Efficient algorithm for approximating Nash equilibrium of distributed aggregative games

Abstract: In this paper, we aim to design a distributed approximate algorithm for seeking Nash equilibria of an aggregative game. Due to the local set constraints of each player, projectionbased algorithms have been widely employed for solving such problems actually. Since it may be quite hard to get the exact projection in practice, we utilize inscribed polyhedrons to approximate local set constraints, which yields a related approximate game model. We first prove that the Nash equilibrium of the approximate game is the $\epsilon$-Nash equilibrium of the original game, and then propose a distributed algorithm to seek the $\epsilon$-Nash equilibrium, where the projection is then of a standard form in quadratic programming. With the help of the existing developed methods for solving quadratic programming, we show the convergence of the proposed algorithm, and also discuss the computational cost issue related to the approximation. Furthermore, based on the exponential convergence of the algorithm, we estimate the approximation accuracy related to $\epsilon$. Additionally, we investigate the computational cost saved by approximation on numerical examples.