Analytically, finding the origins of cooperative behavior in infinite-player games is an exciting topic of current interest. In this paper, we compare three analytical methods, i.e., Nash equilibrium mapping (NEM), Darwinian selection (DS) and Aggregate selection (AS), with a numerical Agent based method (ABM) via the game susceptibility, correlation, and payoff capacity as indicators of cooperative behaviour. While the analytical NEM model shows excellent agreement with the numerical ABM, the other analytical models, like AS and DS, show notable divergence with ABM in the thermodynamic limit for the indicators in question. Previously, cooperative behavior was studied by considering game magnetization and individual players' average payoff as indicators. This paper shows that game susceptibility, correlation, and payoff capacity can aid in understanding cooperative behavior in social dilemmas in the thermodynamic limit. The results obtained via NEM and ABM are in good agreement for all three indicators in question, for both Hawk-Dove and the Public goods games. After comparing the results obtained for all five indicators, we see that individual players' average payoff and payoff capacity serve as the best indicators to study cooperative behavior among players in the thermodynamic limit.

翻译：解析地探寻无限玩家博弈中合作行为的起源是当前备受关注的前沿课题。本文通过博弈敏感性、关联性与收益容量这三种合作行为指标，比较了三种解析方法——纳什均衡映射、达尔文选择与聚合选择——与基于智能体的数值模拟方法。解析的纳什均衡映射模型与数值模拟结果高度吻合，而其他解析模型在热力学极限下对所述指标则表现出显著差异。以往研究多采用博弈磁化强度与个体玩家平均收益作为合作行为指标。本文证明，博弈敏感性、关联性与收益容量能够有效促进对热力学极限下社会困境中合作行为的理解。在鹰鸽博弈与公共物品博弈中，纳什均衡映射与数值模拟方法在全部三项指标上均取得良好的一致性。综合比较五种指标的结果表明，个体玩家平均收益与收益容量是研究热力学极限下玩家间合作行为的最佳指标。