# Random expected utility theory with a continuum of prizes

Random expected utility theory with a continuum of prizes 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 \$\$C_0[0,M]\$\$ C 0 [ 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 \$\$C_0[0,M]\$\$ C 0 [ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Operations Research Springer Journals

# Random expected utility theory with a continuum of prizes

Annals of Operations Research, Volume 271 (2) – Jun 4, 2018
23 pages

/lp/springer-journals/random-expected-utility-theory-with-a-continuum-of-prizes-3JtO90l6qw
Publisher
Springer Journals
Subject
Business and Management; Operations Research/Decision Theory; Combinatorics; Theory of Computation
ISSN
0254-5330
eISSN
1572-9338
D.O.I.
10.1007/s10479-018-2914-z
Publisher site
See Article on Publisher Site

### Abstract

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 \$\$C_0[0,M]\$\$ C 0 [ 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 \$\$C_0[0,M]\$\$ C 0 [ 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.

### Journal

Annals of Operations ResearchSpringer Journals

Published: Jun 4, 2018

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