Thermal Min-Max Games: Unifying Bounded Rationality and Typical-Case Equilibrium

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.