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 $n \times n$ 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 $\varepsilon > 0$ there is an algorithm to compute an $\varepsilon$-approximate Nash equilibrium in time $n^{\xi(m)/\varepsilon}$, where, $\xi (m) = \sum_{i=1}^m \lambda_i / n$ and $\lambda_1, \lambda_2, >..., \lambda_m$ are the positive eigenvalues of the adjacency matrix of the graph. For classes of games for which $\xi (m)$ is a constant, there is a PTAS. Based on the best upper bound derived for $\xi(m)$ so far, the worst case complexity of the method is bounded by the subexponential $n^{\sqrt{m}/\varepsilon}$.