Approximate Nash equilibria in large nonconvex aggregative games

TL;DR

This paper establishes the existence of approximate Nash equilibria in large nonconvex sum-aggregative games with nonconvex costs, and proposes a gradient-proximal algorithm to compute such equilibria efficiently, with applications to electricity demand management.

Contribution

It proves the existence of approximate equilibria in nonconvex aggregative games and introduces a gradient-proximal algorithm for their computation.

Findings

Existence of $rac{1}{n^ ext{gamma}}$-Nash equilibria in large games.

A gradient-proximal algorithm computes $rac{1}{n}$-Nash equilibria efficiently.

Application to demand-side management in electricity systems.

Abstract

This paper shows the existence of $\mathcal{O}(\frac{1}{n^\gamma})$-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 $\gamma$-H\"{o}lder 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 $\gamma$ 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.