Game Theoretic Axioms for Utilities with Random Choices

Abstract

Constructing objective functions for decision making under uncertainty requires prescribing numerical utilities to lotteries. A lottery is a situation without the outcome known beforehand, but with probabilities of these outcomes a priori known to the decision maker. Presuming that the utilities of lottery outcomes are determined, the von Neumann-Morgenstern utility theory recommends to evaluate the utility of a lottery by its mathematical expectation.

Compound lotteries are lotteries, for which the outcomes are lotteries as well. In this case von Neumann-Morgenstern utility theory suggests the stochastic independence of trials, determining the outcomes of ‘inward’ and ‘outward’ lotteries. This rule is explicitly formulated as the ‘Axiom of Utility for Compound Lottery’.

In my model I drop the explicit Axiom of Utility for Compound Lottery and deduce it as a consequence of a system of axioms. In the constructed model compound lotteries appear as the result of an interaction of the agents playing a non-cooperative game. A general mixed extension of non-cooperative games is considered which originates from an arbitrary utility for a compound lottery. The system of axioms is formulated. It contains natural requirements for a general mixed extension. The proposed system of axioms determines the unique utility function for compound lotteries which is equal to the mathematical expectation with respect to the product of independent randomized choices.

This study was partially supported by the grant ACE-91-R02 of European Community which is gratefully acknowledged