[Paper Review] The Complexity of Proper Equilibrium in Extensive-Form and Polytope Games

Brian Hu Zhang, Ioannis Anagnostides+3 more|arXiv (Cornell University)|Feb 10, 2026

Game Theory and Applications0 citations

TL;DR

The paper proves that computing normal-form proper equilibria in extensive-form games is PPAD-complete (two-player) and FIXP_a-complete (multi-player), while proving NP-hardness for computing proper equilibria in polytope games; it also develops efficient methods for extensive-form games and identifies a barrier (KM best responses) that is #P-hard to compute in general.

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.

Motivation & Objective

• Motivate and analyze the computational complexity of Myerson’s proper equilibrium in extensive-form and polytope game representations.
• Resolve open questions about whether proper equilibria in extensive-form games share the same complexity as Nash equilibria.
• Identify algorithmic barriers in standard perturbation schemes (Kohlberg–Mertens) and establish tractable approaches for EFGs.
• Contrast the complexity landscape between extensive-form and polytope games for equilibrium refinements.
• Provide a framework that connects best-response computation to broader complexity classes (PPAD, FIXP_a, NP).

Proposed method

• Introduce epsilon-proper best responses and analyze their computational properties in polytope and extensive-form games.
• Show #P-hardness of computing Kohlberg–Mertens style perturbations (KM best responses) even on the hypercube.
• Prove NP-hardness of computing proper equilibria in polytope games.
• Develop an efficient best-response computation technique for extensive-form games that enables PPAD- and FIXP_a-membership results.
• Leverage the optimization framework of Filos-Ratsikas et al. to structure the best-response oracle as a linear feasibility problem compatible with PPAD/FIXP_a analysis.
• Present formal reductions and formal theorems establishing the complexity classifications (Theorems 1.1–1.3, informal statements).

Experimental results

Research questions

• RQ1 Does computing a proper equilibrium in extensive-form games have the same complexity as computing a Nash equilibrium?
• RQ2 What is the computational complexity of finding proper equilibria in polytope games?
• RQ3 Can the standard Kohlberg–Mertens perturbation approach be efficiently implemented in extensive-form and polytope settings?
• RQ4 Are there efficient algorithms for computing perturbed best responses in extensive-form games that avoid exponential sorting?
• RQ5 How do the complexity classes PPAD and FIXP_a relate to proper equilibrium computation in different game representations (extensive-form vs polytope)?

Key findings

• Computing a proper equilibrium of an extensive-form game has the same complexity as computing a Nash equilibrium (PPAD-complete for two-player, FIXP_a-complete for multi-player).
• Computing a Kohlberg–Mertens perturbed best response is #P-hard even for Bayesian games (the KM barrier).
• Computing a proper equilibrium of a polytope game is NP-hard, marking a natural class where equilibrium refinements do not collapse to Nash equilibria.
• There exists an efficient best-response computation for extensive-form games that enables membership in PPAD and FIXP_a for proper equilibria in EFGs.
• The paper demonstrates a separation: extensive-form game proper equilibria have tractable complexity characterizations while polytope game proper equilibria are NP-hard, highlighting a fundamental complexity distinction.