We study single-player extensive-form games with imperfect recall, such as the Sleeping Beauty problem or the Absentminded Driver game. For such games, two natural equilibrium concepts have been proposed as alternative solution concepts to ex-ante optimality. One equilibrium concept uses generalized double halving (GDH) as a belief system and evidential decision theory (EDT), and another one uses generalized thirding (GT) as a belief system and causal decision theory (CDT). Our findings relate those three solution concepts of a game to solution concepts of a polynomial maximization problem: global optima, optimal points with respect to subsets of variables and Karush-Kuhn-Tucker (KKT) points. Based on these correspondences, we are able to settle various complexity-theoretic questions on the computation of such strategies. For ex-ante optimality and (EDT,GDH)-equilibria, we obtain NP-hardness and inapproximability, and for (CDT,GT)-equilibria we obtain CLS-completeness results.

翻译：我们研究具有非完全记忆的单人扩展形式博弈，例如睡美人问题或心不在焉司机博弈。针对这类博弈，目前已提出两种自然均衡概念作为事前最优性的替代解概念：一种均衡概念采用广义双倍减半（GDH）作为信念系统并结合证据决策理论（EDT），另一种则采用广义三倍缩减（GT）作为信念系统并结合因果决策理论（CDT）。我们的研究发现，博弈的这三种解概念与多项式最大化问题的解概念存在对应关系：全局最优解、关于变量子集的最优点以及Karush-Kuhn-Tucker（KKT）点。基于这些对应关系，我们得以解决关于此类策略计算的若干复杂性理论问题。对于事前最优性和（EDT,GDH）均衡，我们获得了NP难性及不可近似性结论；对于（CDT,GT）均衡，我们则获得了CLS完全性结论。