论文标题
在N-Cluster游戏中寻求分布式NASH平衡，具有完全不协调的恒定步骤
Distributed Nash Equilibrium Seeking in N-Cluster Games with Fully Uncoordinated Constant Step-Sizes
论文作者
论文摘要
最近，在控制社区中，分布式优化和NASH平衡（NE）引起了很多关注。本文研究了一类非合作游戏，称为N-Cluster游戏，该游戏在两个问题中的多个代理中均包含合作和非合作性：在集群中解决分布式优化问题，同时在整个集群中玩不合作游戏。此外，我们考虑了部分决策信息游戏设置，即代理人无法直接访问其他代理商的决策，因此需要通过有向图相互通信。为了解决N群集游戏问题，我们通过共识和梯度跟踪的综合提出了分布式的NE寻求算法。与其他用于N群集游戏的现有离散时间方法不同，所有代理商都公开知道了一个常见的台阶大小，或者只有来自同一集群的代理知道，拟议的算法可以与完全不协调的恒定步骤尺寸一起使用，这允许代理（内部和整个簇）可以选择自己的首选步骤。我们证明，只要台阶尺寸的最大阶梯尺寸和异质性很小，所有代理的决策将线性汇聚到相应的NE。我们通过在Cournot竞赛游戏中的数字示例来验证派生结果。
Distributed optimization and Nash equilibrium (NE) seeking problems have drawn much attention in the control community recently. This paper studies a class of non-cooperative games, known as N-cluster game, which subsumes both cooperative and non-cooperative nature among multiple agents in the two problems: solving distributed optimization problem within the cluster, while playing a non-cooperative game across the clusters. Moreover, we consider a partial-decision information game setup, i.e., the agents do not have direct access to other agents' decisions, and hence need to communicate with each other through a directed graph. To solve the N-cluster game problem, we propose a distributed NE seeking algorithm by a synthesis of consensus and gradient tracking. Unlike other existing discrete-time methods for N-cluster games where either a common step-size is publicly known by all agents or only known by agents from the same cluster, the proposed algorithm can work with fully uncoordinated constant step-sizes, which allows the agents (both within and across the clusters) to choose their own preferred step-sizes. We prove that all agents' decisions converge linearly to their corresponding NE so long as the largest step-size and the heterogeneity of the step-sizes are small. We verify the derived results through a numerical example in a Cournot competition game.
