The Complexity of Proper Equilibrium in Extensive-Form and Polytope Games

2026-02-10

AI Summary

This study resolves the long-standing open problem of the computational complexity of computing proper equilibria in extensive-form and polyhedral games. By developing an efficient perturbed best-response approach that circumvents the #P-hardness inherent in the classical Kohlberg–Mertens perturbation, and through careful complexity-theoretic reductions, the authors establish for the first time that computing a proper equilibrium is PPAD-complete in two-player extensive-form games, FIXP_a-complete in the multi-player setting, and NP-hard in polyhedral games. This work elucidates fundamental differences in the intrinsic complexity of equilibrium refinements across distinct game models and precisely characterizes the computational boundaries of proper equilibrium within major classes of strategic interactions.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 S{\o}rensen (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.

Problem

• proper equilibrium
• extensive-form games
• polytope games
• computational complexity
• equilibrium refinement

Innovation

• proper equilibrium
• extensive-form games
• computational complexity
• polytope games
• best response perturbation