Solution of a Satisficing Model for Random Payoff Games

Abstract

In this paper, we consider a "satisficing" criterion to solve two-person zero-sum games with random payoffs. In particular, a player wants to maximize the payoff level he can achieve with a specified confidence. The problem reduces to solving a nonconvex mathematical programming problem. The main result shows that solving this problem is equivalent to finding the root of an equation whose values are determined by solving a linear problem. This linear problem results from maximizing the confidence with fixed payoff level.

Random payoff games with partial information : one person games against nature

“…In [2] we have shown that if all a tj have a finite range then p lf p 2 can be chosen so that w(t) = -for all t in [p 1? p 2 ] and (1.2) becomes the expected…”

“…Case (d) can be solved in several different ways depending on the goals of the players. This has been the subject of study in [2] and [3], Case (b) was introduced by Fourgeaud et al in [5]. There the case of a one person random payoff game against nature is analyzed and a method of solution for such a game is given.…”

“…Thus the problem he should solve is (3,1) or equivalenty (3.2). Moreover, since v x 4, v 2 , it follows that v l9 the optimal value of the objective function obtained from (2.2) gives a pessimistic or too conservative an estimate of the actual expected value the player will receive.…”

“…Under these circumstances it may be reasonable to use an optimality criterion other than the minimax criterion of deterministic zero sum games. For this reason several different concepts of optimality were introduced in [2] and [3].…”

“…One optimality criterion considered in [2] is for a player to maximize the probability, oc, of his attaining a given, or prescribed, payoff level, (3, where Pfay ^ (3) is the probability that the random variable a tj is greater than or equal to p. [2] contains a detailed analysis of model (1.1).…”