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：本文提出了一种求解PPAD完全问题[Papadimitriou，1994]的方法。给定对称双矩阵博弈$(C, C^T)$的收益矩阵$C$，我们的目标是计算$(C, C^T)$的一个纳什均衡。本文设计了一种非线性复制动力学（其右侧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都将趋近于$(C, C^T)$的一组策略集，使得对于该集合中的每个策略，均可通过求解前述凸数学规划来计算一个对称纳什均衡策略。我们利用分析学（解析隐函数定理）、非线性优化理论（对偶理论、伯奇最大值原理以及罗宾逊[1980]关于参数非线性规划Lipschitz连续性的定理）和动力系统理论（洛瑟特与阿金[1983]关于在更强假设下强于拉萨尔不变性原理的定理）等已有成果证明了收敛性。