Hamiltonian of polymatrix zero-sum games

2025/05/19 by Toshihiro Ota, Ota, Toshihiro, Yuma Fujimoto +1
Decision Sciences · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #Computer Science and Game Theory (cs.GT) #FOS: Computer and information sciences #FOS: Physical sciences #Game Theory and Applications #Machine Learning (cs.LG) #Multiagent Systems (cs.MA) #Opinion Dynamics and Social Influence #Quantum many-body systems

paper · doi:10.48550/arxiv.2505.12609

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The understanding of a dynamical system's properties can be significantly advanced by establishing it as a Hamiltonian system and then systematically exploring its inherent symmetries. By formulating agents' strategies and cumulative payoffs as canonically conjugate variables, we identify the Hamiltonian function that generates the dynamics of poly-matrix zero-sum games. We reveal the symmetries of our Hamiltonian and derive the associated conserved quantities, showing how the conservation of probability and the invariance of the Fenchel coupling are intrinsically encoded within the system. Furthermore, we propose the dissipation FTRL (DFTRL) dynamics by introducing a perturbation that dissipates the Fenchel coupling, proving convergence to the Nash equilibrium and linking DFTRL to last-iterate convergent algorithms. Our results highlight the potential of Hamiltonian dynamics in uncovering the structural properties of learning dynamics in games, and pave the way for broader applications of Hamiltonian dynamics in game theory and machine learning.

Citations

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Related

• Multi-Agent Learning in Network Zero-Sum Games is a Hamiltonian System
• Finite-Sample Guarantees for Learning Dynamics in Zero-Sum Polymatrix Games
• A dynamic that evolves toward a Nash equilibrium
• Zero-sum Polymatrix Markov Games: Equilibrium Collapse and Efficient Computation of Nash Equilibria
• Set Values of Dynamic Nonzero Sum Games and Set Valued Hamiltonians