Pure-Circuit: Tight Inapproximability for PPAD

Abstract

The current state-of-the-art methods for showing inapproximability in PPAD arise from the ε-Generalized-Circuit (ε- GCircuit ) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε- GCircuit is PPAD -hard, and subsequent work has shown hardness results for other problems in PPAD by using ε- GCircuit as an intermediate problem. We introduce Pure-Circuit , a new intermediate problem for PPAD , which can be thought of as ε- GCircuit pushed to the limit as ε → 1, and we show that the problem is PPAD -complete. We then prove that ε- GCircuit is PPAD -hard for all ε < 1/10 by a reduction from Pure-Circuit , and thus strengthen all prior work that has used GCircuit as an intermediate problem from the existential-constant regime to the large-constant regime. We show that stronger inapproximability results can be derived by reducing directly from Pure-Circuit . In particular, we prove tight inapproximability results for computing approximate Nash equilibria and approximate well-supported Nash equilibria in graphical games, for finding approximate well-supported Nash equilibria in polymatrix games, and for finding approximate equilibria in threshold games.