TY - JOUR

T1 - Random expected utility theory with a continuum of prizes

AU - Ma, Wei

N1 - Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - This note generalizes Gul and Pesendorfer’s random expected utility theory, a stochastic reformulation of von Neumann–Morgenstern expected utility theory for lotteries over a finite set of prizes, to the circumstances with a continuum of prizes. Let [0, M] denote this continuum of prizes; assume that each utility function is continuous, let C0[0 , M] be the set of all utility functions which vanish at the origin, and define a random utility function to be a finitely additive probability measure on C0[0 , M] (associated with an appropriate algebra). It is shown here that a random choice rule is mixture continuous, monotone, linear, and extreme if, and only if, the random choice rule maximizes some regular random utility function. To obtain countable additivity of the random utility function, we further restrict our consideration to those utility functions that are continuously differentiable on [0, M] and vanish at zero. With this restriction, it is shown that a random choice rule is continuous, monotone, linear, and extreme if, and only if, it maximizes some regular, countably additive random utility function. This generalization enables us to make a discussion of risk aversion in the framework of random expected utility theory.

AB - This note generalizes Gul and Pesendorfer’s random expected utility theory, a stochastic reformulation of von Neumann–Morgenstern expected utility theory for lotteries over a finite set of prizes, to the circumstances with a continuum of prizes. Let [0, M] denote this continuum of prizes; assume that each utility function is continuous, let C0[0 , M] be the set of all utility functions which vanish at the origin, and define a random utility function to be a finitely additive probability measure on C0[0 , M] (associated with an appropriate algebra). It is shown here that a random choice rule is mixture continuous, monotone, linear, and extreme if, and only if, the random choice rule maximizes some regular random utility function. To obtain countable additivity of the random utility function, we further restrict our consideration to those utility functions that are continuously differentiable on [0, M] and vanish at zero. With this restriction, it is shown that a random choice rule is continuous, monotone, linear, and extreme if, and only if, it maximizes some regular, countably additive random utility function. This generalization enables us to make a discussion of risk aversion in the framework of random expected utility theory.

KW - Expected utility

KW - Independence axiom

KW - Random choice

KW - Random utility

KW - Risk aversion

UR - http://www.scopus.com/inward/record.url?scp=85047964892&partnerID=8YFLogxK

U2 - 10.1007/s10479-018-2914-z

DO - 10.1007/s10479-018-2914-z

M3 - Article

AN - SCOPUS:85047964892

SN - 0254-5330

VL - 271

SP - 787

EP - 809

JO - Annals of Operations Research

JF - Annals of Operations Research

IS - 2

ER -