We study multiplayer turn-based games played on a finite directed graph such that each player aims at satisfying an ω-regular Boolean objective. Instead of the well-known notions of Nash equilibrium (NE) and subgame perfect equilibrium (SPE), we focus on the recent notion of weak subgame perfect equilibrium (weak SPE), a refinement of SPE. In this setting, players who deviate can only use the subclass of strategies that differ from the original one on a finite number of histories. We are interested in the constrained existence problem for weak SPEs. We provide a complete characterization of the computational complexity of this problem: it is P-complete for Explicit Muller objectives, NP-complete for Co-Büchi, Parity, Muller, Rabin, and Streett objectives, and PSPACE-complete for Reachability and Safety objectives (we only prove NP-membership for Büchi objectives). We also show that the constrained existence problem is fixed parameter tractable and is polynomial when the number of players is fixed. All these results are based on a fine-grained analysis of a fixpoint algorithm that computes the set of possible payoff profiles underlying weak SPEs.

我们研究在有限有向图上进行的多人回合制游戏，以使每个人都致力于满足ω-常规布尔目标。代替众所周知的纳什均衡（NE）和子博弈完美均衡（SPE）的概念，我们关注的是最近的弱子博弈完美均衡（weak SPE）的概念，这是对SPE的改进。在这种情况下，有偏差的玩家只能在有限的历史记录中使用与原始策略不同的策略子类。我们对弱SPE的约束存在问题感兴趣。我们提供了此问题的计算复杂度的完整表征：对于显式Muller目标，它是P-完全；对于Co-Büchi，Parity，Muller，Rabin和Streett目标，它是NP-完全；对于可达性和安全性目标，它是PSPACE-完全。 （我们仅证明Büchi目标是NP成员）。我们还表明，约束存在问题是固定参数易处理的，并且当玩家数量固定时是多项式。所有这些结果均基于对定点算法的细粒度分析，该定点算法计算了弱SPE背后可能的收益分布图集。