Approximate Nash equilibria in large nonconvex aggregative games

This paper shows the existence of O(frac1nγ)-Nash equilibria in n-player noncooperative sum-aggregative games in which the players' cost functions, depending only on their own action and the average of all players' actions, are lower semicontinuous in the former while γ-H"older continuous in the latter. Neither the action sets nor the cost functions need to be convex. For an important class of sum-aggregative games, which includes congestion games with γ equal to 1, a gradient-proximal algorithm is used to construct O(frac1n)-Nash equilibria with at most O(n3) iterations. These results are applied to a numerical example concerning the demand-side management of an electricity system. The asymptotic performance of the algorithm when n tends to infinity is illustrated.