On Some Approaches to Find Nash Equilibrium in Concave Games

Abstract

This paper considers finite-dimensional concave games, i.e., noncooperative n-player games in which the objective functionals are concave in their “own” variables. For such games, we investigate the design problem of numerical search algorithms for Nash equilibrium that have guaranteed convergence without additional requirements on the objective functionals such as convexity in the “other” variables or similar hypotheses (weak convexity, quasiconvexity, etc.). Two approaches are described as follows. The first approach, being obvious enough, relies on the Hooke-Jeeves method for residual function minimization and acts as a “standard” for comparing the efficiency of alternative numerical solution methods. To some extent, the second approach can be regarded as “an intermediate” between the relaxation algorithm and the Hooke-Jeeves method of configurations (with proper consideration of all specifics of the objective functions). A rigorous proof of its convergence is the main result of this paper, for the time being in the case of one-dimensional sets of players strategies yet under rather general requirements to objective functionals. The results of some numerical experiments are presented and discussed. Finally, a comparison with other well-known algorithms is given.