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

Yuma Ichikawa

TL;DR

This work introduces thermal min-max games, a thermodynamic relaxation that unifies bounded and perfect rationality by assigning each player a temperature to regulate their rationality level, and develops a nested replica framework for this relaxation.

Abstract

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.

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

TL;DR

This work introduces thermal min-max games, a thermodynamic relaxation that unifies bounded and perfect rationality by assigning each player a temperature to regulate their rationality level, and develops a nested replica framework for this relaxation.

Abstract

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.
Paper Structure (67 sections, 21 theorems, 187 equations, 4 figures)

This paper contains 67 sections, 21 theorems, 187 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Problem Setting
  4. Strategic-Form Min-Max Game
  5. Typical-Case Analysis.
  6. Scaled Min-Max Game.
  7. Thermal Min-Max Games
  8. A Unified View of Bounded Rationality via Thermal Min-Max Games.
  9. Nested Replica Framework for Thermal Min-Max Games
  10. Typical Behavior in Gaussian Thermal Games
  11. Perfect Rationality: Ordered Zero-Temperature Limit
  12. Perturbative Analysis and Linear Response
  13. Perturbation in Payoff Randomness
  14. Perturbation around Equal-Strategy Regime
  15. Numerical Comparison with Finite Strategy Games
  16. ...and 52 more sections

Key Result

Proposition 5.2

At a saddle point ${\bm \Theta}^\star$ of $g({\bm \Theta}; {\bm \Lambda})$, the conjugate variables satisfy and the order parameters obey the moment self-consistency relations where $\langle\cdot\rangle_{z}$ and $\langle\cdot\rangle_{z,\eta}$ denote the Gibbs expectations under the one-site measures induced by $Z_x(z)$ and $Z_y(z,\eta)$, with $\langle\cdot\rangle_{\eta|z}$ defined as follows: f

Figures (4)

  • Figure 1: Replica-overlap structure under the RS and 1RSB ansatz. Left: RS overlap matrix $Q^x_{ab}$ for the $n$ outer replicas of ${\bm x}$, with diagonal entries $Q_x$ ($a=b$) and off-diagonal entries $q_x$ ($a\neq b$). Right: 1RSB overlap matrix $Q^y_{abls}$ for the ${\bm y}$ replicas indexed by $(a,l)$, showing $k$ inner replicas within each outer replica $a$: $Q_y$ on $(a=b,l=s)$, $q_1$ on $(a=b,l\neq s)$, and $q_0$ on $(a\neq b)$.
  • Figure 2: Comparison of theoretical predictions with finite-size LP equilibrium at $M=200$. Left: normalized Nash value $\nu=E_0/L$ as a function of the aspect ratio $\gamma=N/M$. Middle: support fractions $\rho_x$ and $\rho_y$. Right: second moments $q_x={\mathbb E}_{{\bm C}}[\|{\bm x}\|^{2}/N]$ and $q_y={\mathbb E}_{{\bm C}}[\|{\bm y}\|^{2}/M]$ of the rescaled equilibrium strategies. Markers denote disorder-averaged measurements obtained from the LP solver, with error bars showing standard errors over 10 random seeds; solid curves show the corresponding RS predictions from Claim \ref{['claim:oztl']}.
  • Figure 3: Schematic of the RS/1RSB hierarchy induced by the nested replica indices: RS across outer replicas $a$ for $x$, and a 1RSB block structure in the inner replicas $l$ for each fixed $a$ for $y$.
  • Figure 4: Finite-temperature normalized free energy $\nu$ as a function of $|k|=\beta_{\min}/\beta_{\max}$: markers denote finite-size estimates and solid curves are the replica predictions, shown for $\beta_{\max}\in\{0.1,0.5,1.0\}$ and $\gamma\in\{0.8,1.0,1.2\}$.

Theorems & Definitions (43)

  • Definition 2.1: Thermodynamic Limit
  • Definition 3.1: Thermal Min-Max Games
  • Claim 5.1
  • Proposition 5.2
  • Proposition 5.3
  • Claim 5.4
  • Proposition 5.5
  • Proposition 6.1
  • Proposition 6.2
  • proof
  • ...and 33 more