Tractable finite approximation of continuous noncooperative games on a product of linear strategy functional spaces

Abstract

A method of the finite approximation of continuous noncooperative games is presented. The method is based on sampling the functional spaces, which serve as the sets of pure strategies of the players. The pure strategy is a linear function of time, in which the trend-defining coefficient is variable. The spaces of the players’ pure strategies are sampled uniformly so that the resulting finite game is a finite game whose payoff matrices are hypercubic. The presented method of finite approximation makes solutions tractable so that they can be easily implemented and practiced. The approximation procedure starts with not a great number of intervals, for which the respective finite game is built and solved. Then this number is gradually increased, and new, bigger, finite games are solved until an acceptable solution becomes sufficiently close to the same-type solutions at the preceding iterations. The closeness is expressed as the absolute difference between the trend-defining coefficients of the strategies from the neighboring solutions. These distances should be decreasing once they are smoothed with respective polynomials of degree 2.