A value for n-person games

Abstract

Introduction At the foundation of the theory of games is the assumption that the players of a game can evaluate, in their utility scales, every “prospect” that might arise as a result of a play. In attempting to apply the theory to any field, one would normally expect to be permitted to include, in the class of “prospects,” the prospect of having to play a game. The possibility of evaluating games is therefore of critical importance. So long as the theory is unable to assign values to the games typically found in application, only relatively simple situations—where games do not depend on other games—will be susceptible to analysis and solution. In the finite theory of von Neumann and Morgenstern difficulty in evaluation persists for the “essential” games, and for only those. In this note we deduce a value for the “essential” case and examine a number of its elementary properties. We proceed from a set of three axioms, having simple intuitive interpretations, which suffice to determine the value uniquely. Our present work, though mathematically self-contained, is founded conceptually on the von Neumann—Morgenstern theory up to their introduction of characteristic functions. We thereby inherit certain important underlying assumptions: (a) that utility is objective and transferable; (b) that games are cooperative affairs; (c) that games, granting (a) and (b), are adequately represented by their characteristic functions.