ(Ai)independent 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 22 Grant, Roorda, and Yang Theoretical Economics 20 (2025) of i. From Proposition 2, it follows now that (hY Rx)E¯ x∼(¯ xAx)E¯ x. This means that this indifference in fact only prescribes the ratio μ(R)/μ(A)for any E∈R. More generally, we know from Axiom 4that the ratio μ(¯ RY f)/μ(RY f)must be the same in all acts of the form hY RY pfthat are on the same indifference curve as f, being all indifferent to hY RY p[¯ xAx]for some A∈R. Since we defined U(Y,p)from the rule hY RY px∼¯ xRpx (when hYf)andhY RY p ¯ x∼¯ xRpx(when fhY), we can determine this ratio as μ¯ RY f μRY f=⎧ ⎪ ⎨ ⎪ ⎩ p q=U(Y,p)+pif hYf q−1+p q=U(Y,p)+potherwise. So, the contribution of each outcome set Yto the EBUU equation is μRY fμ¯ RY f μRY f(1−p)+1−μ¯ RY f μRY f(−p)=μRY fU(Y,p), as desired. Necessity of the axioms Axiom 1follows from the fact that for any f∈F,thereexistsa unique p∈[0, 1]such that (2) holds true for V(f)=p. Let Rdenote the domain of μ,whichisaσ-algebra. To show that the events in Rare decomposable, consider an arbitrary R∈R, and fix a pair of acts f,g∈F,withf∼xR¯ x∈ Xand μ(R)=p.IfgRff,then  Y∈X() U(Y,p)μg−1(Y)∩R> Y∈Xf() U(Y,p)μf−1(Y)∩R.(7) And fRgfimplies that  Y∈Xg() U(Y,p)μg−1(Y)∩\R> Y∈Xf() U(Y,p)μf−1(Y)∩\R.(8) Adding inequalities (7)and(8), we get  Y∈Xg() U(Y,p)μg−1(Y)>0. That is, gf.So,Ronly contains decomposable events. Next, we show that any event outside Ris not decomposable. Consider an event E⊂but E/∈Rμ.Let [E]∗(resp., [\E]∗) denote the inner sleeve of E(resp., \E), and define ˜ E:=\(E∗∪[\E]∗). Since E/∈R,r:=μ(˜ E)>0. By Lemma A2 (p. 22) and the proof of B11 (p. 31) in GP, we can partition E\[E]∗(resp., (\E)\[\E]∗) into two nonnull events B11 and B12 (resp., B21 and B22). Notice by construction none of the four events B11,B12,B21,andB22 contain any nonnull measurable event. To show Eis not decomposable, observe that U({¯ x},0 )>U ({x},0 )=0andU({¯ x}, 0)≥U({x,¯ x},0 )≥U({x},0 )=0. Hence at least one of following two inequalities: (i) U({¯ x,x},0 )>0 and (ii) U({¯ x},0 )>U({¯ x,x},0 )must hold. 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Theoretical Economics 20 (2025) Expected balanced uncertain utility 23 Consider first, the case U({¯ x,x},0 )>U ({x},0 )and the act f=¯ xB11∪B21 x. Since xEf=¯ xB21 xand fEx=¯ xB11 x, it follows that f={x}Ef=fE{x}(i.e., the envelope of each of those three acts are all the same). Since by construction, the measure μis a coherent belief for the preferences generated by the (implicitly defined) EBUU functional, this means f∼xEf∼fEx. However, since 1−μ[E]∗−μ[\E]∗u{¯ x,x},0 +μ[E]∗+μ[\E]∗u{x},0  >1−μ[E]∗−μ[\E]∗u{x},0 +μ[E]∗+μ[\E]∗u{x},0 (=0), it follows that fx,sayf∼¯ xRpxfor some Rpwith μ(Rp)=p>0. To arrive at a violation of the decomposability criterion, choose a measurable event R⊂˜ Ewith 0<μ (R)<p,sothatff:=xRfx. Since the envelopes of f Exand xEfare the same as those of respectively fExand xEf,wehavef Exfand xEff,yetfx. So, Eis not decomposable. So, now consider the case U({¯ x},0 )>U({¯ x,x},0 )and the pair of acts f=¯ xB11∪B21 x and f=x[E]∗∪[\E]∗¯ x. Since fEf=xB11∪B21∪B22 ¯ xand f Ef=xB11∪B12∪B22 ¯ x,f=fEf=f Ef, that is, all three acts come from the same indifference set, and hence f∼fEfand f∼ f Ef. However, since 1−μ[E]∗−μ[\E]∗u{¯ x},0 +μ[E]∗+μ[\E]∗u{x},0  >1−μ[E]∗−μ[\E]∗u{¯ x,x},0 +μ[E]∗+μ[\E]∗u{x},0 (=0), it follows ff. A violation of the decomposability criterion for Ecan be established as above. So, also in this case, Eis not decomposable, and hence Ris the set of all decomposable events. Axiom 2now follows directly from the assumption that Ris a σ-algebra. The necessity of the rest of the axioms follows straightforwardly from the EBUU representation combined with the fact that for any pair of acts fand gwith respective envelopes fand g, and any decomposable event Rin R, the envelope of gRfis gRf. In particular, the envelope of hRfhas outcome-set h()on R,andequalsfoutside R. The necessity of Axiom 4 is now obvious. Axioms 3and 5follow directly from the corresponding properties of U. Axiom 6follows from the fact that μis convex-ranged. Proof of Proposition 1 Let be characterized by μ,U, and satisfy Axiom 7.Givenh,h∈Hwith h()+, h()+⊂(,m),h≥h.Let  hn=h+εn∈Hwith εn>0andlimn→∞ εn=0. Axiom 7 implies  hnhfor all n. Since  hnconverges to huniformly with | hn()=h()|for all n, Axiom 5.1 implies  hnconverges to hin preference, that is, hh. Choose h=xDy∈H with D∈Dand x,y∈(,m)with x>y. Lemma A2 of Gul and Pesendorfer (2014)implies there are disjoint D1,D2∈Dwith D1∪D2=D. Similarly, there are disjoint D 1,D 2∈D with D 1∪D 2=Dc.Foranyzwith x>z>y, define h=xD1zD2yand h =xDzD 1y.We have h ≥h≥hand so h hh. Since h()=h(),h ∼h,thatis,h ∼h∼h. The same argument gives that for all h,h∈H,h∼hwhen maxh()=maxh(), minh()=minh()and h(),h()⊂(,m). 15557561, 2025, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5404 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 24 Grant, Roorda, and Yang Theoretical Economics 20 (2025) Let h,h∈Hwith h()={m,}and h()={m,x1,,xn,}with m>x 1>,,> xn>. Given a positive decreasing sequence {εn}such that m−x1>ε n,xn−>ε n,and for all nand εngoes to 0 as n→∞. Define hn,h nas follows: hn(ω)=h(ω)−εnif ω∈h−1(m)h n(ω)=h(ω)−εnif ω∈h−1(m) hn(ω)=h(ω)+εnif ω∈h−1(l)h n(ω)=h(ω)+εnif ω∈h−1(l) hn(ω)=h(ω)otherwise h n(ω)=h(ω)otherwise and so hn∼h nfor all n. Since hn,h nconverges uniformly to h,hrespectively with |hn()|=|h()|and |h n()|=|h()|for all n.Axiom5implies h∼h. Therefore, for all h,h∈H,h∼hif maxh()=max h()and min h()=min h().Andsoforall outcome-sets Yand all p∈(0, 1),U(Y,p)=U({minx∈Yx,maxy∈Yy},p). References Chew, Soo Hong (1983), “A generalization of the quasilinear mean with applications to meaurement of income inequality and decision theory resolving the Allais paradox.” Econometrica, 51, 1065–1092. [0002,0012,0015] Dekel, Eddie (1986), “An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom.” Journal of Economic Theory, 40, 304–318. [0002, 0012,0015] Dempster, Arthur P. 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