Some Topics in Continuous Games

Abstract

In Chapter 9 we examined two-person constant sum games with finite numbers of pure strategies. These games can be represented by matrices where the rows designate one player’s pure strategies and the columns the other’s. The number of pure strategies available to each player may be super-astronomical (as, for example, in chess); so that determining the optimal strategies by standard algorithms, e.g. the simplex method, is out of the question. However, limits on the games that can actually be solved in this way need not imply limits on the theoretical conclusions valid for all finite games of a given type. The conclusions do not, however, necessarily hold for games with infinite numbers of strategies. It is easy to define a game of this sort. Let each of two players choose a point in the interval [0, 1]. If Row chooses ξ and Column η, the outcome (ξ, η) is determined, which can be represented by a point in the interior or on the boundary of a unit square. We can represent Row’s payoff as K(ξ, η),a function of two variables. If the game is zero sum, Column’s payoff will be —K(ξ, η).