On the tractability of Nash equilibrium

In this paper, we propose a method for solving a PPAD-complete problem [Papadimitriou, 1994]. Given is the payoff matrix $C$ of a symmetric bimatrix game $(C, C^T)$ and our goal is to compute a Nash equilibrium of $(C, C^T)$. In this paper, we devise a nonlinear replicator dynamic (whose right-hand-side can be obtained by solving a pair of convex optimization problems) with the following property: Under any invertible $0 \leq C \leq 1$, every orbit of our dynamic starting at an interior strategy of the standard simplex approaches a set of strategies of $(C, C^T)$ such that, for each strategy in this set, a symmetric Nash equilibrium strategy can be computed by solving the aforementioned convex mathematical programs. We prove convergence using previous results in analysis (the analytic implicit function theorem), nonlinear optimization theory (duality theory, Berge's maximum principle, and a theorem of Robinson [1980] on the Lipschitz continuity of parametric nonlinear programs), and dynamical systems theory (a theorem of Losert and Akin [1983] related to the LaSalle invariance principle that is stronger under a stronger assumption).