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.

翻译：近十年前, Azriririri和Shmaya 引入了 $ lambda$- Lipschitz 类游戏, 其中每个玩家的回报功能是 $\ lambda$- Lipschitz 相对于其他玩家的动作来说, $\ lambda$- Lipschitz 。 显示这些游戏接纳了 $\ epsilon$- 纯 Nash equilibria, 某些设置的 $\ epsilon$ 和 $\ lambda$ 。 然而, 问题是如何找到更简单的平衡。 在这项工作中, 我们开发了一个从更普通的游戏到利普西茨游戏 。 我们使用这种更普通的游戏 $- libschitz 的直径直率值, 提供了一个比普通的利普利比 利比萨利伯茨 的直率值 。