Uniform sampling of the infinite noncooperative game on unit hypercube and reshaping ultimately multidimensional matrices of players’ payoff values

Abstract

There is suggested a method of obtaining the approximate solution of the infinite noncooperative game on the unit hypercube. The method is based on sampling uniformly the players’ payoff functions with the constant step along each of the hypercube dimensions. There are stated conditions for the sufficiently accurate sampling. There is suggested a method of reshaping the multidimensional matrix of the player’s payoff values, being the former player’s payoff function before its sampling, into a matrix with minimally possible number of dimensions, where also maintenance of one-to-one indexing has been provided. Requirements for finite NE-strategy from NE-solution of the finite game as the initial infinite game approximation are given as definitions of the approximate solution consistency. The approximate solution consistency ensures its relative independence upon the sampling step within its minimal neighborhood or the minimally decreased sampling step. The ultimate reshaping of multidimensional matrices of players’ payoff values to the minimal number of dimensions, being equal to the number of players, stimulates shortened computations.