This paper proposes a distributed algorithm to find the Nash equilibrium in a class of non-cooperative convex games with partial-decision information. Our method employs a distributed projected gradient play approach alongside consensus dynamics, with individual agents minimizing their local costs through gradient steps and local information exchange with neighbors via a time-varying directed communication network. Addressing time-varying directed graphs presents significant challenges. Existing methods often circumvent this by focusing on static graphs or specific types of directed graphs or by requiring the stepsizes to scale with the Perron-Frobenius eigenvectors. In contrast, we establish novel results that provide a contraction property for the mixing terms associated with time-varying row-stochastic weight matrices. Our approach explicitly expresses the contraction coefficient based on the characteristics of the weight matrices and graph connectivity structures, rather than implicitly through the second-largest singular value of the weight matrix as in prior studies. The established results facilitate proving geometric convergence of the proposed algorithm and advance convergence analysis for distributed algorithms in time-varying directed communication networks. Numerical results on a Nash-Cournot game demonstrate the efficacy of the proposed method.

翻译：本文提出了一种分布式算法，用于求解一类具有部分决策信息的非合作凸博弈的纳什均衡。我们的方法采用分布式投影梯度博弈策略并结合一致性动力学，其中每个智能体通过梯度步骤最小化其局部成本，并通过时变有向通信网络与邻居进行局部信息交换。处理时变有向图带来了重大挑战。现有方法通常通过专注于静态图或特定类型的有向图，或要求步长按Perron-Frobenius特征向量缩放来规避这些挑战。相比之下，我们建立了新的结果，为与时变行随机权重矩阵相关的混合项提供了收缩性质。我们的方法基于权重矩阵特性和图连通性结构显式地表达了收缩系数，而非如先前研究那样通过权重矩阵的第二大奇异值隐式表达。所建立的结果有助于证明所提算法的几何收敛性，并推动了时变有向通信网络中分布式算法的收敛性分析。在纳什-古诺博弈上的数值结果验证了所提方法的有效性。