Pure-Circuit: Strong 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 < 0.1 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 ε-well-supported Nash equilibria in two-action polymatrix games, as well as for finding approximate equilibria in threshold games.

Tight Inapproximability for Graphical Games

“…Theorem 1 ( (Deligkas et al 2022a)). PURE-CIRCUIT is PPAD-complete, even when every node of the interaction graph has in-degree at most 2 and total degree at most 3.…”

“…This was recently improved by a result of Deligkas et al (Deligkas et al 2022a). They showed that it is PPADcomplete to find a 1/48-NE of a two-action polymatrix game, and it is PPAD-complete to find an ε-WSNE of a two-action polymatrix game for all ε < 1/3 (the latter result being tight).…”

“…We present a polynomial time algorithm to compute a 1 − 2 2 d +1 -WSNE in any two-action graphical game with maximum in-degree d ≥ 2. The algorithm relies on a simple and natural approach that has been used for similar problems by Liu et al (Liu, Li, and Deng 2021) and Deligkas et al (Deligkas et al 2022a).…”

“…• If u plays one, then v gets 1 from zero and 0 from one. This gadget appeared in (Deligkas et al 2022a), but we include it here for completeness. We claim that this gadget works correctly.…”

“…We note that our lower bounds, both for ε-WSNE and ε-NE, also hold for graphical games with more than two actions. 2 All of our lower bounds are shown via reductions from the PURE-CIRCUIT problem that was recently introduced by Deligkas et al (Deligkas et al 2022a). In that paper, PURE-CIRCUIT was used to show the aforementioned lower bounds for polymatrix games.…”