A distributed Tikhonov-regularization algorithm for optimal Nash equilibrium computation in aggregative games

This paper aims to design a distributed coordination algorithm for solving a multi-agent decision-making problem with a hierarchical structure. The primary goal is to search the Nash equilibrium of a non-cooperative game where each player minimizes its private object with others’ strategies unchanged. Meanwhile, a specific social cost is taken into account during decision-making and is optimized within the equilibria of the underlying game. Such an optimal Nash equilibrium problem can be modeled as a distributed optimization problem with variational inequality constraints. We consider the scenario where the objective functions of both the underlying game and social cost optimization problem have a special aggregation structure. When each player has access only to its local objectives and cannot directly know the decisions of all players, a distributed algorithm is highly recommended. By utilizing the Tikhonov regularization and dynamical average tracking technique, we propose a distributed coordination algorithm by introducing an incentive term to the gradient-based Nash equilibrium seeking, so as to intervene in players’ decisions to improve the system efficiency. We prove its convergence to the optimal Nash equilibrium of a monotone aggregative game, and carry out numerical experiments to empirically demonstrate the algorithm performance.