Graphon Games with Multiple Nash Equilibria: Analysis and Computation

Abstract

A graphon game can be seen either as a limit of a sequence of network games when the number of players tends to infinity, or a stochastic model for sampling network games. We show that, under mild conditions which allow for multiple best-responses, every graphon game has a Nash equilibrium. We then show that every convergent sequence of Nash equilibria of network games sampled from a graphon game converges to a Nash equilibrium of the graphon game. Further, we show that every Nash equilibrium of a graphon game is a limit of a subsequence of ϵ-Nash equilibria of network games sampled from the graphon game. Finally, we provide methods for computing the Nash equilibria of two broad classes of graphon games modeling commonly observed interaction structures: graphon games with (i) a low-rank graphon and a polynomial best response function, and (ii) a stochastic block model graphon and finite strategy sets.

Keywords: Graphon games, network games, large population games, Nash equilibria convergence