On Search of a Nash Equilibrium in Quasiconcave Quadratic Games

The Nash equilibrium problem with nonconcave quadratic payoff functions is considered. We analyze conditions that provide quasiconcavity of payoff functions in their own variables on the respective strategy sets and hence guarantee the existence of an equilibrium point. One such condition is that the matrix of every payoff function has exactly one positive eigenvalue; this condition is viewed as a basic assumption in the paper. We propose an algorithm that either converges to an equilibrium point or declares that the game has no equilibria. It is shown that some stages of the algorithm are noticeably simplified for quasiconcave games. The algorithm is tested on small-scale instances.