Approximate Nash Equilibria in Large Nonconvex Aggregative Games

This paper shows the existence of $\mathcal{O}\left(1/n^{\gamma }\right)$-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, whereas γ-Hölder 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 one, a gradient-proximal algorithm is used to construct $\mathcal{O}\left(1/n\right)$-Nash equilibria with at most $\mathcal{O}(n^3)$ 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.