扩展型游戏和多面体游戏中适当均衡复杂性的研究
The Complexity of Proper Equilibrium in Extensive-Form and Polytope Games
AI总结：
本文研究了扩展型游戏和多面体游戏中适当均衡的计算复杂性，证明了其在不同游戏类型中的不同难度，揭示了均衡改进计算的复杂性不依赖于纳什均衡。
摘要：
适当均衡，由Myerson于1978年提出，是纳什均衡的经典改进，被称为“所有改进的母体”。对于正常形式游戏，计算适当均衡已知在双人游戏中的复杂性为PPAD完全问题，在三人或更多玩家游戏中为FIXP_a完全问题。然而，超出正常形式游戏的复杂性——特别是对于扩展型游戏（EFGs）——一直是长期存在的开放问题，首次由Miltersen和Sørensen（SODA '08）提出。在本文中，我们通过建立正常形式适当均衡在双人和多人EFG中的PPAD和FIXP_a成员资格（从而证明其完全性）来解决这一问题。我们的主要成分是一种能够高效计算扰动（适当）最佳反应的技术。尽管如此，我们证明使用Kohlberg和Mertens基于置换体的经典扰动计算最佳反应在贝叶斯游戏中是#P难的。相比之下，我们证明在多面体游戏中计算适当均衡是NP难的。这标志着第一个自然类别，其中计算均衡改进的复杂性不坍缩到纳什均衡的复杂性，也是第一个问题，其中多面体游戏中的均衡计算严格更难——除非复杂性层次结构发生崩溃——相对于扩展型游戏。
英文摘要：
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.
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