Coalitional Expected Multi-Utility Theory

This paper begins by observing that any reflexive binary (preference) relation (over risky prospects) which satisfies the Independence Axiom admits a form of expected utility representation. We refer to this representation notion as 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 are dropped from its statement. This approach also allows us to weaken transitivity in this theorem, rather than eliminating 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). 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, existence of Nash equilibrium, preference for portfolio diversification, and possibility of the preference reversal phenomenon.