并发停止博弈中的均衡算法
Algorithms for Equilibria in Concurrent Stopping Games
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中文总结 AI 辅助
研究并发停止博弈中均衡算法，通过放宽精确性考虑近似约束存在问题，给出指数时间算法并证明其\(\PSPACE\)-难下界，还通过转向极端风险敏感均衡，证明其约束存在问题在并发博弈中是\(\NP\)-完全的。
中文摘要 AI 辅助
并发博弈是多智能体系统的标准模型，以纳什均衡作为核心解概念。相关的“约束存在问题”，即博弈是否存在一个纳什均衡，使得每个玩家的期望收益在规定区间内，是不可判定的，即使对于10人“停止”博弈也是如此，在这种博弈中，在每个策略组合下几乎肯定会达到终端状态。我们给出了两种可处理性的途径。首先放宽精确性，考虑由\(\varepsilon\)-NE参数化的近似约束存在问题，即确定是否存在具有规定收益的\(\varepsilon\)-纳什均衡。该算法运行时间为指数时间，且仅在\(\varepsilon\)的比特大小上是多项式时间。我们用一个对于回合制博弈以及纯均衡也成立的\(\PSPACE\)-难下界对其进行补充。然后放宽解概念，转向最近为回合制随机博弈引入的“极端风险敏感均衡”（XRSE）。在这里，玩家被分为乐观主义者和悲观主义者，他们分别通过以正概率可获得的最佳和最差收益来评估策略组合，而不是期望收益。我们证明，对于并发博弈，XRSE的约束存在问题与回合制博弈一样是\(\NP\)-完全的。
英文摘要
Concurrent games are a standard model for multi-agent systems, with Nash equilibrium as their central solution concept. The associated \emph{constrained existence problem}---does a game admit a Nash equilibrium whose expected payoff lies within a prescribed interval for every player?---is undecidable, and remains so even for 10-player \emph{stopping} games, in which a terminal state is reached almost surely under every strategy profile. We give two routes to tractability. We first relax exactness and consider the problem of approximate constrained existence problem, parametrised by $\varepsilon$-NE, which decides whether an \(\varepsilon\)-Nash equilibrium with the prescribed payoffs exists. The algorithm runs in exponential time, and only polynomially in the bit-size of \(\varepsilon\). We complement it with a \PSPACE-hardness lower bound that holds already for turn-based games, and for pure equilibria as well. We then relax the solution concept, turning to \emph{extreme risk-sensitive equilibria} (XRSE), recently introduced for turn-based stochastic games. Here the players are partitioned into optimists and pessimists, who evaluate a strategy profile by the best, respectively the worst, payoff attainable with positive probability, instead of the expected payoff. We prove that the constrained existence problem for XRSE is \NP-complete on concurrent games, as for turn-based games.
