Lower Bounds for the Query Complexity of Equilibria in Lipschitz Games

Abstract

Nearly a decade ago, Azrieli and Shmaya introduced the class of λ-Lipschitz games in which every player's payoff function is λ-Lipschitz with respect to the actions of the other players. They showed that such games admit -approximate pure Nash equilibria for certain settings of and λ. 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 -approximate pure Nash equilibria of n-player, binaryaction, λ-Lipschitz games that is exponential in nλ/ . 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 average of the individual Lipschitz parameters. Finally, we provide an exponential lower bound on the deterministic query complexity of finding -approximate Nash equilibria of n-player, m-action, λ-Lipschitz games for strong values of , motivating the consideration of explicitly randomized algorithms in the above results.

Near-optimal no-regret learning for correlated equilibria in multi-player general-sum games

“…Furthermore, Hadiji et al [2023] recently investigated the first-order query complexity of computing ǫ-Nash equilibria in m × m two-player zero-sum games (cf. [Goldberg and Katzman, 2023, Fearnley and Savani, 2016, Babichenko, 2016, Fearnley et al, 2015, Maiti et al, 2023). They showed that Ω(m) (first-order) queries are needed when ǫ = 0, and roughly Ω(log( 1 mǫ )) when ǫ = O( 1 m 4 ), thereby leaving a substantial gap with the upper bound of O( log m ǫ ) attained via OMWU.…”