无限重复量子博弈与策略效率
Infinitely Repeated Quantum Games and Strategic Efficiency
AI总结：
本文建立一般重复量子博弈概念并证明量子民间定理，揭示强纠缠下量子囚徒困境中经典 Pareto 最优与相互合作未必构成均衡，并给出具体均衡策略。
摘要：
重复量子博弈理论研究选择量子策略的参与者之间的长期关系。在传统量子博弈理论中，单轮量子博弈或至多有限重复博弈已得到广泛研究，但人们对无限重复量子博弈知之甚少。研究无限重复博弈至关重要，因为有限重复博弈与单轮博弈并无太大差异。在本工作中，我们建立了一般重复量子博弈的概念，并证明了 Quantum Folk Theorem（量子民间定理）；该定理声称，通过迭代一个博弈，人们可以找到该博弈的一种均衡策略，并获得相应单轮量子博弈的 Nash 均衡无法获得的奖励。重复量子囚徒困境与重复经典囚徒困境之间的一个显著差异在于，当纠缠足够强时，经典 Pareto 最优解并不总是重复量子博弈的均衡。当纠缠足够强且奖励较小时，相互合作不可能成为重复量子博弈的均衡。此外，我们给出了重复量子囚徒困境的若干具体均衡策略。
英文摘要：
Repeated quantum game theory addresses long term relations among players who choose quantum strategies. In the conventional quantum game theory, single round quantum games or at most finitely repeated games have been widely studied, however less is known for infinitely repeated quantum games. Investigating infinitely repeated games is crucial since finitely repeated games do not much differ from single round games. In this work we establish the concept of general repeated quantum games and show the Quantum Folk Theorem, which claims that by iterating a game one can find an equilibrium strategy of the game and receive reward that is not obtained by a Nash equilibrium of the corresponding single round quantum game. A significant difference between repeated quantum prisoner's dilemma and repeated classical prisoner's dilemma is that the classical Pareto optimal solution is not always an equilibrium of the repeated quantum game when entanglement is sufficiently strong. When entanglement is sufficiently strong and reward is small, mutual cooperation cannot be an equilibrium of the repeated quantum game. In addition we present several concrete equilibrium strategies of the repeated quantum prisoner's dilemma.
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