Approximate Nash equilibria in large nonconvex aggregative games

Abstract

This paper shows the existence of $\mathcal{O}(\frac{1}{n^3})$-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 ${\hat{I}}^3$-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 ${\hat{I}}^3$ equal to 1, a gradient-proximal algorithm is used to construct $\mathcal{O}(\frac{1}{n})$-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.

28 pages,2 figures