具有单个控制器的马尔可夫博弈中计算粗相关均衡的复杂性
The Complexity of Computing Coarse Correlated Equilibria in Markov Games with a Single Controller
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中文总结 AI 辅助
研究两人单控制器随机博弈中计算近似平稳马尔可夫粗相关均衡的复杂性，证明其为PPAD - 完全，不依赖均衡坍缩或类似纳什结构，通过构造单控制器小装置，在不同精度要求下得出难性结论。
中文摘要 AI 辅助
我们研究在折扣单控制器随机（马尔可夫）博弈中计算平稳马尔可夫粗相关均衡（CCE）的复杂性，这是一类基本的随机博弈，其中所有玩家可能影响奖励，但只有一个玩家控制状态转移。先前工作通过基于回合的构造在两人一般和随机博弈中确立了计算平稳马尔可夫CCE的PPAD - 难，该构造中每个状态由单个玩家控制且跨状态交替控制，这使得每个马尔可夫CCE坍缩为纳什均衡（NE），所以NE的难直接转移到CCE。当单个玩家控制所有转移时（不存在这种坍缩的情况），其难性是否持续仍未解决。我们解决了这个问题：在两人单控制器随机博弈中计算近似平稳马尔可夫CCE是PPAD - 完全的，即使有固定折扣因子和二元动作。对于完美概念（每个状态的均衡约束）在恒定精度下无条件成立；对于非完美概念，我们在PCP - for - PPAD假设下证明了恒定精度难性，并且无条件地证明了逆多项式精度难性。据我们所知，我们的结果首次表明在不依赖均衡坍缩现象或通过类似纳什结构的其他途径的情况下计算CCE的难性。相反，我们构造了单控制器小装置，其局部激励约束即使在强相关平稳策略下也迫使求解纯电路实例。
英文摘要
We study the complexity of computing stationary Markov coarse correlated equilibria (CCE) in discounted single-controller stochastic (Markov) games [PR81, FV97], a fundamental subclass of stochastic games in which all players may affect rewards, but only one player controls the state transitions. Prior work [DGZ23, JMS23, HN25] established PPAD-hardness for computing stationary Markov CCE in two-player general-sum stochastic games via turn-based constructions in which each state is controlled by a single player, with control alternating across states. This structure forces every Markov CCE to collapse to a Nash equilibrium (NE), so hardness for NE transfers immediately to CCE. It remained open whether hardness persists when a single player controls all transitions--a setting where no such collapse occurs. We resolve this question: computing an approximate stationary Markov CCE in two-player single-controller stochastic games is PPAD-complete, even with a fixed discount factor and binary actions. For the perfect notion (equilibrium constraints at every state) this holds unconditionally at constant accuracy; for the non-perfect notion, we prove constant-accuracy hardness under the PCP-for-PPAD hypothesis [BPR16, DFHM26] and inverse-polynomial-accuracy hardness unconditionally. To the best of our knowledge, our result is the first to show hardness for computing CCE without relying on equilibrium collapse phenomena or other routes through Nash-like structure [FGK23, AKSZ24, PR24]. Instead, we construct single-controller gadgets whose local incentive constraints force a solution of a Pure-Circuit instance even under strongly correlated stationary policies.
