Hamiltonian of polymatrix zero-sum games

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.

28 pages. v2: minor corrections. v3: minor improvements and references added. Published version

References in corpus (6)

• Quantum Games
• Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds
• Hamiltonian Evolutionary Games
• Piecewise Linear Hamiltonian Flows Associated to Zero-Sum Games: Transition Combinatorics and Questions on Ergodicity
• Conservative Replicator and Lotka-Volterra Equations in the context of Dirac$\backslash$ big-isotropic Structures
• Learning in Quantum Common-Interest Games and the Separability Problem