Distributed Strategy Seeking for Aggregative Games Over Digraphs Based on the Out-Degree Information
研究有向图上带耦合约束和局部可行性约束的聚合博弈，通过估计拉普拉斯矩阵右特征向量消除有向图不平衡，提出分布式投影算法并证明渐近收敛到纳什均衡，无约束时可达指数收敛。
In this article, we investigate aggregative games with coupling constraints and local feasibility constraints over digraphs. It is noted that the imbalance introduced by the digraph leads to an inaccurate estimation of the global aggregation function and increases the difficulty of handling coupling constraints. To overcome this issue, a consensus dynamics is designed to estimate the right eigenvector associated with the zero eigenvalue of the Laplacian matrix constructed using the nodes’ out-degree information. By normalizing with the estimated right eigenvector, the imbalance is eliminated, enabling accurate estimation of the global aggregation function. Based on this, a distributed projection-based algorithm is developed, and through the aid of the proposed consensus dynamics, the asymptotic convergence to the Nash equilibrium (NE) is rigorously proven via singular perturbation theory. In particular, when either local feasibility constraints or coupling constraints are absent, the proposed algorithm achieves exponential convergence to the NE. Finally, the effectiveness of the proposed algorithms is validated through simulations on the location problem and Nash–Cournot games.