FINITE APPROXIMATION OF NONCOOPERATIVE 2-PERSON GAMES PLAYED IN STAIRCASE-FUNCTION CONTINUOUS SPACES

Abstract

V. Romanuke
 Finite approximation of noncooperative 2-person games played in staircase-function continuous spaces
 Background. There is a known method of approximating continuous noncooperative 2-person games, wherein an approximate solution (an equilibrium situation) is considered acceptable if it changes minimally by changing the sampling step minimally. However, the method cannot be applied straightforwardly to a 2-person game played with staircase-function strategies. Besides, the independence of the player’s sampling step selection should be taken into account.
 Objective. The objective is to develop a method of finite approximation of 2-person games played in staircase-function continuous spaces by taking into account that the players are likely to independently sample their pure strategy sets.
 Methods. To achieve the said objective, a 2-person game, in which the players’ strategies are staircase functions of time, is formalized. In such a game, the set of the player’s pure strategies is a continuum of staircase functions of time, and the time is thought of as it is discrete. The conditions of sampling the set of possible values of the player’s pure strategy are stated so that the game becomes defined on a product of staircase-function finite spaces. In general, the sampling step is different at each player and the distribution of the sampled points (function-strategy values) is non-uniform.
 Results. A method of finite approximation of 2-person games played in staircase-function continuous spaces is presented. The method consists in irregularly sampling the player’s pure strategy value set, finding the best equilibria in “smaller” bimatrix games, each defined on a subinterval where the pure strategy value is constant, and stacking the equilibrium situations if they are consistent. The stack of the “smaller” bimatrix game equilibria is an approximate equilibrium in the initial staircase game. The (weak) consistency of the approximate equilibrium is studied by how much the payoff and equilibrium situation change as the sampling density minimally increases by the three ways of the sampling increment: only the first player’s increment, only the second player’s increment, both the players’ increment. The consistency is decomposed into the payoff, equilibrium strategy support cardinality, equilibrium strategy sampling density, and support probability consistency. It is practically reasonable to consider a relaxed payoff consistency.
 Conclusions. The suggested method of finite approximation of staircase 2-person games consists in the independent samplings, solving “smaller” bimatrix games in a reasonable time span, and stacking their solutions if they are consistent. The finite approximation is regarded appropriate if at least the respective approximate (stacked) equilibrium is -payoff consistent.
 Keywords: game theory; payoff functional; staircase-function strategy; bimatrix game; irregular sampling; approximate equilibrium consistency.