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).

翻译：本文提出一种求解PPAD完全问题的方法[Papadimitriou, 1994]。给定对称双矩阵博弈$(C, C^T)$的支付矩阵$C$，我们的目标是计算$(C, C^T)$的纳什均衡。本文设计了一种非线性复制子动力学（其右端项可通过求解一对凸优化问题获得），该动力学具有以下性质：对于任意可逆的$0 \leq C \leq 1$，从标准单纯形内部策略出发的动力学轨道趋近于$(C, C^T)$的一组策略集，使得对于该集合中的每个策略，可通过求解前述凸规划问题计算出一个对称纳什均衡策略。我们利用分析学中的经典结论（解析隐函数定理）、非线性优化理论（对偶理论、Berge最大值原理及Robinson [1980]关于参数化非线性规划Lipschitz连续性的定理）以及动力系统理论（Losert与Akin [1983]提出的与LaSalle不变性原理相关且在更强假设下结论更强的定理）证明了收敛性。