Strategic-form min-max game theory examines the existence, multiplicity, selection of equilibria, and the worst-case computational complexity under perfect rationality. However, in many applications, games are drawn from an ensemble, and players exhibit bounded rationality. We introduce thermal min-max games, a thermodynamic relaxation that unifies bounded and perfect rationality by assigning each player a temperature to regulate their rationality level. To analyze typical behavior in the large-strategy limit, we develop a nested replica framework for this relaxation. This theory provides tractable predictions for typical equilibrium values and mixed-strategy statistics as functions of rationality strength, strategy-count aspect ratio, and payoff randomness. Numerical experiments demonstrate that these asymptotic predictions accurately align with the equilibrium of finite games of moderate size.

翻译：策略型最小-最大博弈理论研究在完全理性假设下均衡的存在性、多重性、选择问题以及最坏情况的计算复杂度。然而，在许多实际应用中，博弈往往从特定分布中抽取，且参与者表现出有限理性特征。本文提出热力学最小-最大博弈——一种通过为每位参与者设定温度参数来调节其理性水平的热力学松弛框架，从而统一有限理性与完全理性。为分析大策略极限下的典型行为，我们为此松弛系统构建了嵌套复本理论框架。该理论可推导关于典型均衡值及混合策略统计量随理性强度、策略数量纵横比和收益随机性变化的可计算预测。数值实验表明，这些渐近预测能够准确吻合中等规模有限博弈的均衡特性。