We study distributed algorithms for finding a Nash equilibrium (NE) in a class of non-cooperative convex games under partial information. Specifically, each agent has access only to its own smooth local cost function and can receive information from its neighbors in a time-varying directed communication network. To this end, we propose a distributed gradient play algorithm to compute a NE by utilizing local information exchange among the players. In this algorithm, every agent performs a gradient step to minimize its own cost function while sharing and retrieving information locally among its neighbors. The existing methods impose strong assumptions such as balancedness of the mixing matrices and global knowledge of the network communication structure, including Perron-Frobenius eigenvector of the adjacency matrix and other graph connectivity constants. In contrast, our approach relies only on a reasonable and widely-used assumption of row-stochasticity of the mixing matrices. We analyze the algorithm for time-varying directed graphs and prove its convergence to the NE, when the agents' cost functions are strongly convex and have Lipschitz continuous gradients. Numerical simulations are performed for a Nash-Cournot game to illustrate the efficacy of the proposed algorithm.

翻译：我们研究在部分信息条件下，针对一类非合作凸博弈寻找纳什均衡（NE）的分布式算法。具体而言，每个智能体仅能获取自身光滑的局部代价函数，并可通过时变有向通信网络从邻居处接收信息。为此，我们提出一种利用局部分布式信息交换计算纳什均衡的分布式梯度博弈算法。在该算法中，每个智能体执行梯度步以最小化自身代价函数，同时与邻居进行本地信息共享与获取。现有方法需强假设条件，如混合矩阵的平衡性、网络通信结构的全局知识（包括邻接矩阵的Perron-Frobenius特征向量及其他图连通性常数）。相比之下，我们的方法仅依赖混合矩阵行随机性这一合理且广泛使用的假设。我们针对时变有向图分析了该算法，并证明当智能体代价函数强凸且梯度Lipschitz连续时，算法收敛至纳什均衡。通过纳什-古诺博弈的数值仿真验证了所提算法的有效性。