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.

翻译：Myerson (1978) 提出的适当均衡是纳什均衡的经典精炼，被称为“所有精炼之母”。对于标准式博弈，已知计算适当均衡在双人博弈中是 PPAD-完全的，在至少三人博弈中是 FIXP$_a$-完全的。然而，超越标准式博弈——特别是扩展式博弈（EFGs）——的复杂性，是 Miltersen 和 Sørensen (SODA '08) 首次强调的一个长期悬而未决的问题。本文通过分别确立双人与多人 EFGs 中标准式适当均衡的 PPAD- 与 FIXP$_a$-成员性（从而完备性），解决了这一问题。我们的核心要素是一种计算扰动（适当）最优反应的技术，该技术可在 EFGs 中高效计算。尽管我们证明，即使是在贝叶斯博弈中，使用基于置换多面体的 Kohlberg 和 Mertens 经典扰动来计算最优反应也是 #P-困难的，但我们的技术仍可实现高效计算。与此形成鲜明对比的是，我们证明计算多面体博弈中的适当均衡是 NP-困难的。这标志着计算均衡精炼的复杂性并未坍缩至纳什均衡复杂性的第一个自然类别，也是第一个多面体博弈中的均衡计算严格难于扩展式博弈的问题——除非复杂性层次结构发生坍缩。