School of Physical Sciences, National Institute of Science Education and Research, HBNI, Jatni 752050, India.
Chaos. 2020 Sep;30(9):093117. doi: 10.1063/5.0015655.
The emergence of cooperation in the thermodynamic limit of social dilemmas is an emerging field of research. While numerical approaches (using replicator dynamics) are dime a dozen, analytical approaches are rare. A particularly useful analytical approach is to utilize a mapping between the spin-1/2 Ising model in 1D and the social dilemma game and calculate the magnetization, which is the net difference between the fraction of cooperators and defectors in a social dilemma. In this paper, we look at the susceptibility, which probes the net change in the fraction of players adopting a certain strategy, for both classical and quantum social dilemmas. The reason being, in statistical mechanics problems, the thermodynamic susceptibility as compared to magnetization is a more sensitive probe for microscopic behavior, e.g., observing small changes in a population adopting a certain strategy. In this paper, we find the thermodynamic susceptibility for reward, sucker's payoff, and temptation in classical Prisoner's Dilemma to be positive, implying that the turnover from defect to cooperate is greater than vice versa, although the Nash equilibrium for the two-player game is to defect. In the classical hawk-dove game, the thermodynamic susceptibility for resource suggests that the number of players switching to hawk from dove strategy is dominant. Entanglement in Quantum Prisoner's Dilemma has a non-trivial role in determining the behavior of thermodynamic susceptibility. At maximal entanglement, we find that sucker's payoff and temptation increase the number of players switching to defect. In the zero-temperature limit, we find that there are two second-order phase transitions in the game, marked by a divergence in the susceptibility. This behavior is similar to that seen in type-II superconductors wherein also two second-order phase transitions are seen.
社会困境的热力学极限中的合作出现是一个新兴的研究领域。虽然数值方法（使用复制者动力学）比比皆是，但分析方法却很少。一种特别有用的分析方法是利用一维自旋-1/2 Ising 模型与社会困境游戏之间的映射，并计算磁化强度，即社会困境中合作者与叛徒之间的分数净差。在本文中，我们研究了易感性，它探测了采用某种策略的玩家分数的净变化，对于经典和量子社会困境都是如此。原因是，在统计力学问题中，与磁化强度相比，热力学磁化率是对微观行为的更敏感探针，例如，观察采用某种策略的人群中发生的微小变化。在本文中，我们发现经典囚徒困境中的奖励、 sucker 的收益和诱惑的热力学易感性为正，这意味着从缺陷到合作的转变大于反之亦然，尽管两人游戏的纳什均衡是缺陷。在经典鹰鸽博弈中，资源的热力学易感性表明，从鸽策略切换到鹰策略的玩家数量占主导地位。量子囚徒困境中的纠缠在确定热力学易感性的行为方面起着重要作用。在最大纠缠时，我们发现 sucker 的收益和诱惑增加了转向缺陷的玩家数量。在零温极限下，我们发现游戏中有两个二阶相变，这在易感性上表现出发散。这种行为类似于在 II 型超导体中看到的行为，其中也观察到两个二阶相变。
