PPAD-Complete Pure Approximate Nash Equilibria in Lipschitz Games
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Lipschitz games, in which there is a limit λ (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 pure ϵ-approximate Nash equilibria, for ϵ≥λ√(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 pure ϵ-approximate equilibria is PPAD-complete, for suitable pairs of values (ϵ(n), λ(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.
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