Hoping for the best while preparing for the worst in the face of uncertainty: A new typemof incomplete preferences
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Bardier, Pierre; Xuan, Bach Dong; Nguyen, Van-Quy Working Paper Hoping for the best while preparing for the worst in the face of uncertainty: A new typemof incomplete preferences Center for Mathematical Economics Working Papers, No. 701 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Bardier, Pierre; Xuan, Bach Dong; Nguyen, Van-Quy (2025) : Hoping for the best while preparing for the worst in the face of uncertainty: A new typemof incomplete preferences, Center for Mathematical Economics Working Papers, No. 701, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-30009222 This Version is available at: https://hdl.handle.net/10419/312917 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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/4.0/
701 January 2025 Hoping for the best while preparing for the worst in the face of uncertainty: a new type of incomplete preferences Pierre Bardier, Bach Dong-Xuan and Van-Quy Nguyen Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en
Hoping for the best while preparing for the worst in the face of uncertainty: a new type of incomplete preferences∗ Pierre Bardier†Bach Dong-Xuan‡Van-Quy Nguyen§ January 2025 Abstract We propose and axiomatize a new model of incomplete preferences under uncertainty, which we call hope-and-prepare preferences. An act is considered more desirable than another when, and only when, both an optimistic evaluation, computed as the welfare level attained in a best-case scenario, and a pessimistic one, computed as the welfare level attained in a worst-case scenario, rank the former above the latter. Our comparison criterion involves multiple priors, as best and worst cases are determined among sets of probability distributions. We make the case that, compared to existing incomplete criteria under ambiguity, hope-and-prepare preferences address the trade-off between conviction and decisiveness in a new way, which is more favorable to decisiveness. Keywords: Decision theory; Incomplete preference; Multiple-selves; Non-obvious manipulability. JEL classification: D01; D81; D90 ∗We deeply thank Marc Fleurbaey, Antonin Macé, William Thomson, Xiangyu Qu, and Frank Riedel for their support and their advice. We also thank the participants to the theory seminar of the University of Rochester, economic theory lunch seminar of the Bielefeld University, the TOM seminar of the Paris School of economics, the theory seminar of the Karlsruhe Technical University, and the participants to the 17th Meeting of the Society for Social Choice and Welfare (Paris), and the Time, Uncertainties & Strategies X conference (Paris). †Paris School of Economics, École Normale Supérieure de Paris (Paris Sciences et Lettres). E-mail: [email protected] ‡Center for Mathematical Economics, Bielefeld University. Financial support by the German Research Foundation (DFG) [RTG 2865/1 – 492988838] is gratefully acknowledged. E-mail: [email protected] §Université Paris-Saclay, Univ Evry, EPEE, France. Faculty of Mathematical Economics, National Economics University, Vietnam. E-mail: nguyenv[email protected] 1
1 Introduction “Hoping for the best, prepared for the worst, and unsurprised by anything in between.” - Maya Angelou, I Know Why the Caged Bird Sings. The complexity of economic decisions is likely to result in agents’ inability or unwillingness to decide over the uncertain options they are supposed to compare. In this regard, the restrictiveness of the assumption that individual preferences be complete was early acknowledged,1and was recently highlighted by empirical studies.2We propose and characterize a new incomplete decision criterion according to which, in the face of Knightian uncertainty (Knight (1921)), agents both hope for the best and prepare for the worst. We study preferences over acts f:S→X, which are mappings from states of the world to outcomes, and we introduce and axiomatize preferences admitting the following representation: fg⇐⇒ minp∈C∫u(f)dp > minp∈C∫u(g)dp maxp∈D∫u(f)dp > maxp∈D∫u(g)dp ,(1) where uis a numerical representation of preferences over outcomes, and Cand Dare sets of probability distributions over the states, interpreted as sets of different scenarios.3Thus, a decision maker (DM) following such a criterion ranks an act fabove an act gif and only if fprovides a higher expected utility than gin the worst-case scenario in Cas well as in the best-case scenario in D. Our criterion is based on the conjunction of an optimistic (or ambiguity-seeking) assessment and of a pessimistic (or ambiguity-averse) assessment.4We then interpret a DM with such a preference as hoping for the best scenario to realize, while also preparing for the worst one to happen, when evaluating each option: we thus refer to a preference relation admitting such a representation as a hope-and-prepare preference. As a brief illustration, think 1For instance, Aumann (1962) wrote: “Of all the axioms of utility theory, the completeness axiom is perhaps the most questionable. Like others of the axioms, it is inaccurate as a description of real life; but unlike them, we find it hard to accept even from a normative viewpoint.”Schmeidler (1989), commenting on his characterization of the maxmin criterion, depicted the completeness axiom as “the most restrictive and demanding assumption.” 2See Cettolin and Riedl (2019), Nielsen and Rigotti (2022). 3The function u:X→Ris non-constant, affine and unique up to positive affine transformation. The sets Cand Dare unique, non-disjoint, compact and convex. 4Cand Dbeing non-disjoint, the expected utility in the best-case scenario is higher. 2
of a company considering launching a new product. Typically, such a dual policy of decision making would favor investment or production strategies that present promising profit opportunities, in case the product captures an important market share, and a substantial safeguard, in case the product does not. We shall give special attention in our analysis to the concordant case in which C=D —for which we also provide an axiomatization. Acts are then evaluated according to the interval of all expected utility levels that they induce across all possible scenarios. More precisely, an act fis preferred to an act gif and only if any expected utility level that is attainable from gbut not from fis below any expected utility level that is attainable from f, and there exists at least one level that is indeed attainable from gbut not from f.This intuitive criterion for comparing ranges of expected utility levels works as a strict version of the strong set order, which is, arguably, the most common way to compare intervals. Importantly, hope-and-prepare preferences treat the optimistic and the pessimistic assessments symmetrically: therefore, they do not systematically display a particular attitude toward ambiguity, which is consistent with extensive empirical evidence (see Trautmann and van de Kuilen (2015) for a survey). The conjunction of a best-case evaluation and of a worst-case evaluation at play in our criterion is akin to the one at play in the notion of obvious manipulation (Troyan and Morrill (2020)), defined for revelation games in which the uncertainty faced by an agent concerns others’ messages. Accordingly, the significant practical relevance of the notion of obvious manipulation provides support for our criterion within uncertain strategic environments. This notion gives an explanation, for instance, of untruthful reporting strategies that have been consistently observed in the Immediate Acceptance mechanism, used to match students with schools.5 The scope for applications of our criterion goes beyond strategic interactions. The idea that both worst-case and best-case scenarios serve as reference points is recognized for various social and economic domains where ambiguity is present. In this regard, let us simply mention the evaluation of financial assets (Bossaerts et al. (2010), Schröder (2011), Ahn et al. (2014)), or the evaluation of different medical treatments by physicians and patients (Back et al. (2003), Taylor et al. (2017)); we discuss a third example in more detail. It is not unusual for practitioners, reporters or fans to evaluate “young prospects” participating in the annual Draft in North-American sports leagues —we take the example of the 5See Pathak and Sönmez (2008) and Dur et al. (2018). 3
National Basketball Association league (NBA)— according to “ceiling and floor scenarios.”6 This can be formulated in our framework. There is potentially a myriad of parameters that the agent considers relevant for the evaluation of prospects: a state is a particular configuration of parameters.7In this complex environment, the agent faces ambiguity and must compare prospects on the basis of a set Cof probability distributions over configurations of parameters. Loosely speaking, each player is identified with an act f, indicating their overall performance in each state, which is then evaluated according to a utility function u, and, for every scenario p∈C, the agent can compute the expectation of u(f)according to p. Then, the incompleteness of a criterion such as ours reflects the necessity to have sufficient conviction when declaring that a player is more promising than an other one. On the other hand, a criterion should not be too incomplete; let us illustrate this point by comparing our criterion to two alternative ones. Given uand C, the agent could require, for a “player f” to be declared more promising than a “player g”, that, for each scenario in C, the expected utility associated with gbe lower than the expected utility level associated with f(Bewley (2002)). One could even require that any expected utility level attainable from gbe lower than any expected utility level attainable from f(Echenique et al. (2022)) —that the “ceiling” of gbe lower than the “floor” of f. Both of these conditions are stronger than condition (1), expressing a more demanding notion of sufficient conviction. However, it may very well be the case that only “generational talents” such as Victor Wembanyama,8 (who was present in the 2023 Draft) be distinguished from other players on the basis of these more conservative criteria, and that for rather homogeneous cohorts such as the 2024 cohort, the agent fail to rank any player above an other one.9In practical terms, according to our criterion, a “player f” is declared more promising than a “player g” if and only if anything that gcould achieve and that fcould not is considered worse than anything fcould achieve. With hope-and-prepare preferences, which, in this case, compare players on the basis of the associated ranges of expected utility, in a way that is reminiscent of the strong set order, the trade-off between decisiveness and conviction is addressed in a way that is more favorable to 6See, for example, James Hansen, “What makes an NBA Draft prospect high ceiling or high floor?”, SLC Dunk, June 2023, and Kyle Boone, “NBA Draft 2024 ceiling and floor scenarios: The best or worst case projections for five top prospects”, CBS Sports, June 2024. We note that the use of the expressions “ceiling” and “floor” suggests that any case lying “in between” is considered possible. 7A state may thus encompass the roasters of coaches and players, at the beginning of the season and after the winter “trade” period, of each franchise, their financial capacities, the performance of players already in the league, the progression of each of these prospects, the approach to officiating favored by the league’s executives, etc. 8See, for example, Sam Harris, “Why ‘alien’ Wembanyama is France’s next big thing - literally”, BBC Sports, July 2024. 9See, for example, Adam Finkelstein, “No stars have revealed themselves in the 2024 NBA Draft, but history tells us they’re hiding in plain sight”, CBS Sports, June 23 2024. 4
decisiveness. The two original axioms involved in our characterization are interpreted along this line: we propose in Axiom 6a relatively strong sufficient condition for incomparability —so that Axiom 6is satisfied by the vast majority of incomplete criteria defined on single acts proposed in the literature— and a relatively weak sufficient condition for comparability in Axiom 7. Our axiomatization maintains the assumption that preferences are complete over constant acts, deemed as the simplest ones. Axiom 6underscores the role of constant acts as benchmarks for decision making: if the DM is unable to compare the act gto the constant act x whenever she is unable to compare xto f, then she is not able to compare fand g. According to Axiom 7, if i) the DM cannot compare fto the constant x, while she declares xmore desirable than gand, on the other hand, ii)she cannot compare gto the constant act y, while she declares fmore desirable than y, then she declares fmore desirable than g. Thus, two specific aligned pieces of evidence are enough to conclude that an act is better than an other one, and Axiom 7may be seen as formulating a minimal departure from the completeness of a standard expected utility preference relation —we refer the reader to Section 3.1.1 for a more precise discussion. Furthermore, in order to account for typical situations in which agents have to choose between two options, even if they lack conviction to express a clear preference between them in the first place, we study the completion of hope-and-prepare preferences.10 We demonstrate that the invariant biseparable complete extension of a hope-and-prepare preference admits an asymmetric11 α-maxmin expected utility (α-MEU) representation —and a standard α-maxmin representation if the hope-and-prepare preference is concordant. Notably, the asymmetric α-MEU retains much of the tractability of the standard α-MEU —which is beneficial for applications— while remaining flexible enough to accommodate mixed ambiguity attitudes (Chandrasekher et al. (2022)).12 Importantly, in the representation we obtain, the weight αdoes not depend on the considered acts, and is unique whenever the extended hope-and-prepare preference is incomplete. Finally, answering two natural questions of comparative statics that emerge from the proposition of a new type of incomplete preference under ambiguity, we compare the degree of incompleteness of our criterion to that of Bewley preferences (Bewley (2002)) and of twofold preferences (Echenique et al. (2022)), and we provide a way to compare the ambiguity 10From a theoretical point of view, studying a completion of an incomplete preference relation enables to use standard mathematical tools, for example for utility maximization and welfare analysis. 11“Asymmetric” refers to the fact that best and worst cases may be taken on different sets of scenarios. 12Specifically, it captures ambiguity-averse behavior for large/moderate-likelihood events, ambiguityseeking behavior for small-likelihood events, and source-dependent ambiguity attitudes (Chandrasekher et al. (2022)). 5
attitudes of two hope-and-prepare preferences. Our paper is organized as follows: we define the formal framework and introduce our criterion in Section 2. In Section 3, we give the main representation result and explore the case in which the sets of scenarios used in the two assessments are equal. In Section 4, we investigate the completion of our criterion. Section 5is dedicated to the comparative statics questions mentioned above. Section 6provides an illustrative comparison of concordant hope-and-prepare preferences to Bewley preferences, in the context of the aggregation of opinions of experts. The conclusions are presented in Section 7. All proofs can be found in the appendix. 1.1 Related literature A DM hoping for the best while also preparing for the worst responds to uncertainty by combining opposite ambiguity attitudes. In this perspective, one may interpret a DM with a hope-and-prepare preference as requiring that her optimistic (ambiguity loving) self and her pessimistic (ambiguity averse) self be unanimous for her to rank some act above an other one. The idea that the DM consists of multiple (strategic) selves appears frequently in behavioral economics, in particular in models of dynamic choice or choice within risky environments.13 In recent works, Chandrasekher et al. (2022) and Xia (2020) provided axiomatizations for preferences involving two selves, called by the former dual-self expected utility. Their representation differs from ours in that the agent’s final decision is to be interpreted as the result of a specific leader-follower game between an optimistic self and a pessimistic self, whereas, in our representation, it is induced by a requirement of unanimity imposed by the agent herself on the assessments of her two selves.14 Our representation is also motivated by the concept of obvious manipulation proposed in the context of mechanism design by Troyan and Morrill (2020). A revelation mechanism is said to be non-obviously manipulable if, for any agent and any potential untruthful report from her, revealing her own type leads to a more desirable outcome in both of the following cases: when the others’ reports are the most favourable to her, and when they are the least favourable. In our model, in the same spirit, an option —such as an untruthful report in the previous example— is only abandoned for an alternative if this alternative leads to preferred 13Thaler and Shefrin (1981), Bénabou and Pycia (2002), Fudenberg and Levine (2006), Brocas and Carrillo (2008). 14At a hight-level, the difference of our approach with that of “Preparing for the Worst but Hoping for the Best: Robust (Bayesian) Persuasion” (Dworczak and Pavan (2022)) is similar to the difference with Chandrasekher et al. (2022): in the criteria studied in both of these papers, one of the (pessimistic or optimistic) evaluations constrains the other. It is not the case with hope-and-prepare preferences in which both evaluations are treated symmetrically. 6
outcomes in both the best and the worst scenarios among given sets of probability measures. The relation of our contribution to the concept of non-obvious manipulation mirrors that of Echenique et al. (2022) to the concept of obvious dominance, due to Li (2017): informally, when the set of scenarios according to which all acts are evaluated is the simplex, act fis preferred to act gby a twofold multi-prior preference if and only if fobviously dominates g, and, on the other hand, fis preferred to gby a hope-and-prepare preference if and only if f dominates gin the sense of Troyan and Morrill (2020).15 Hope-and-prepare preferences define a partial order on acts. Pioneering work by Aumann (1962), Bewley (2002) and Dubra et al. (2004) studied the representation of incomplete preferences under risk and uncertainty. Incomplete preferences in non-deterministic environments have been the object of a growing literature: see, for example, Nascimento and Riella (2011)Galaabaatar and Karni (2012), Efe et al. (2012), Faro (2015), Minardi and Savochkin (2015), Hill (2016), Karni (2020), Cusumano and Miyashita (2021) and Echenique et al. (2022). The closest model of incomplete preference to ours, apart from those studied in Bewley (2002) and Echenique et al. (2022), both compared to ours in the introduction, is introduced in Nascimento and Riella (2011). As a special case of their main result, they study a criterion in which the DM considers several sets of scenarios, in each of which the performance of an act is evaluated according to the worst-case expected utility level. Then, an act is preferred to an other one if and only if it performs better in each set of scenarios. Hope-and-prepare preferences enable to capture a different type of ambiguity attitude, through the consideration of the optimistic assessment. We discuss in more details how our work relates to Bewley (2002), Nascimento and Riella (2011) and Echenique et al. (2022) in the next sections. In line with Hurwicz’s approach for decision making under complete ignorance (Hurwicz (1951)), the α-MEU model was proposed to capture the idea that, under ambiguity, worst and best expected utility levels, over one set of probability measures, can serve as sufficient statistics for the DM: she then computes an α-weighted average of these levels (Marinacci (2002), Kopylov (2002), Ghirardato et al. (2004)).16 Among the recent explorations of (variants of) the α-maxmin model,17 the one of Frick et al. (2022) is particularly important for the way we characterize the asymmetric α-maxmin model as representing the completion of 15With this same set of scenarios, one can also recover the concept of strategy-proofness from Bewley preferences (Bewley (2002)). 16Let us mention two alternatives to the standard α-MEU model. The geometric α-MEU model (Binmore (2009)) uses a geometric weighted average. More recently, Grant et al. (2020) introduced and characterized a general aggregation of best-case and worst-case expected utility representations, referred to as ordinal Hurwicz expected utility. 17Chateauneuf et al. (2007), Eichberger et al. (2011), Gul and Pesendorfer (2015), Frick et al. (2022), Klibanoff et al. (2022), Hartmann (2023), Hill (2023), Chateauneuf et al. (2024). 7
for uncertainty: when the DM has sufficient conviction to declare two uncertain acts less desirable than the constant act x, then the DM also considers with sufficient conviction that an act obtained through hedging between the two is less desirable than x. Axiom 5. For all f, g ∈ F, if f(s)g(s)for all s∈S, then fg. According to Axiom 5, if the outcome of an act is considered more desirable than the outcome of an other act in each state of the world, then the first act is preferred to the second one. In other words, according to a preference relation satisfying Axiom 5, the state-wise dominance of an act fover an act gprovides sufficient conviction to rank fabove g. In the perspective of the trade-off between decisiveness and conviction, we see this property as an intuitive limitation of incomparability. While it is imposed in most approaches close to ours, the strong degree of conservatism, or indecisiveness, of twofold preferences is rooted in the fact that they violate it.32 We propose in Axiom 6a relatively strong sufficient condition for incomparability — equivalently, a relatively weak necessary condition for comparability— so that Axiom 6is satisfied by almost all (the asymmetric part of) the incomplete criteria comparing single acts mentioned in Section 1.1, that is, almost all the incomplete criteria defined in a classical Anscombe-Aumann framework mentioned in Section 1.1. More precisely, the (asymmetric part of) the criteria proposed in Bewley (2002), Nascimento and Riella (2011), Efe et al. (2012), Faro (2015), Cusumano and Miyashita (2021) and Echenique et al. (2022) all satisfy Axiom 6(see Appendix A).33 On the other hand, we impose a relatively weak sufficient condition for comparability in Axiom 7. We jointly discuss these axioms after we briefly present them. Axiom 6. For all f, g ∈ F, if for all x∈X,f’ximplies g’x, then f’g. Axiom 6underscores the role of constant acts as benchmark acts based on which comparisons of more complex acts are made: for the DM to express a preference between the acts fand g,it is necessary that there exists a constant act xthat the DM prefers to either for g, while she cannot compare xwith the other act.34 Axiom 7. For all f, g ∈ F, and for all x, y ∈X, if f’x,xg,g’yand fythen fg. 32See Cusumano and Miyashita (2021) and Echenique et al. (2022). 33There is one incomplete criterion mentioned in Section 1.1 that is defined on single acts and that may not satisfy Axiom 6, the one proposed in Hill (2016). 34Note that we do not impose that whenever there exists x∈Xsuch that f’xand g’x, then f’g (which is Axiom 5 in Echenique et al. (2022)). Actually, our criterion does not satisfy this property in general. 14
While the DM cannot compare fto the constant x, she declares xmore desirable than g. On the other hand, while she cannot compare gto the constant act y, she declares fmore desirable than y. Axiom 7implies that in the presence of such consonant conclusions as to the comparison of fand g, the DM considers f, with sufficient conviction, more desirable than g. As we already highlighted, given the complexity involved in the evaluation of uncertain acts, constant acts, which are the simplest acts, are likely to be used as comparison devices. A straightforward way to use them in comparing two acts, when preferences may be incomplete, then consists in looking for a constant act that is incomparable to one of them and comparable to the other one. Each such constant act then provides a piece of evidence as to the comparison between the two uncertain acts —the question is then to determine what are sufficient pieces of evidence. Consequently, given two acts fand g, the DM we model compares to gall constant acts that are incomparable to f, and vice versa. This process gives rise to three possible cases: (i) for all x∈Xsuch that f’x,g’x; (ii) for all x∈Xsuch that g’x,f’x; and (iii) there are x, y ∈Xsuch that [f’xand gand xare comparable] and [g’yand fand yare comparable]. In the first two cases, there is no piece of evidence on which the DM may base her comparison: Axiom 6implies that fand gare incomparable. In the last case, there are four possible situations; it suffices to consider the following two, to which the other ones are symmetric: (a) [f’xand xg]combined with [yfand g’y]; and (b) [f’xand xg]combined with [fyand g’y]. In case (a), the first piece of evidence favors fwhile the second favors g. In contrast, in case (b), the two pieces of evidence go in the same direction, favoring f: according to Axiom 7, this is sufficient to conclude that fis more desirable than g. There is a sense in which Axiom 7expresses, for an asymmetric and incomplete preference relation, a minimal departure from the completeness of weak orders for which all acts admit acertainty equivalent.35 Using the previous formulation, for these weak orders, one piece of 35That is, binary relations Áwhich are reflexive, transitive and complete, such that, for all f∈ F, there is x∈Xsuch that f∼x. 15
evidence is sufficient: if f∈ F has a certainty equivalent x∈Xand xis strictly preferred to g∈ F, then fis strictly preferred to g. For an asymmetric and incomplete preference relation, Axiom 7involves no more than one piece of evidence based on a constant act incomparable to fand one piece of evidence based on a constant act incomparable to g. Axiom 7is violated by twofold preferences, Bewley preferences and N&R preferences in general. Through the satisfaction of both Axioms 6and 7, in particular, hope-and-prepare preferences address the trade-off between decisiveness and conviction in a new way. We sometimes refer to the classical Axioms 2,3and 5as continuity, certainty independence and monotonicity. 3.1.2 First characterization theorem Theorem 1. A binary relation satisfies Axioms 1-7if and only if there exist •a non-constant affine function u:X→R, unique up to positive affine transformation, •a unique pair (C, D)of non-disjoint convex compact subsets of ∆, such that, for all f, g ∈ F, fg⇔ minp∈C∫u(f)dp > minp∈C∫u(g)dp maxp∈D∫u(f)dp > maxp∈D∫u(g)dp , that is, admits the hope-and-prepare representation (u, C, D), where Cand Dare unique, and uis unique up to positive affine transformation. We now give a brief sketch of the proof and highlight some interesting properties of that we derive.36 First of all, Axioms 1-3guarantee that there exists a non-constant affine function u:X→R, unique up to affine transformation, representing on X. The proof consists in defining two binary relations on F, denoted pand o, such that for any f, g ∈ F,fgif and only if fpgand fog—we provide the precise definitions of these relations below. In that perspective, the following two lemmas are crucial. Lemma. For all f∈ F, the set {x∈X:x’f}is non-empty. Lemma. For all f∈ F, and x, y, z ∈X, if x’f,fy, and zf, then zxy. This second result has an interesting interpretation. While the DM cannot assert with sufficient conviction that fis more desirable than the constant act x, she considers with 36The following lemmas are not presented here in the order in which they are proved. 16
sufficient conviction that fis more desirable than the constant act yand worse than the constant act z. We show that in such a case, the DM considers, with sufficient conviction, that zis more desirable than x, and that xis more desirable than y. From the original relation , we define two preference relations on Fas follows: gpf⇐⇒ gxand x’ffor some x∈X, gof⇐⇒ x’gand xffor some x∈X. The subscripts pand oare used to denote respectively a pessimistic and an optimistic assessment, based on , where these two terms are justified given the way the incomparability to a constant act is treated. Let us describe the interpretation of p: this relation is pessimistic in the sense that for the default act f, whenever there is a constant act x such that fcannot be compared with sufficient conviction to x, while gis considered more desirable than xwith sufficient conviction, then pdeclares fto be worse than g. We then proceed by showing that pand oare asymmetric and negatively transitive. This enables us to define ∼pby f∼pgif and only if fčpgand gčpf, for all f, g ∈ F, and to define Ápby fÁpgif and only if either fpgor f∼pg, for all f, g ∈ F. We define in the same way ∼oand Áo. Then it is clear that Ápand Áoare weak orders,37 and we show that they are continuous and monotone, that they satisfy the classical properties of certainty independence, and, respectively, aversion to ambiguity and preference for ambiguity.38 As a consequence, Ápcan be represented by the function f7→ minp∈C∫up(f)dp, and Áocan represented by the function f7→ maxp∈D∫uo(f)dp, where Cand Dare non-empty convex compact subsets of ∆, and upand uoare two affine functions on X. We conclude that there is no loss of generality in assuming up=uo=u, and that C∩D6=∅, using the separating hyperplane theorem on these subsets of ∆endowed with the weak* topology. Note that in this sketch of proof, the relation between and the two weak orders pand ois established before the minmax and maxmax representations of pand o: Axioms 1-3 and Axioms 5-7are necessary and sufficient for a general representation that we describe in Appendix B. 37They are non-trivial asymmetric and negatively transitive binary relations. 38Definitions of these properties for weak orders are provided in the appendix. 17
3.2 Characterization of concordant hope-and-prepare preferences 3.2.1 Axioms The necessary and sufficient conditions identified in Echenique et al. (2022) for the identity C=Dto hold in their twofold multiprior preference representation are also necessary and sufficient in our representation.39 Before introducing them, let us specify that, as suggested in the sketch of the proof of Theorem 1, when satisfies Axioms 1-3, we define on Xthe relation Áby xÁyif and only if yčxfor all x, y ∈X. Clearly, Áon Xis asymmetric and negatively transitive; and ’is equivalent to ∼, the symmetric part of Á, on X. We use the notion of complementary acts (Siniscalchi (2009)) to identify comparisons that are, under Axioms 1-7, characteristic of the uncertainty aversion of the agent’s pessimistic evaluation, and of the preference for uncertainty of her optimistic evaluation, respectively. Two acts fand gare complementary if they perfectly hedge against each other in the sense that their equal-weight-mixture is equivalent to a constant act: 1 2f(s) + 1 2g(s)∼1 2f(s′) + 1 2g(s′)for all s, s′∈S. Axiom 8. If fand gin Fare complementary, then f1 2f+1 2gimplies 1 2f+1 2gg. Consider two complementary f, g ∈ F, and a preference with representation (u, C, D) on F. Assume f1 2f+1 2g, as in Axiom 8and let x∈Xdenote a constant act such that x∼1 2f+1 2g. It cannot be the case that g1 2f+1 2g, because this would imply fxand gx, and thus 1 2f+1 2gx—a contradiction. In other words, if f1 2f+1 2g, then either 1 2f+1 2g’gor 1 2f+1 2gg. Axiom 8 requires that the second case hold, and this requirement is interpreted as a consequence of the simplicity of constant acts. Indeed, by transitivity, in this second case, one has, by transitivity, fg, so that Axiom 8states that whenever f1 2f+1 2g, one has fg, that is, it should always be easier for the DM to assess whether fis more desirable than the essentially constant act 1 2f+1 2gthan to assess whether fis more desirable than g. The interpretation of Axiom 9is similar: it states that for complementary acts f, g ∈ F, it should always be easier for the DM to assess whether the essentially constant act 1 2f+1 2g is more desirable than gthan to assess whether fis more desirable than g. Axiom 9. If fand gin Fare complementary, then 1 2f+1 2ggimplies f1 2f+1 2g. 39The proof of the following result is a direct adaptation of the proof of Proposition 1 in their paper. 18
3.2.2 Second characterization theorem Theorem 2. The following statements hold: (i) A hope-and-prepare preference , with unique representation (u, C, D), satisfies Axiom 8if and only if D⊆C. (ii) A hope-and-prepare preference , with unique representation (u, C, D), satisfies Axiom 9if and only if C⊆D. In particular, a binary relation satisfies Axioms 1-9if and only if there exist •a non-constant affine function u:X→R, unique up to positive affine transformation, •a unique convex compact subset of ∆, denoted C, such that, for all f, g ∈ F, fg⇐⇒ minp∈C∫u(f)dp > minp∈C∫u(g)dp maxp∈C∫u(f)dp > maxp∈C∫u(g)dp . When admits a concordant representation, acts are evaluated according to the minimum and the maximum expected utility level attained on a common set of scenarios. On the other hand, when satisfies both Axiom 8and 9, for any simple complementary acts f and g,f1 2f+1 2gif and only if 1 2f+1 2gg. In other words, for complementary acts, it is always as easy to determine whether their equal-weight-mixture is more desirable than one of them as it is to determine whether one of them is more desirable than the mixture. As a recall, according to a concordant hope-and-prepare preference relation, acts are evaluated according to the interval of all expected utility levels that they induce across all scenarios in a given set. More precisely, an act fis preferred to an act gif and only if any expected utility level that is attainable from gbut not from fis below any expected utility level that is attainable from f, and there exists at least one level that is indeed attainable from gbut not from f. 4 Complete extension of hope-and-prepare preferences In this section, we will explore the extension of hope-and-prepare preferences to complete preferences. We will focus on the invariant biseparable complete extension; these define a broad class of complete preferences that nests the majority of preferences studied in the literature. 19
We refer to an asymmetric complete and negatively transitive binary relation on Fsatisfying Axioms 2,3and 5as invariant biseparable.40 When it is, in addition, a weak order, it satisfies the axioms characterizing expected utility, apart from the independence axiom, which is weakened to the certainty independence property introduced in Gilboa and Schmeidler (1989). Definition 5. A preference relation on Fadmits an asymmetric α-MEU representation if there exist α∈[0,1], two non-disjoint compact convex subsets Cand Dof ∆, and a non-constant affine function u:X→Rsuch that for all f, g ∈ F, fg⇐⇒ αmin p∈C∫u(f)dp + (1 −α) max p∈D∫u(f)dp > α min p∈C∫u(g)dp + (1 −α) max p∈D∫u(g)dp. We will refer to such representation as a (u, C, D, α)representation. Remarkably, Chandrasekher et al. (2022) show that the asymmetric α-MEU, while retaining the tractability property of the standard α-MEU, is flexible enough to accommodate ambiguity-averse for large/moderate-likelihood events but ambiguity-seeking for smalllikelihood events and source-dependent ambiguity attitudes. Standard α-MEU criteria are obtained if C=Din Definition 5, and the following result, as a particular case, characterizes them as invariant bi-separable extensions of concordant hope-and-prepare preferences. Theorem 3. The following conditions are equivalent when is a hope-and-prepare preference with unique representation (u, C, D): (i) ∗is an invariant biseparable preference and an extension of . (ii) ∗admits an α-maxmin expected utility representation (u, C, D, α)in which αis unique whenever is not complete. 40Ghirardato et al. (2004) originally used the expression “invariant biseparable preferences” when studying weak-orders. For an asymmetric complete and negatively transitive binary relation on F, as for pand oin Section 3, we define ∼by f∼gif and only if fčgand gčf, for all f, g ∈ F, and Áby fÁgif and only if either fgor f∼g, for all f, g ∈ F. Then, in the proof of Theorem 3, we show that Á∗is an “invariant biseparable preference” in the sense of Ghirardato et al. (2004). 20
5 Comparison of incomplete criteria 5.1 Degree of incompleteness We have stated that with hope-and-prepare preferences, in comparison to Bewley preferences and twofold preferences, the trade-off between decisiveness and conviction is addressed in a way that is more favorable to decisiveness. The criterion we use to determine whether a binary relation is more conservative than an other one pertains to their respective degree of incompleteness. Definition 6. Given two preference relations 1and 2on F, we say that 1is more conservative than 2if 2is an extension of 1, that is, for all f, g ∈ F, f1gimplies f2g. The next proposition identifies necessary and sufficient conditions under which a hopeand-prepare preference relation is an extension of a Bewley or of a twofold preference relation. Proposition 1. Let Hbe a hope-and-prepare preference with unique representation (u, CH, DH). Let Tbe a twofold multiprior preference with unique representation (u, CT, DT). Let B be a Bewley preference with unique representation (u, CB). Then, (i) the preference relation Bis more conservative than Hif and only if CH∪DH⊆CB; (ii) the preference relation Tis more conservative than Hif and only if CH⊆CTand DH⊆DT. Remark 1. A direct consequence of this proposition and Proposition 4 in Echenique et al. (2022) is that if CH∪DH⊆CB⊆CT∩DT, in particular if CH=DH=CB=CT=DT, then Tis more conservative than B, which is more conservative than H. 5.2 Ambiguity attitudes We are able to compare ambiguity attitudes displayed by different hope-and-prepare preferences using the classical comparative statics notions of Ghirardato and Marinacci (2002). Definition 7. Given two preference relations 1and 2on F, (i) 1is more ambiguity averse than 2if, for all f∈ F and x∈X,f1ximplies f2x. 21
(ii) 1is more ambiguity loving than 2if, for all f∈ F and x∈X,x1fimplies x2f. An agent is more ambiguity averse than an other one if she is less inclined to choose an uncertain act fover a constant act x. On the other hand, an agent is more uncertainty loving than an other one if she is more inclined to stick to an uncertain act fthan to switch to a constant act x. The next result characterizes ambiguity attitudes for hope-and-prepare preferences. Proposition 2. Let 1and 2be two hope-and-prepare preference relations with unique representation (u, C1, D1)and (u, C2, D2), respectively. Then, (i) 1is more ambiguity averse than 2if and only if C2⊆C1. (ii) 1is more ambiguity loving than 2if and only if D2⊆D1. For a hope-and-prepare representation (u, C, D), the two sets of priors Cand Drepresent the level of pessimism and optimism related to the DM’s ambiguity attitudes. More precisely, the relationship C2⊆C1means that, in the worst scenario, the level of welfare attained by the agent if she has preference relation 1is lower than the one attained if she has preference relation 2. Similarly, D2⊆D1means that, in the best scenario, the level of welfare attained by the agent if she has preference relation 1is higher than the one attained if she has preference relation 2. Based on Proposition 2(i), by comparing the concordant preference with representation (u, C, C)to the non-concordant preference 1with representation (u, C1, C), with C1⊂C, we can say that 1is more ambiguity averse than it is ambiguity loving. Similarly, the non-concordant representation 2with representation (u, C, D2), with D2⊂C,can be said to be more ambiguity loving than it is ambiguity averse. Then, a DM with concordant preferences is as ambiguity loving as she is ambiguity averse, or, in other words, her pessimistic evaluation is as pessimistic as her optimistic evaluation is optimistic. We end this subsection by briefly discussing the relation between the degree of conservatism of a hope-and-prepare preference relation and the attitude towards ambiguity that it displays. It is easy to see that if 1and 2are hope-and-prepare preferences, and if 1 is more conservative than 2, then 1is both more ambiguity averse and more ambiguity loving than 2. Does the converse statement hold ? This question is all the more natural that if 1and 2are twofold preferences, then 1is more conservative that 2if, and only if,1is more ambiguity averse and more ambiguity loving than 2.41 An example in Appendix Cshows that the answer is negative for hope-and-prepare preferences. 41See Corollary 1 in Echenique et al. (2022). 22
6 Aggregating the opinion of experts with hope-andprepare preferences Numerous economic decisions under uncertainty, such as those related to fiscal policy and those addressing climate change, often hinge on the guidance provided by groups of experts, who frequently hold conflicting “opinions.” We propose a simple illustration, in the context of the aggregation of conflicting opinions among experts, in which the fact that the planner’s decisions are taken according to a hope-and-prepare preference relation rather than according to a Bewley one reflects her preference for decisiveness. Due to the complexity of the issue at hand, the opinions of experts may encompass several probability distributions (scenarios) over payoff-contingent states. Following Danan et al. (2016), we assume that experts have Bewley preferences, which expresses, given a set of plausible scenarios, the need of experts to have a strong conviction in order to report to the planner (the DM) that an option is better than an other one:42 “[...] a given individual may also consider more than one model to be plausible—or have an imprecise belief. For such an individual, which of two policies yields the highest expected utility may depend on the model considered. When a policy yields a higher expected utility than another one for all plausible models, we say that the individual unambiguously prefers the former policy to the latter. Unambiguous preferences are thus robust to belief imprecision.” Let N={1,2, . . . , n}be a finite set of experts. Expert j∈Nhas a preference j on F. We use 0to denote the DM’s preference on F. We suppose that, for all i∈N, expert i’s preference is a Bewley preference with unique representation (u, Ci). We thus assume in particular that there is no diversity of preferences over outcomes, which is a distinctive element of the theory of the aggregation of opinions, compared to the theory of the aggregation of preferences. We study how 0should depend on (j)j∈Nand impose the two following conditions: Axiom 10 (Pareto).For all f, g ∈F, if figfor all i∈N, then f0g. Axiom 11 (Caution for incomparability).For all f∈Fand x∈X, if there exists i∈Nsuch that f’ix, then f’0x. The Pareto condition is the standard one. It asserts that the DM should follow the comparisons expressed by experts when they are unanimous: if all experts prefer act fto act g, 42In particular, given this set of scenarios, the condition under which they have sufficient conviction that an option is better than an other one is stronger than if they had a concordant hope-and-prepare preference relation. 23
holds: fxg. Let y∈Xsuch that g’y. From Lemma 1,xy, implying fy. But then, fpg, which is a contradiction. Therefore, pis negatively transitive. Step 2. fgif and only if fpgand fog. Let us first prove that for f, g ∈ F such that fg, one has fpgand fog, giving the explicit argument exclusively for fpg, as fogis proved symmetrically. By contradiction, assume that fčpg, then for all x’g, one has fčx, that is, either f’x or xf. But if xf, then xgby transitivity, which contradicts x’g. Thus, for all x∈X, if x’g, then x’f. By Axiom 6,g’f, a contradiction. We have thus proved fpg. Suppose now fpgand fog, and let us show fg. By definition of pand o, there exist x∈Xsuch that fxand x’g, and y∈Xsuch that y’fand yg. Axiom 7then implies fg. Define ∼pby f∼pgif and only if fčpgand gčpf, for all f, g ∈ F, and define Ápby fÁpgif and only if either fpgor f∼pg, for all f, g ∈ F. The relations ∼oand Áoare similarly defined. It is clear that Ápand Áoare complete and transitive. We say that Áp (resp. Áo) is continuous if p(resp. o) is continuous, which is equivalent to the closedness of {α∈[0,1] : αf + (1 −α)gÁph}and {α∈[0,1] : hÁpαf + (1 −α)g}. Step 3. Ápand Áoare continuous and satisfy monotonicity and certainty independence.45 We only provide the proof that Ápis continuous and satisfies monotonicity and certainty independence, where monotonicity, when allowing for indifference, means that for all f, g ∈ F such that f(s)Ápg(s)for all s∈S,fÁpg, and certainty independence means that for all f, g ∈ F, all x∈X, and all α∈(0,1),fÁpgif and only if αf + (1−α)xÁpαg +(1 −α)x. We first show that Ápis continuous. Let f, g, h ∈ F and x∈X; denote Axthe set of α∈[0,1] such that αf + (1 −α)gxand x’h. Either Axis empty or it coincides with {λ∈[0,1] : λf+(1−λ)gx}. Then Axis open by Axiom 2. Therefore, {α∈[0,1] : αf+(1− α)gph}=∪x∈XAxis open. Similarly, one can show that {α∈[0,1] : hpαf + (1 −α)g} is open; thus, Ápis continuous. Next, we prove that Ápsatisfies monotonicity. Let f, g ∈ F such that f(s)Ápg(s)for all s∈S, which clearly implies f(s)Ág(s)for all s∈S. Suppose gpf, which means that there exists x∈Xsuch that gxand x’f. This is a direct contradiction as, by Lemma 4, for any x′∈Xsuch that x′’f,gčx′. Thus, fÁpg. 45The definition of these properties for a weak order are reminded in the following lines. 30
Lastly, we establish that Ápsatisfies certainty independence. Let f, g ∈ F,x∈X, and α∈(0,1). We first show that fÁpgimplies αf + (1 −α)xÁpαg + (1 −α)x. Since Ápis a weak order, fÁpgis equivalent to gčpf, which holds if, and only if, for all y∈Xsuch that gy,fand yare comparable. Under Axiom 3, it is sufficient to prove that, for all y∈Xsuch that αg + (1 −α)xy,αf + (1 −α)xand yare comparable. Let y∈Xsuch that αg + (1 −α)xyand suppose by contradiction αf + (1 −α)x’y. Claim: For such y∈X, there is z∈ {z:z’f}with u(z) = inf{u(z) : z’f}such that yÁαz + (1 −α)x. We have shown in Step 2that there exists z∈ {z∈X:z’f}such that u(z) = inf{u(z) : z’f}. Since ’satisfies certainty independence (Lemma 2), αf +(1−α)x’αz +(1−α)x. We claim yÁαz + (1 −α)x. Indeed, if there exists z∈Xsuch that zz, then it follows from Axiom 3and Lemma 1that fzβ:= βz + (1 −β)zfor all β∈(0,1). Using Axiom 3again yields αf + (1 −α)xαzβ+ (1 −α)x. It then follows from Lemma 1that yαzβ+(1−α)xfor all β∈(0,1). Letting βtend to 1, one concludes, since uis affine, that yÁαz +(1−α)x. If zÁzfor all z∈X, then αf(s)+(1−α)xÁαz +(1−α)xfor all s∈S (otherwise, Axiom 3implies zf(s), which is a contradiction). Since αf + (1 −α)x’y, Lemma 4implies αz + (1 −α)xčy, which is equivalent to yÁαz + (1 −α)x. Since αg + (1 −α)xyand yÁαz + (1 −α)x, Lemma 3implies αg + (1 −α)xαz + (1 −α)x, which is equivalent to gzby Axiom 3. Hence, by definition of z,gpf, a contradiction. Therefore, αf +(1−α)xand yare comparable, which yields αf +(1−α)xÁpαg +(1−α)x. We now show the converse implication. For α∈(0,1), and any two f, g ∈ F,αf + (1 − α)xÁpαg +(1−α)xif, and only if, for all y∈Xsuch that αg +(1−α)xy,αf +(1−α)x and yare comparable. Let y∈Xsuch that gy, then one has αg+(1−α)xαy+(1−α)x by Axiom 3. Thus, αf + (1 −α)xand αy + (1 −α)xare comparable, implying that fand y are comparable by Axiom 3. Hence, gčpf, which is equivalent to fÁpg. We have proved that Ápsatisfies certainty independence. Step 4. An agent with preferences Ápon Fis averse to ambiguity, i.e., for all f, g ∈ F, f∼pgimplies αf + (1 −α)gÁpf. An agent with preferences Áoon Floves ambiguity, i.e., for all f, g ∈ F,f∼pgimplies fÁoαf + (1 −α)g. We only prove that Ápdisplays ambiguity aversion. Let f, g ∈ F such that f∼pg, i.e.,fčpgand gčpf. In other words, for all x∈Xwith fx,xis comparable with g, and for all x∈Xwith gx,xis comparable with f. Let x∈Xsuch that fx. 31
If xg, then fg, and then, by Step 2,fpg, which is a contradiction; thus, one must have gx. This implies {x∈X:fx}⊆{x∈X:gx}. Analogously, {x∈X:gx} ⊆ {x∈X:fx}; therefore, {x∈X:fx}={x∈X:gx}. Let α∈(0,1), we claim that αf +(1−α)gÁpf. Since Ápis a weak order, it is sufficient to prove fčpαf +(1−α)g, which holds if, for all x∈Xsuch that fx,αf +(1−α)gx. Yet, we have just proved that fxif and only if gx. Axiom 4then directly entails αf + (1 −α)gx; which concludes. Conclusion. It is well-known since Gilboa and Schmeidler (1989) that a weak order defined on Fsatisfying the properties stated in Step 3can be represented by f7→ minp∈C∫up(f)dp if it displays ambiguity aversion, such as Áp, and by f7→ maxp∈D∫uo(f)dp if it displays love for ambiguity, such as Áo, where Cand Dare unique non-empty convex compact subsets of ∆,upand uoare two affine functions on X, unique up to positive affine transformation. Clearly, for all x, y ∈X,xÁpyif and only if xÁy, and xÁoyif and only if xÁy. Thus, upand uoare positive affine transformations of u, and one may assume that up=uo=u. Finally, it remains to prove that C∩D6=∅. Claim: Cand Dare non-disjoint if, and only if, for all f∈ F,minp∈C∫u(f)dp ≤ maxp∈D∫u(f)dp. We only prove the if part, the other direction being trivial. We proceed by contraposition. Suppose that C∩D=∅. By the separating hyperplane theorem, there exists a bounded measurable function φ:S→Rsuch that minp∈C∫φdp > maxp∈D∫φdp. Yet, there exists a sequence of simple functions {φn}that converges (in supnorm topology) to φ. Since both ˜φ7→ minp∈C∫˜φdp and ˜φ7→ maxp∈D∫˜φdp are continuous, there is n∈Nsuch that minp∈C∫φndp > maxp∈D∫φndp. As aφn+balso satisfies this last inequality for all a > 0 and b∈R, one can choose a > 0and b∈Rsuch that aφn(s) + b∈u(X)for all s∈S, which implies φn=u(f)for some f∈ F: min p∈C∫u(f)dp > max p∈D∫u(f)dp. As a consequence, the fact that, for all f∈ F,minp∈C∫u(f)dp ≤maxp∈D∫u(f)dp, implies C∩D6=∅. Based on this claim, it remains to prove that minp∈C∫u(f)dp ≤maxp∈D∫u(f)dp for all f∈ F in order to conclude that Cand Dare not disjoint. Let us show that the inequality minp∈C∫u(f)dp ≤maxp∈D∫u(f)dp holds for all f∈ F if and only if, for all x∈X, for all f∈ F,fpximplies fox. Suppose that for all x∈X, for all f∈ F,fpximplies fox. Suppose, by 32
contradiction, that there is f∈ F such that minp∈C∫u(f)dp > maxp∈D∫u(f)dp. Clearly, one has u(x∗)≤minp∈C∫u(f)dp ≤u(x∗), where x∗and x∗are defined as in the proof of Lemma 5. Then, since u(X)is convex, minp∈C∫u(f)dp belongs to u(X). Similarly, one can deduce that maxp∈D∫u(f)dp lies in u(X). Then, the convexity of u(X)implies that there exists x∈Xsuch that min p∈C∫u(f)dp > u(x)>max p∈D∫u(f)dp, which is a contradiction as it implies, as minp∈C∫u(x)dp = maxp∈D∫u(x)dp =u(x),fpx and xof. The other direction of the equivalence is trivial. It remains to show that, indeed, fpximplies ox, for all x∈X, and all f∈ F. Yet, fpximplies fx. Indeed, fpxif and only if there exists y∈Xsuch that fyand y’x; then Lemma 3implies fx. By Step 2, we conclude that fox. We have thus proved that minp∈C∫u(f)dp ≤maxp∈D∫u(f)dp for all f∈ F, and, thus, that Cand Dare non-disjoint. If part. Assume that admits a hope-and-prepare representation. One can readily check that Axioms 1to 5are satisfied. For all f∈ F, denote pf∈arg maxp∈D∫u(f)dp and pf∈arg minp∈C∫u(f)dp. Define also the constant acts f=∫fdpfand f=∫fdpf. Clearly, f’fand f’f; moreover, fx⇐⇒ u(f)> u(x), xf⇐⇒ u(x)> u(f), f’x⇐⇒ u(f)≥u(x)≥u(f) .(2) We prove that Axiom 6is verified by contradiction. Consider f, g ∈ F such that for all x∈X,f’ximplies g’x. If fg, then u(f)> u(g). However, by assumption, g’f, which implies u(g)≥u(f)≥u(g), a contradiction. The same argument applies to prove that gfcannot hold. Therefore, f’g. Axiom 7easily obtains from the comparisons in (2). Indeed, let f, g ∈ F and x, y ∈X such that f’x,g’y,xg, and fy. Using (2), one gets u(f)≥u(x)≥u(f), u(g)≥u(y)≥u(g), u(x)> u(g), u(f)> u(y) .(3) 33
Then u(f)≥u(x)> u(g)and u(f)> u(y)≥u(g), that is fg, by definition of a hope-and-prepare preference. D.2 Proof of Theorem 2 By assumption, fg⇐⇒ minp∈C∫u(f)dp > minp∈C∫u(g)dp maxp∈D∫u(f)dp > maxp∈D∫u(g)dp , where uis an affine function defined on X, unique up to affine transformation, Cand Dare two unique compact and convex subsets of ∆with C∩D6=∅. It remains to prove that admitting such a representation satisfies Axiom 8and 9if and only if C=D. We show in a very similar way to Echenique et al. (2022) that it satisfies Axiom 8if and only if D⊆C —the other inclusion being equivalent to Axiom 9is shown in a symmetric way. Only-if part. Suppose by contraposition that DĘC: there is some p∗∈Dsuch that p∗/∈ C. Then, by the separating hyperplane theorem and the argument given in the Conclusion step of the proof of Theorem 1, there is an act ψand k∈Rsuch that min p∈C∫u(ψ)dp > k > ∫u(ψ)dp∗.(4) By scaling ψand kappropriately, as uis affine, one can find f, h ∈ F and x∈Xsuch that u(f) = 1 2u(ψ), u(h) = −u(ψ)and u(x) = 2k.46 Let g=1 2h+1 2x: u´1 2f+1 2g¯=1 4u(ψ) + 1 2ˆ−1 2u(ψ) + k˙=k 2, that is, fand gare complementary, and 1 2f(s) + 1 2g(s)∼yfor some y∈Xsuch that u(y) = k 2, for all s∈S. Since u(f) = 1 2u(ψ), Equation (4) implies min p∈C∫u(f)dp=1 2min p∈C∫u(ψ)dp > k 2=u(y). In addition, as D∩C6=∅,maxp∈D∫u(f)dp≥minp∈C∫u(f)dp>u(y). As a consequence, fy. 46We abuse notation in a standard way when writing u(f) = t, for t∈R, to actually denote u(f(s)) = t for all s∈S. 34
Futhermore, u(g) = u`1 2h+1 2x˘=−1 2u(ψ) + k. Since p∗∈D, Equation (4) implies max p∈D∫u(g)dp≥∫u(g)dp∗=−1 2∫u(ψ)dp∗+k > −k 2+k=k 2=u(y), from which yčg. One thus has 1 2f(s) + 1 2g(s)∼yfor all s∈S, f y, and yčg, which is a violation of Axiom 8. If part. Suppose that D⊆C. Consider two complementary acts f, g ∈ F such that 1 2f(s) + 1 2g(s)∼xfor some x∈X, for all s∈S, or, 1 2u(f) + 1 2u(g) = k, with u(x) = k. Assume fx, which is is equivalent to minp∈C∫u(f)dp > k maxp∈D∫u(f)dp > k ⇐⇒ 1 2minp∈C∫u(f)−u(g)dp > 0 1 2maxp∈D∫u(f)−u(g)dp > 0 ⇐⇒ 1 2maxp∈C∫u(g)−u(f)dp < 0 1 2minp∈D∫u(g)−u(f)dp < 0 . Since D⊆C, the last inequalities yield 1 2maxp∈D∫u(g)−u(f)dp < 0 1 2minp∈C∫u(g)−u(f)dp < 0 . Plugging u(f) = 2k−u(g), one obtains 2 maxp∈D∫u(g)−kdp < 0 2 minp∈C∫u(g)−kdp < 0 ⇐⇒ maxp∈D∫u(g)dp < k minp∈C∫u(g)dp < k . As k=u(x), this means xg. Therefore, satisfies Axiom 8. D.3 Proof of Proposition 1 i)Let Hbe a hope-and-prepare preference with unique representation (u, CH, DH), and let Bbe Bewley preference with unique representation (u, CB). First, suppose that CH∪DH⊆CB. If fBg, then for all p∈CH∪DH, ∫u(f)dp > ∫u(g)dp, 35
which implies, as CHand DHare not disjoint, minp∈CH∫u(f)dp > minp∈CH∫u(g)dp, maxp∈DH∫u(f)dp > maxp∈DH∫u(g)dp. Therefore, fHg. Thus, Bis more conservative than H. Conversely, suppose Bis more conservative than Hand suppose, by contradiction, that there exists p∗∈CH\CB. By the separation argument we already used in the Conclusion step of the proof of Theorem 1, there are f∈ F and x∈Xsuch that ∫u(f)dp∗> u(x)>max p∈CB∫u(f)dp. It follows that xBfbut xčHf, a contradiction. Similarly, suppose there exists p∗∈DH\CB. Then there are f∈ F and x∈Xsuch that min p∈CB∫u(f)dp > u(x)>∫u(f)dp∗. In this case, we have fBxbut fčHx, an other contradiction. Therefore, CH∪DH⊆CB. ii)Let Hbe a hope-and-prepare preference with unique representation (u, CH, DH), and Tbe a twofold multiprior preference with unique representation (u, CT, DT). First, suppose that CH⊆CTand DH⊆DT. Since DT∩CT6=∅and DH∩CH6=∅, CH∩DT6=∅and DH∩CT6=∅. If fTg, then min p∈CT∫u(f)dp > max p∈DT∫u(g)dp, which implies min p∈CH∫u(f)dp ≥min p∈CT∫u(f)dp > max p∈DT∫u(g)dp ≥max p∈DH∫u(g)dp, max p∈DH∫u(f)dp ≥min p∈CT∫u(f)dp > max p∈DT∫u(g)dp ≥max p∈DH∫u(g)dp. Since DH∩CH6=∅, one gets max p∈DH∫u(f)dp ≥min p∈CH∫u(f)dp > max p∈DH∫u(g)dp ≥min p∈CH∫u(g)dp. Therefore, fHg. Thus, Tis more conservative than H. 36
Conversely, suppose Tis more conservative than Hand suppose, by contradiction, that there exists p∗∈CH\CT. There are f∈ F and x∈Xsuch that min p∈CT∫u(f)dp > u(x)>∫u(f)dp∗, from which it follows that fTxbut fčHx, a contradiction. To prove that DH⊆DT, suppose there exists p∗∈DH\DT. There are f∈ F and x∈Xsuch that ∫u(f)dp∗> u(x)>max p∈DT∫u(f)dp. In this case, xTfbut xčHf, an other contradiction. D.4 Proof of Proposition 2 Clearly, for each i∈ {1,2}, and all x∈X,fixif and only if fip x, where ip is the pessimistic relation defined, as in the proof of Theorem 1, by fip gif and only if fiy and y’igfor some y∈X. Thus, 1is more ambiguity averse than 2if and only if 1p is more ambiguity averse than 2p. When proving Theorem 1, we have shown that ip is represented by a maxmin expected utility functional; therefore, 1pis more ambiguity averse than 2pif and only if C2⊆C1. Similarly, for each i∈ {1,2}, and all x∈X,xifif and only if xio f, where io is the optimistic relation defined, as in the proof of Theorem 1, by fio gif and only if f’iyand yigfor some y∈X. As we have proved that io admits a maxmax expected utility representation, one obtains that 1is more ambiguity loving than 2if and only if D2⊆D1. D.5 Proof of Theorem 3 We will only prove that (i)implies (ii), the inverse implication being routine. Lemma 6. A weak order relation on Fsatisfies Axioms 2,3and 5if and only if there exists a monotonic, constant-linear functional I:B0(Σ) →Rand a non-constant affine function u:X→Rsuch that, for all f, g ∈ F, fg⇐⇒ I(u(f)) > I(u(g)). Moreover, Iis unique and uis unique up to positive affine transformation. 37
Proof. As before, define Áby fÁgif and only if gčffor all f, g ∈ F. Clearly, Á is complete and transitive, and ’is an equivalence relation (see Theorem 2.1 in Fishburn (1970)). The weak order Áis continuous if, for all f, g, h ∈ F,{α∈[0,1] : αf +(1−α)gÁh} and {α∈[0,1] : hÁαf + (1 −α)g}are closed. Clearly, Áis continuous and non-trivial. It is monotone if and only if, for all f, g ∈ F, if f(s)Ág(s)for all s∈S, then fÁg. Since Lemma 4holds, in particular, for a weak order of which the asymmetric part satisfies Axioms 2,3and 5, and since ’is an equivalence relation, Áis monotone. Now, we check that that Ásatisfies certainty independence: for all f, g ∈ F and x∈X, fÁg⇐⇒ gčf ⇐⇒ αg + (1 −α)xčαf + (1 −α)x ⇐⇒ αf + (1 −α)xÁαg + (1 −α)x. As a consequence, by Lemma 1 in Ghirardato et al. (2004),47 there exists a monotonic, constant-linear functional I:B0(Σ) →Rand a non-constant affine function u:X→Rsuch that, for all f, g ∈ F, fÁg⇐⇒ I(u(f)) ≥I(u(g)). Moreover, Iis unique and uis unique up to positive affine transformation. Lemma 7. Suppose that I, I′, I′′ :B0(Σ) →Rare monotonic and constant-linear with I′≤I′′. Then the following statements are equivalent: (i) For all ϕ, φ ∈B0(Σ), if I′(ϕ)> I′(φ)and I′′(ϕ)> I′′(φ), then I(ϕ)> I(φ). (ii) There exists α∈[0,1] such that, for all φ∈B0(Σ),I(φ) = αI′(φ) + (1 −α)I′′(φ). Proof. The following proof closely follows the proof of Lemma A.3 of Frick et al. (2022). We only prove that (i)implies (ii); the other implication is easily checked. By (i), there is an increasing function W:{(I′(φ), I′′(φ)) : φ∈B0(Σ)} → Rsuch that W(I′(φ), I′′(φ)) = I(φ). Let φ∈B0(Σ) be such that I′(φ) = I′′(φ) = k. We will show that I(φ) = k. Since I′ and I′′ are monotonic and constant-linear, k+ε=I′(k+ε)> I′(φ)> I′(k−ε) = k−εand k+ε=I′′(k+ε)> I′′(φ)> I′′(k−ε) = k−ε. Thus, by (i),k+ε=I(k+ε)> I(φ)> I(k−ε) = k−ε. Let εconverge to 0, then I(φ) = k. Thus, I(φ) = k, which implies that I(φ) = αI′(φ) + (1 −α)I′′(φ)for all α∈R. 47Axiom 2implies the “Archimedean axiom” in Ghirardato et al. (2004). 38
Now, consider φ∈B0(Σ) such that I′(φ)< I′′(φ). There exists α(φ)∈Rsuch that I(φ) = α(φ)I′(φ) + (1 −α(φ))I′′(φ). By a simple computation, one obtains α(φ) = I(φ)−I′′(φ) I′(φ)−I′′(φ)=−I(ϕ) = −W(I′(ϕ), I′′(ϕ)), where ϕ=φ−I′′ (φ) I′′ (φ)−I′(φ). Clearly, I′(ϕ) = −1and I′′(φ) = 0. Thus, α(φ) = −W(−1,0), which is independent of φ. Let α=−W(−1,0). Then, I(φ) = αI′(φ) + (1 −α)I′′(φ)for all φ∈B0(Σ). We now prove that α∈[0,1]. By contradiction, assume that α < 0. For any φ∈B0(Σ) such that I′(φ)< I′′(φ), we have I(φ)> I′′(φ). There exists ε > 0such that I(φ)> I′′(φ)+ε. Moreover, I′′(φ) + ε=I′(I′′(φ) + ε)> I′(φ)and I′′(φ) + ε=I′′(I′′(φ) + ε)> I′′(φ). By (i), I′′(φ) + ε=I(I′′(φ) + ε)> I(φ), which is a contradiction. Thus, α≥0. One can similarly show that α≤1. Assume that is a hope-and-prepare preference and ∗is an invariant biseparable extension of . Let u:X→Rbe a non-constant affine function, and let Cand Dbe two compact convex subsets of ∆with C∩D6=∅such that fg⇐⇒ minp∈C∫u(f)dp > minp∈C∫u(g)dp maxp∈D∫u(f)dp > maxp∈D∫u(g)dp . From the uniqueness result of Theorem 1,uis unique up to positive affine transformation, and Cand Dare unique. It follows from Lemma 6that there exist a monotonic, constant-linear functional I: B0(Σ) →Rand a non-constant affine function u′:X→Rsuch that, for all f, g ∈ F, f∗g⇐⇒ I(u′(f)) > I(u′(g)). Moreover, Iis unique and u′is unique up to positive affine transformation. It trivially follows from the extension property that, for all x, y ∈X,u(x) = u(y)if and only if u′(x) = u′(y), which implies that uis a positive affine transformation of u′. Thus, one can assume without loss of generality u=u′. Define I′:B0(Σ) →Rand I′′ :B0(Σ) →Rby I′(φ) = minp∈C∫φdp and I′′(φ) = maxp∈D∫φdp for all φ∈B0(Σ). Clearly, I′and I′′ are monotonic, constant-linear functionals; and since C∩D6=∅,I′′ ≥I′. Now, let ϕ, φ ∈B0(Σ) such that I′(ϕ)> I′(φ)and I′′(ϕ)> I′′(φ). We denote by B0(Σ, u(X)) the set of all functions in B0(Σ) that take values in u(X). Since u(X)is an 39