PPAD-complete approximate pure Nash equilibria in Lipschitz games

Lipschitz games, in which there is a limit A (the Lipschitz value of the game) on how much a player's payoffs may change when some other player deviates, were introduced about 10 years ago by Azrieli and Shmaya. They showed via the probabilistic method that n-player Lipschitz games with m strategies per player have e-approximate pure Nash equilibria, for root e >= A 8nlog(2mn). Here we provide the first hardness result for the corresponding computational problem, showing that even for a simple class of Lipschitz games (Lipschitz polymatrix games), finding e-approximate pure equilibria is PPAD-complete, for suitable pairs of values e(n), A(n). Novel features of this result include both the proof of PPAD hardness (in which we apply a population game reduction from unrestricted polymatrix games) and the proof of containment in PPAD (by derandomizing the selection of a pure equilibrium from a mixed one). In fact, our approach implies containment in PPAD for any class of Lipschitz games where payoffs from mixed-strategy profiles can be deterministically computed. When instead considering games where only payoffs from action profiles can be deterministically computed, we provide two equivalent definitions of "randomized PPAD" and show that the generalized problem belongs to this class.