Efficient Algorithm for Approximating Nash Equilibrium of Distributed Aggregative Games

In this article, we aim to design a distributed approximate algorithm for seeking Nash equilibria (NE) of an aggregative game. Due to the local set constraints of each player, projection-based 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 NE of the approximate game is the
$\epsilon $
-NE of the original game and then propose a distributed algorithm to seek the
$\epsilon $
-NE, where the projection is then of a standard form in quadratic optimization with linear constraints. With the help of the existing developed methods for solving quadratic optimization, 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 $
. In addition, we investigate the computational cost saved by approximation in numerical simulation.