A Graph Spectral Approach for Computing Approximate Nash Equilibria

We present a new methodology for computing approximate Nash equilibria for two-person non-cooperative games based upon certain extensions and specializations of an existing optimization approach previously used for the derivation of fixed approximations for this problem. In particular, the general two-person problem is reduced to an indefinite quadratic programming problem of special structure involving the n × n adjacency matrix of an induced simple graph specified by the input data of the game, where n is the number of players’ strategies. Using this methodology and exploiting certain properties of the positive part of the spectrum of the induced graph, we show that for any ε > 0 there is an algorithm to compute an ε-approximate Nash equilibrium in time n, where, ξ(m) = ∑m i=1 λi/n and λ1, λ2, . . . , λm are the positive eigenvalues of the adjacency matrix of the graph. For classes of games for which ξ(m) is a constant, there is a PTAS. Based on the best upper bound derived for ξ(m) so far, the worst case complexity of the method is bounded by the subexponential n √