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Inverse Learning in $2\times2$ Games: From Synthetic Interactions to Traffic Simulation

Daniela Aguirre Salazar, Firas Moatemri, Tatiana Tatarenko

TL;DR

This work tackles inverse game-theoretic learning for 2x2 normal-form games, aiming to recover latent utilities from observed joint actions in domains such as traffic. It introduces two maximum-likelihood frameworks: CE-ML, which leverages the closed-form 2x2 CE polytope to fit utilities and CE mixtures, and LBR-ML, which treats behavior as the stationary distribution of logit best-response dynamics to capture bounded rationality. The methods are evaluated in synthetic Chicken-Dare games and SUMO-based traffic simulations, showing that CE-ML excels when data align with correlated equilibrium and LBR-ML is more robust under uncoordinated or noisy conditions, with ICE serving as a weaker baseline in non-CE regimes. The findings highlight a practical trade-off between interpretability and dynamic realism, and point to potential extensions to larger-scale, heterogeneous, or online-estimation settings for traffic and multi-agent systems.

Abstract

Understanding how agents coordinate or compete from limited behavioral data is central to modeling strategic interactions in traffic, robotics, and other multi-agent systems. In this work, we investigate the following complementary formulations of inverse game-theoretic learning: (i) a Closed-form Correlated Equilibrium Maximum-Likelihood estimator (CE-ML) specialized for $2\times2$ games; and (ii) a Logit Best Response Maximum-Likelihood estimator (LBR-ML) that captures long-run adaptation dynamics via stochastic response processes. Together, these approaches span the spectrum between static equilibrium consistency and dynamic behavioral realism. We evaluate them on synthetic "chicken-dare" games and traffic-interaction scenarios simulated in SUMO, comparing parameter recovery and distributional fit. Results reveal clear trade-offs between interpretability, computational tractability, and behavioral expressiveness across models.

Inverse Learning in $2\times2$ Games: From Synthetic Interactions to Traffic Simulation

TL;DR

This work tackles inverse game-theoretic learning for 2x2 normal-form games, aiming to recover latent utilities from observed joint actions in domains such as traffic. It introduces two maximum-likelihood frameworks: CE-ML, which leverages the closed-form 2x2 CE polytope to fit utilities and CE mixtures, and LBR-ML, which treats behavior as the stationary distribution of logit best-response dynamics to capture bounded rationality. The methods are evaluated in synthetic Chicken-Dare games and SUMO-based traffic simulations, showing that CE-ML excels when data align with correlated equilibrium and LBR-ML is more robust under uncoordinated or noisy conditions, with ICE serving as a weaker baseline in non-CE regimes. The findings highlight a practical trade-off between interpretability and dynamic realism, and point to potential extensions to larger-scale, heterogeneous, or online-estimation settings for traffic and multi-agent systems.

Abstract

Understanding how agents coordinate or compete from limited behavioral data is central to modeling strategic interactions in traffic, robotics, and other multi-agent systems. In this work, we investigate the following complementary formulations of inverse game-theoretic learning: (i) a Closed-form Correlated Equilibrium Maximum-Likelihood estimator (CE-ML) specialized for games; and (ii) a Logit Best Response Maximum-Likelihood estimator (LBR-ML) that captures long-run adaptation dynamics via stochastic response processes. Together, these approaches span the spectrum between static equilibrium consistency and dynamic behavioral realism. We evaluate them on synthetic "chicken-dare" games and traffic-interaction scenarios simulated in SUMO, comparing parameter recovery and distributional fit. Results reveal clear trade-offs between interpretability, computational tractability, and behavioral expressiveness across models.
Paper Structure (21 sections, 2 theorems, 19 equations, 5 figures, 6 tables)

This paper contains 21 sections, 2 theorems, 19 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation and Equilibrium Concepts
  3. Inverse Game-Theoretic Learning
  4. Equilibrium Concepts
  5. Notion I: Correlated Equilibrium
  6. Notion II: Stationary State of Logit Best Response Dynamics
  7. Methodology for Inverse Game-Theoretic Learning in 2$\times$ 2 Games
  8. Approach 1: Closed-form 2$\times$2 CE Maximum-Likelihood (CE-ML)
  9. Approach 2: Logit Best Response Learning (LBR-ML)
  10. Experimental Setup
  11. Experimental Domains
  12. Synthetic "Chicken-Dare" Game
  13. Traffic Interaction Game in SUMO
  14. Scenario design
  15. Kinematic-dependent utilities
  16. ...and 6 more sections

Key Result

Theorem 1

Let the players in $\Gamma_N^w$ choose LBR strategies simultaneously over time according to eq:LD with finite rationality levels $\lambda_i<\infty$ and starting by any mixed strategy $\sigma^0\in\Delta(A)$. Then there exists a unique stationary distribution $\sigma^\star(\lambda,w)$ such that $\lim_

Figures (5)

  • Figure 1: Uncontrolled intersection used in SUMO. Distances $d_A,d_B$ and velocities $v_A,v_B$ define times $\tau_i=d_i/v_i$ and offset $\delta=\tau_2-\tau_1$.
  • Figure 2: Probability distribution of equilibria per joint action, T = 2000.
  • Figure 3: Probability distribution of equilibria per joint action, T = 2000.
  • Figure 4: Probability distribution of equilibria per joint action, T=500
  • Figure 5: Probability distribution of equilibria per joint action, T=500

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Remark 1: see CalvoArmengol2003_CE2x2