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

TL;DR

This paper establishes the computational complexity of finding proper equilibria in extensive-form and polytope games, showing PPAD- and FIXP$_a$-completeness for EFGs and NP-hardness for polytope games, resolving long-standing open problems.

Contribution

It proves the PPAD- and FIXP$_a$-completeness of proper equilibria in extensive-form games and demonstrates NP-hardness in polytope games, revealing new complexity distinctions.

Findings

Proper equilibria in two-player EFGs are PPAD-complete.

Proper equilibria in multi-player EFGs are FIXP$_a$-complete.

Computing proper equilibrium in polytope games is NP-hard.

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.