Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games

Nearly a decade ago, Azrieli and Shmaya introduced the class of $\lambda$-Lipschitz games in which every player's payoff function is $\lambda$-Lipschitz with respect to the actions of the other players. They showed that such games admit $\epsilon$-approximate pure Nash equilibria for certain settings of $\epsilon$ and $\lambda$. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding $\epsilon$-approximate pure Nash equilibria of $n$-player, binary-action, $\lambda$-Lipschitz games that is exponential in $\frac{n\lambda}{\epsilon}$. In addition, we introduce ``Multi-Lipschitz games,'' a generalization involving player-specific Lipschitz values, and provide a reduction from finding equilibria of these games to finding equilibria of Lipschitz games, showing that the value of interest is the sum of the individual Lipschitz parameters. Finally, we provide an exponential lower bound on the deterministic query complexity of finding $\epsilon$-approximate correlated equilibria of $n$-player, $m$-action, $\lambda$-Lipschitz games for strong values of $\epsilon$, motivating the consideration of explicitly randomized algorithms in the above results. Our proof is arguably simpler than those previously used to show similar results.