We study the decidability and complexity of non-cooperative rational synthesis problem (abbreviated as NCRSP) for some classes of probabilistic strategies. We show that NCRSP for stationary strategies and Muller objectives is in 3-EXPTIME, and if we restrict the strategies of environment players to be positional, NCRSP becomes NEXPSPACE solvable. On the other hand, NCRSP_>, which is a variant of NCRSP, is shown to be undecidable even for pure finite-state strategies and terminal reachability objectives. Finally, we show that NCRSP becomes EXPTIME solvable if we restrict the memory of a strategy to be the most recently visited t vertices where t is linear in the size of the game.

翻译：本文研究了针对若干概率策略类别的非合作理性综合问题（简称NCRSP）的可判定性与计算复杂性。我们证明：对于具有平稳策略与Muller目标的情形，NCRSP属于3-EXPTIME复杂度类；若将环境参与者的策略限制为位置性策略，则NCRSP可在NEXPSPACE内求解。另一方面，我们证明了NCRSP的变体NCRSP_>即使在纯有限状态策略与终端可达性目标下也是不可判定的。最后，我们证明当将策略的记忆限制为最近访问的t个顶点（其中t与博弈规模呈线性关系）时，NCRSP可在EXPTIME内求解。