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

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.

💡 Research Summary

The paper investigates the computational complexity of Myerson’s proper equilibrium—a refinement of Nash equilibrium—in two classes of games that go beyond the normal‑form representation: extensive‑form games (EFGs) and polytope games. Proper equilibrium, introduced in 1978, imposes a “tremble” axiom that players are less likely to make costly mistakes, and it has long been regarded as the “mother of all refinements.” While the complexity of computing proper equilibria in normal‑form games is well‑understood (PPAD‑complete for two‑player games and FIXPₐ‑complete for three or more players), the situation for EFGs remained an open problem for more than two decades, first highlighted by Miltersen and Sørensen (SODA ’08).

The authors first show that the classic perturbation technique of Kohlberg and Mertens, which relies on the permutahedron and requires sorting an exponential number of pure strategies, is infeasible for EFGs. They formalize this barrier by proving that computing a Kohlberg‑Mertens perturbed best response is #P‑hard even in Bayesian games (Theorem 3.2). This result demonstrates that the naïve extension of the normal‑form approach cannot be used in sequential settings.

To overcome this obstacle, the paper introduces a new perturbation method that works directly at the level of coordinates rather than whole strategies. The key insight is illustrated on the d‑dimensional hypercube: by exploiting linear dependencies among vertices, one can construct a full‑support probability distribution whose expectation matches a desired point while satisfying the ε‑proper ordering condition (if utility of vertex s is lower than that of s′, then its probability is at most ε times the probability of s′). Crucially, the algorithm only needs to sort the d coordinate indices, not the 2ᵈ vertices, yielding a polynomial‑time procedure for ε‑proper best responses.

The authors generalize this construction to arbitrary extensive‑form trees. They formulate a Best‑Response Oracle (BRO) as a linear feasibility problem that encodes the coordinate‑wise ordering constraints (u