[论文解读] Studies in the Theory of Quantum Games

Azhar Iqbal|ArXiv.org|Mar 21, 2005

Quantum Mechanics and Applications参考文献 74被引用 32

一句话总结

本论文研究了量化如何改变经典博弈论中的解概念，特别关注演化稳定性和均衡精炼。研究表明，量子策略可以在不改变经典纳什均衡的情况下改变演化稳定策略（ESS），并提出了利用量子波粒二象性和EPR型纠缠实现非经典博弈结果的新物理实现方式。

ABSTRACT

Theory of quantum games is a new area of investigation that has gone through rapid development during the last few years. Initial motivation for playing games, in the quantum world, comes from the possibility of re-formulating quantum communication protocols, and algorithms, in terms of games between quantum and classical players. The possibility led to the view that quantum games have a potential to provide helpful insight into working of quantum algorithms, and even in finding new ones. This thesis analyzes and compares some interesting games when played classically and quantum mechanically. A large part of the thesis concerns investigations into a refinement notion of the Nash equilibrium concept. The refinement, called an evolutionarily stable strategy (ESS), was originally introduced in 1970s by mathematical biologists to model an evolving population using techniques borrowed from game theory. Analysis is developed around a situation when quantization changes ESSs without affecting corresponding Nash equilibria. Effects of quantization on solution-concepts other than Nash equilibrium are presented and discussed. For this purpose the notions of value of coalition, backwards-induction outcome, and subgame-perfect outcome are selected. Repeated games are known to have different information structure than one-shot games. Investigation is presented into a possible way where quantization changes the outcome of a repeated game. Lastly, two new suggestions are put forward to play quantum versions of classical matrix games. The first one uses the association of De Broglie's waves, with travelling material objects, as a resource for playing a quantum game. The second suggestion concerns an EPR type setting exploiting directly the correlations in Bell's inequalities to play a bi-matrix game.

研究动机与目标

• 分析量化对博弈论中解概念的影响，特别是演化稳定性和纳什均衡的精炼。
• 研究量子策略是否能在保持经典纳什均衡不变的情况下改变演化稳定性。
• 提出利用量子波特性与EPR型纠缠实现量子博弈的新物理实现方式。
• 探讨量子关联与纠缠在重新定义双矩阵博弈中的收益结构与均衡中的作用。
• 考察量子力学对重复博弈及子博弈完美结果的影响，特别是在囚徒困境中的表现。

提出的方法

• 采用Eisert-Wilkens-Lewenstein方案与Marinatto-Weber方案对矩阵博弈进行量化。
• 将演化博弈论概念，特别是演化稳定策略（ESS），应用于量子博弈。
• 使用密度矩阵与量子测量形式化（包括POVM）来建模量子态与收益。
• 提出一种基于衍射的装置，利用de Broglie波实现通过电子干涉的量子博弈策略。
• 提出一种EPR型实验装置，其中来自贝尔不等式违背态的量子关联定义博弈收益。
• 分析量子博弈中的复制子动力学与吸引子结构，以评估均衡的稳定性。

实验结果

研究问题

• RQ1 量化是否可以在不改变相应经典纳什均衡的情况下改变演化稳定策略（ESS）？
• RQ2 EPR型设置中的量子关联如何影响双矩阵博弈的结构与结果？
• RQ3 量子力学是否能在两阶段囚徒困境中实现合作，而经典子博弈完美均衡要求背叛？
• RQ4 量子波粒二象性与衍射装置中的干涉在多大程度上能产生如囚徒困境等博弈中的非经典均衡？
• RQ5 量化如何影响如Stackelberg双寡头这类序贯博弈中的逆向归纳结果？

主要发现

• 量化可将经典鞍点转换为中心点，反之亦然，但如果在经典形式中该点并非吸引子或排斥子，则在量子版本中也无法成为吸引子或排斥子。
• 在三玩家对称合作博弈中，量子纠缠可用于准备一种初始态，使得结成联盟不再具有优势，这与经典版本不同。
• 在Stackelberg双寡头的量子版本中，即使领导者行动已知，经典Cournot纳什均衡仍作为逆向归纳结果被保留，从而消除了对跟随者而言的经典劣势。
• 在两阶段囚徒困境中，量子策略可实现子博弈完美结果，使第一阶段发生合作——这是无经典类比的结果。
• 所提出的基于衍射的装置在囚徒困境中实现了非经典均衡，当de Broglie波长趋近于零时，经典行为随之出现。
• 在基于EPR的博弈模型中，收益与纳什均衡直接依赖于量子关联的性质：仅当关联满足贝尔不等式时才会出现经典博弈，而量子关联则产生根本不同的解结构。