将多矩阵零和博弈建模为哈密顿系统，揭示其对称性与收敛机制。
Hamiltonian of polymatrix zero-sum games
- 用策略与累计收益作为共轭变量构建哈密顿函数。
- 证明了耗散型FTRL动态可收敛至纳什均衡。
- 为博弈学习中的结构分析提供新视角，适合理论研究者。
通过将参与者策略与累积收益视为共轭变量，我们确立了多矩阵零和博弈的动力学哈密顿函数。揭示了该哈密顿系统的对称性并推导出相应的守恒量，表明概率守恒与Fenchel耦合的不变性内在编码于系统中。进一步提出耗散型FTRL（DFTRL）动态，通过引入耗散Fenchel耦合的扰动，证明其收敛至纳什均衡，并将DFTRL与最后迭代收敛算法联系起来。结果凸显了哈密顿动力学在揭示博弈学习动态结构性质方面的潜力，为哈密顿方法在博弈论与机器学习中的广泛应用铺平道路。
原文摘要 · Abstract (English)
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.
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