What happens when an infinite number of players play a quantum game? In this paper, we will answer this question by looking at the emergence of cooperation in the presence of noise in a one-shot quantum Prisoner's dilemma (QuPD). We will use the numerical Agent-based model (ABM) and compare it with the analytical Nash equilibrium mapping (NEM) technique. To measure cooperation, we consider five indicators, i.e., game magnetization, entanglement susceptibility, correlation, player's payoff average, and payoff capacity, respectively. In quantum social dilemmas, entanglement plays a non-trivial role in determining the players' behavior in the thermodynamic limit, and we consider the existence of bipartite entanglement between neighboring players. For the five indicators in question, we observe \textit{first}-order phase transitions at two entanglement values, and these phase transition points depend on the payoffs associated with the QuPD game. We numerically analyze and study the properties of both the \textit{Quantum} and the \textit{Defect} phases of the QuPD via the five indicators. The results of this paper demonstrate that both ABM and NEM, in conjunction with the chosen five indicators, provide insightful information on cooperative behavior in the thermodynamic limit of the one-shot quantum Prisoner's dilemma.

翻译：当无限数量的玩家进行量子博弈时会发生什么？本文通过研究存在噪声情况下一次性量子囚徒困境（QuPD）中合作行为的涌现来回答这一问题。我们将采用数值化的基于智能体模型（ABM），并将其与解析的纳什均衡映射（NEM）技术进行比较。为度量合作程度，我们分别考虑五个指标：博弈磁化强度、纠缠敏感性、相关性、玩家平均收益以及收益容量。在量子社会困境中，纠缠在热力学极限下决定玩家行为时扮演着重要角色，我们假设相邻玩家之间存在二分体纠缠。针对所讨论的五个指标，我们在两个纠缠值处观察到一级相变，且这些相变点依赖于与QuPD博弈相关的收益值。我们通过这五个指标对QuPD的"量子相"和"缺陷相"的性质进行了数值分析与研究。本文结果表明，ABM与NEM方法结合所选的五个指标，能够为一次性量子囚徒困境在热力学极限下的合作行为提供富有洞见的信息。