Coalitional Expected Multi‐Utility Theory

Abstract

This paper begins by observing that any reflexive binary (preference) relation (over risky prospects) that satisfies the independence axiom admits a form of expected utility representation. We refer to this representation notion as the coalitional minmax expected utility representation. By adding the remaining properties of the expected utility theorem, namely, continuity, completeness, and transitivity, one by one, we find how this representation gets sharper and sharper, thereby deducing the versions of this classical theorem in which any combination of these properties is dropped from its statement. This approach also allows us to weaken transitivity in this theorem, rather than eliminate it entirely, say, to quasitransitivity or acyclicity. Apart from providing a unified dissection of the expected utility theorem, these results are relevant for the growing literature on boundedly rational choice in which revealed preference relations often lack the properties of completeness and/or transitivity (but often satisfy the independence axiom). They are also especially suitable for the (yet overlooked) case in which the decision‐maker is made up of distinct individuals and, consequently, transitivity is routinely violated. Finally, and perhaps more importantly, we show that our representation theorems allow us to answer many economic questions that are posed in terms of nontransitive/incomplete preferences, say, about the maximization of preferences, the existence of Nash equilibrium, the preference for portfolio diversification, and the possibility of the preference reversal phenomenon.

Completeness and transitivity of preferences on mixture sets

“… See Aumann (1962, Footnote 25). We invite the reader to use Propositions 1, 2 and Footnote 5 above to see the relationship between the continuity assumptions of this paper and those inAumann (1962, (4.1) and (4.2)).22 See Karni (2014) for a recent survey andHara et al (2015) for the state-of-the-art results in this line of literature.…”

Boolean Representations of Preferences under Ambiguity

“…Nevertheless, standard arguments imply that the utility u under BEU is unique up to positive affine transformation. Moreover, Supplementary Appendix S.1 shows that the belief-set collection P is unique up to "half-space closure," analogous to recent representations featuring collections of sets of utilities (e.g., Hara, Ok, and Riella, 2019).…”

Dual-self Representations of Ambiguity Preferences

“…Related to the structure of DSEU, several recent papers employ belief-set or utility-set collections in other contexts. While we maintain the weak order axiom and focus on relaxing independence, Lehrer and Teper (2011), Nascimento and Riella (2011), Nishimura and Ok (2016), Hara, Ok, and Riella (2019), and Aguiar, Hjertstrand, and Serrano (2020) study preferences that violate completeness and/or transitivity. 6 Whereas DSEU is a utility representation, these papers provide generalized unanimity representations à la Bewley (2002) and Dubra, Maccheroni, and Ok (2004), and the resulting proof methods are quite different.…”

“…The uniqueness of P up to half-space closure parallels the identification result in Hara, Ok, and Riella (2019), who represent independent (but possibly incomplete and intransitive) preferences over lotteries using a collection of utility-sets. Analogous to Hara, Ok, and Riella (2019), the idea is that for any P ∈ P, the closed half-spaces containing P capture all information about P that is relevant to the representation. Indeed, in determining how any given utility act φ ∈ R S is evaluated by the representation, the only relevant feature of P is the worst-case expectation λ P,φ := min µ∈P E µ [φ], and this worst-case expectation is shared by the closed half-space H φ,λ P,φ ⊇ P .…”