### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 933-980 |

Number of pages | 48 |

Journal | Econometrica |

Volume | 87 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2019 |

### Fingerprint

### Keywords

- Affine binary relations
- existence of mixed strategy Nash equilibrium
- justifiable preferences
- nontransitive and incomplete expected utility representations
- preference for portfolio diversification
- preference reversal phenomenon

### ASJC Scopus subject areas

- Economics and Econometrics

### Cite this

*Econometrica*,

*87*(3), 933-980. https://doi.org/10.3982/ECTA14156

**Coalitional Expected Multi-Utility Theory.** / Hara, Kazuhiro; Ok, Ahmet; Riella, Gil.

Research output: Contribution to journal › Article

*Econometrica*, vol. 87, no. 3, pp. 933-980. https://doi.org/10.3982/ECTA14156

}

TY - JOUR

T1 - Coalitional Expected Multi-Utility Theory

AU - Hara, Kazuhiro

AU - Ok, Ahmet

AU - Riella, Gil

PY - 2019/5/1

Y1 - 2019/5/1

N2 - 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.

AB - 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.

KW - Affine binary relations

KW - existence of mixed strategy Nash equilibrium

KW - justifiable preferences

KW - nontransitive and incomplete expected utility representations

KW - preference for portfolio diversification

KW - preference reversal phenomenon

UR - http://www.scopus.com/inward/record.url?scp=85065923776&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065923776&partnerID=8YFLogxK

U2 - 10.3982/ECTA14156

DO - 10.3982/ECTA14156

M3 - Article

VL - 87

SP - 933

EP - 980

JO - Econometrica

JF - Econometrica

SN - 0012-9682

IS - 3

ER -