The complexity of computing a (quasi-)perfect equilibrium for an n-player extensive form game of perfect recall

Abstract

We study the complexity of computing or approximating refinements of Nash equilibrium for a given finite n-player extensive form game of perfect recall (EFGPR), where n ≥ 3.Our results apply to a number of well-studied refinements: to sequential equilibrium (SE), which refines both Nash and subgame-perfect equilibrium; to extensive-form perfect equilibrium (PE), which refines SE; to normal-form perfect equilibrium (NF-PE); and to quasi-perfect equilibrium (QPE) which refines both SE and NF-PE. Of these, the two most refined notions are PE and QPE, which are incompatible with each other. By a classic result of Selten (and by a result of van Damme) a PE (respectively, a QPE) exists for any EFGPR.We show that, for all these notions of equilibrium, approximating an equilibrium for a given EFGPR, to within a given desired precision, is FIXP a -complete. We also consider the complexity of corresponding "almost" equilibrium notions, and show that they are PPAD-complete. In particular, we define δ-almost ǫ-(quasi-)perfect equilibrium, and show that computing one for a given EFGPR, given δ > 0 and ǫ > 0, is PPAD-complete. We show that these notions refine the notion of δ-almost subgame-perfect equilibrium for EFGPRs, which is PPAD-complete.Thus, approximating one such (δ-almost) equilibrium for n-player EFGPRs, n ≥ 3, is P-time equivalent to approximating a (δ-almost) NE for a normal form game with 3 or more players. Normal form games are trivially encodable as EFGPRs without blowup in size. Thus our results extend the celebrated complexity results for Nash equilibrium in normal form games to various refinements of equilibrium in the more general setting of EFGPRs.For 2-player EFGPRs, analogous complexity results follow from the algorithms of Koller, Megiddo, and von Stengel [17,42], von Stengel, van den Elzen, and Talman [43], and Mitersen and Sørensen [25,26]. For n-player EFGPRs, an analogous result for Nash equilibrium and for subgame-perfect equilibrium was given by Daskalakis, Fabrikant, and Papadimitriou [9]. No analogous results were known for the more refined notions of equilibrium for EFGPRs with 3 or more players.