Ігри двох осіб на добутку неперервних і скінченних просторів сходинкових функцій

Abstract

A tractable method of solving two-person games defined on a product of staircase-function spaces is presented. The spaces can be finite and continuous as well. The method is based on stacking equilibria of “short” two-person games, each defined on an interval where the pure strategy value is constant. First, a two-person game is formalized, in which the players’ strategies are staircase functions. In such a game, the set of the player’s pure strategies is a continuum of staircase functions of time. The time can be thought of as it is discrete. The four theorems allowing to fulfill the stacking are proved for the case of pure-strategy equilibria. Second, the set of possible values of the player’s pure strategy is discretized so that the game becomes defined on a product of staircase-function finite spaces. To formalize a method of solving two-person games defined on a product of staircase-function finite spaces, it is then proved that the game is solved as a stack of respective equilibria in the “short” bimatrix games. The equilibria in this case are considered in general terms, so they can be in mixed strategies as well. The stack is any combination (succession) of the respective equilibria of the “short” bimatrix games. Apart from the stack, there are no other equilibria in this “long” bimatrix game. The stack is always possible, even when only time is discrete (and the set of pure strategy possible values is continuous). An example is presented to show how the stacking is fulfilled for a case of when every “short” bimatrix game has a single pure-strategy equilibrium. The presented method, further “breaking” the initial (“long”) game defined on a product of staircase-function finite spaces, makes it completely tractable.