This paper studies the rational synthesis problem for multi-player games played on graphs when rational players are following subgame perfect equilibria. In these games, one player, the system, declares his strategy upfront, and the other players, composing the environment, then rationally respond by playing strategies forming a subgame perfect equilibrium. We study the complexity of the rational synthesis problem when the players have {\omega}-regular objectives encoded as parity objectives. Our algorithm is based on an encoding into a three-player game with imperfect information, showing that the problem is in 2ExpTime. When the number of environment players is fixed, the problem is in ExpTime and is NP- and coNP-hard. Moreover, for a fixed number of players and reachability objectives, we get a polynomial algorithm.

翻译：本文研究了多玩家图博弈中的理性综合问题，其中理性玩家遵循子博弈完美均衡策略。在这些博弈中，一位称为“系统”的玩家预先声明其策略，而构成“环境”的其他玩家则通过选择形成子博弈完美均衡的策略进行理性响应。我们分析了当玩家具有以奇偶目标编码的ω正则目标时，理性综合问题的计算复杂度。我们的算法基于对具有不完美信息的三玩家博弈的编码，证明该问题属于2ExpTime复杂度类。当环境玩家数量固定时，该问题属于ExpTime复杂度类，且同时具有NP难与coNP难特性。此外，对于固定玩家数量和可达性目标的情形，我们给出了多项式时间算法。