Multi-Agent Learning in Network Zero-Sum Games is a Hamiltonian System

Zero-sum games are natural, if informal, analogues of closed physical systems where no energy/utility can enter or exit. This analogy can be extended even further if we consider zero-sum network (polymatrix) games where multiple agents interact in a closed economy. Typically, (network) zero-sum games are studied from the perspective of Nash equilibria. Nevertheless, this comes in contrast with the way we typically think about closed physical systems, e.g., Earth-moon systems which move perpetually along recurrent trajectories of constant energy. We establish a formal and robust connection between multiagent systems and Hamiltonian dynamics-the same dynamics