We study the complexity of problems related to subgame-perfect equilibria (SPEs) in infinite duration non zero-sum multiplayer games played on finite graphs with parity objectives. We present new complexity results that close gaps in the literature. Our techniques are based on a recent characterization of SPEs in prefix-independent games that is grounded on the notions of requirements and negotiation, and according to which the plays supported by SPEs are exactly the plays consistent with the requirement that is the least fixed point of the negotiation function. The new results are as follows. First, checking that a given requirement is a fixed point of the negotiation function is an NP-complete problem. Second, we show that the SPE constrained existence problem is NP-complete, this problem was previously known to be ExpTime-easy and NP-hard. Third, the SPE constrained existence problem is fixed-parameter tractable when the number of players and of colors are parameters. Fourth, deciding whether some requirement is the least fixed point of the negotiation function is complete for the second level of the Boolean hierarchy. Finally, the SPE-verification problem -- that is, the problem of deciding whether there exists a play supported by a SPE that satisfies some LTL formula -- is PSpace-complete, this problem was known to be ExpTime-easy and PSpace-hard.
翻译:我们研究了与无期限的非零和多玩者游戏有关的复杂问题,在无限的时期里,在带有均等目标的限定图形上播放的次游戏中,我们提出了新的复杂结果,缩小了文献中的差距。我们的技术基于基于要求和谈判概念的对前置独立游戏中SPE的描述,根据SPE所支持的剧本恰恰是符合谈判功能中最低固定点的要求的剧本。新的结果如下。首先,检查某一要求是否为谈判功能的一个固定点是一个NP-完整的问题。第二,我们表明SPE所限制的存在问题是完整的,以前人们知道这个问题是耗时和NP-硬的。第三,SPEE所支持的存在问题是固定的,当玩家数量和颜色参数是参数时,SPEE所支持的游戏功能是否最固定点是谈判功能的完成点。最后,SPEP-V-卡利(S-IF)是某种已知的P-S-Flaimiality(P-IL)版本问题,这是一个已知的P-Sliflical-IL(P-IFIL)问题。