Games of fixed rank: a hierarchy of bimatrix games

Abstract

We propose and investigate bimatrix games, whose (entry-wise) sum of the pay-off matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank k-games strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games. We show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we can provide a deterministic polynomial time algorithm for finding an e-approximation (whose running time is polynomial in 1\e) as well as a randomized polynomial time approximation algorithm (whose running time is similar), but which offers the possibility of finding an exact solution in polynomial time if a conjecture is valid. The latter algorithm is based on a new application of random sampling methods to quadratic optimization problems of fixed rank.