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 multi-agent systems and Hamiltonian dynamics -- the same dynamics that describe conservative systems in physics. Specifically, we show that no matter the size, or network structure of such closed economies, even if agents use different online learning dynamics from the standard class of Follow-the-Regularized-Leader, they yield Hamiltonian dynamics. This approach generalizes the known connection to Hamiltonians for the special case of replicator dynamics in two agent zero-sum games developed by Hofbauer. Moreover, our results extend beyond zero-sum settings and provide a type of a Rosetta stone (see e.g. Table 1) that helps to translate results and techniques between online optimization, convex analysis, games theory, and physics.

研究了多智能体系统和哈密顿动力学之间的联系，无论网络结构、复杂性以及使用的算法，都可以将零和网络博弈中的收敛动态映射到哈密顿动力学中。此外，该方法可以将在线优化，凸分析，博弈论和物理学之间的结果和方法进行互相转化。

网络零和博弈中的多智能体学习是哈密顿系统