The Complexity of Proper Equilibrium in Extensive-Form and Polytope Games
Brian Hu Zhang, Ioannis Anagnostides, Kiriaki Fragkia, Maria-Florina Balcan, Tuomas Sandholm
TL;DR
The paper resolves the complexity of computing normal-form proper equilibria in extensive-form games, showing $ extsf{PPAD}$-completeness for two-player cases and $ extsf{FIXP}_a$-completeness for multi-player cases via an efficient perturbed best-response construction. It identifies a fundamental barrier from Kohlberg–Mertens perturbations based on the permutahedron, proving ${ t extsf{#P}}$-hardness even in the hypercube and Bayesian settings, and proves $ extsf{NP}$-hardness for computing proper equilibria in polytope games, marking the first natural class where refinement complexity does not collapse to Nash. Despite this, the authors show that efficient proper best responses exist for extensive-form games, enabling a polynomial-time procedure to compute a sequentially proper BR and, hence, a normal-form proper equilibrium, aligning EF results with Nash computation. The work thus reveals a sharp separation between the complexity of equilibrium refinements in polytope games and extensive-form games and offers a constructive framework to compute refinements in EF representations. It opens avenues for extending these results to other refinements and to broader representations beyond EF, with implications for algorithmic game theory and economic modeling.
Abstract
The proper equilibrium, introduced by Myerson (1978), is a classic refinement of the Nash equilibrium that has been referred to as the "mother of all refinements." For normal-form games, computing a proper equilibrium is known to be PPAD-complete for two-player games and FIXP$_a$-complete for games with at least three players. However, the complexity beyond normal-form games -- in particular, for extensive-form games (EFGs) -- was a long-standing open problem first highlighted by Miltersen and Sørensen (SODA '08). In this paper, we resolve this problem by establishing PPAD- and FIXP$_a$-membership (and hence completeness) of normal-form proper equilibria in two-player and multi-player EFGs respectively. Our main ingredient is a technique for computing a perturbed (proper) best response that can be computed efficiently in EFGs. This is despite the fact that, as we show, computing a best response using the classic perturbation of Kohlberg and Mertens based on the permutahedron is #P-hard even in Bayesian games. In stark contrast, we show that computing a proper equilibrium in polytope games is NP-hard. This marks the first natural class in which the complexity of computing equilibrium refinements does not collapse to that of Nash equilibria, and the first problem in which equilibrium computation in polytope games is strictly harder -- unless there is a collapse in the complexity hierarchy -- relative to extensive-form games.