发表机构
Formal Methods Group, Université Libre de Bruxelles; Department of Computer Science, Rice University（布鲁塞尔自由大学; 莱斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对并发多人可达性博弈，提出含数值约束的无记忆纳什与ε均衡可实现性的数值方法，修正了精确均衡相关构造，揭示了可实现性复杂性与策略数值复杂性的关联。
AI 中文摘要
在形式方法领域，概率无贴现的基于状态的系统的均衡存在性问题被称为可实现性问题，已被证明是极具挑战性的研究场景。本文聚焦于同时满足无记忆性和数值约束的均衡，研究经典可实现性问题的受限版本。文献中对无记忆策略的限制较为常见，但数值约束据我们所知是一种新的有效方法。首先，我们考虑所有涉及数值必须为预定义有限大小的有理数时的无记忆均衡存在性问题；随后，将该分析扩展至由根式代数生成元构建的有理数域扩张。当为后者提供基时，精确纳什均衡与ε纳什均衡的两类可实现性问题均为NP完全问题。最后，我们考虑无约束数值场景：尽管精确纳什均衡可实现性问题的特征为ETR完全是该领域最著名的成果之一，但我们证明该成果所基于的构造存在缺陷；我们修正了这些构造以保留ETR上界，并指出文献中的下界不适用于ε均衡场景。总体而言，本文揭示了可实现性问题的复杂性与策略表示中涉及数值的“复杂性”之间的有趣关联。
英文摘要
The existence of equilibria, which is called the realizability problem in the formal methods community, for probabilistic undiscounted state-based systems has proven to be an incredibly challenging research setting. In this paper, we consider a restricted version of the classic realizability problem by focusing on equilibria that are both memoryless and numerically constrained. While the restriction to memoryless strategies is relatively common in the literature, numerical constraints, to the best of our knowledge, represent a new and powerful approach. First, we consider the existence of memoryless equilibria when all numbers involved must be rational numbers of a pre-defined limited size. Then, we extend this analysis to field extensions of the rational numbers created through radical algebraic generators. When the basis is provided for the latter, both realizability problems are NP-complete for both exact and epsilon Nash equilibria. Finally, we consider an unconstrained numerical setting. While the characterization of the exact Nash equilibria realizability problem as ETR-complete is one of the most celebrated results in the literature, we demonstrate that the constructions underlying this result are flawed as presented. We then mend said constructions to preserve the ETR upper bound, and note that the lower bound in the literature does not apply to the epsilon-equilibrium setting. Overall, this paper demonstrates an interesting relationship between the complexity of the realizability problem and the "complexity" of the numbers involved in the representations of strategies.
