CHAPTER 3 - Probabilistic Extension of Games

Abstract

As discussed in the preceding chapter, the definition of a Nash equilibrium in a nonantagonistic game may give rise to incomparable solutions. This problem does not exist in antagonistic games. These games, however, may have no Nash equilibrium or no saddled point. Therefore, the definition of a Nash equilibrium is useless in such situations. Von Neumann and Morgenstern worked out a way to surmount this difficulty. They assumed that every player can consider a probabilistic mixture of his/her pure strategies. This method of choosing pure strategies at random is called a probabilistic extension of games. It will be discussed in Section 3.1. In Section 3.2 we shall show that the solution of this kind of game can be reduced to a Unear programming problem. This solution always exists if the game is antagonistic.