Distributed ϵ-Nash equilibrium seeking in aggregative games with approximation

In this paper, we aim to design a distributed approximate algorithm for seeking Nash equilibria of an aggregative game. Because players’ actions are constrained by local feasible sets, one of the most popular methods is to employ projection operators. However, it may be hard to get the exact projection points in practice due to complex set constraints. Inspired by the advantage of the projection on hyperplanes, we promote to use inscribed polyhedrons to approximate players’ local sets, which yields a related approximate game model. Then we propose a distributed algorithm to seek the Nash equilibrium of the approximate game. The projection in the algorithm is replaced by a standard quadratic program with linear constraints, which can reduce the computational complexity. Moreover, we show that the equilibrium of the proposed algorithm induces an ϵ-Nash equilibrium of the original game.