Chapter 20 Zero-sum two-person games

This chapter discusses the zero-sum two-person games. Given a two-person game with just one move for each player, the players independently and simultaneously select one among the finitely many actions resulting in a payoff for each player. If i , j are their independent choices in a play, then the game is defined by a pair of real matrices A ( a ij ), B ( b ij ), where a ij is the payoff to player I and b ij is the payoff to player II. The game is called zero sum if a ij b ij 0. Thus, in zero-sum games, what one player gains, the opponent loses. In such games, A suffices to determine the payoff. The chapter describes the equivalence of the minimax theorem and the duality theorem. To prove minimax theorems in greater generality, Kakutani's theorem is further extended to arbitrary locally convex topological vector spaces. These are real vector spaces with a Hausdorff topology admitting convex bases, where vector operations of addition and scalar multiplication are continuous.