# Hamiltonian of polymatrix zero-sum games

arXiv: https://arxiv.org/abs/2505.12609 · HTML: https://mlsanity.com/paper/2505.12609/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 53 references that are in the map.
- First posted on arXiv 19 May 2025.

## Builds on (most cited first, 21 of 21)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Fast Convergence of Regularized Learning in Games](https://mlsanity.com/paper/1507.00407/) (2015)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization](https://mlsanity.com/paper/1807.04252/) (2018)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [Fast Extra Gradient Methods for Smooth Structured Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2106.02326/) (2021)
- [Negative Momentum for Improved Game Dynamics](https://mlsanity.com/paper/1807.04740/) (2018)
- [Differentiable Game Mechanics](https://mlsanity.com/paper/1905.04926/) (2019)
- [Continuous Time Analysis of Momentum Methods](https://mlsanity.com/paper/1906.04285/) (2019)
- [Multi-Agent Learning in Network Zero-Sum Games is a Hamiltonian System](https://mlsanity.com/paper/1903.01720/) (2019)
- [On the Suboptimality of Negative Momentum for Minimax Optimization](https://mlsanity.com/paper/2008.07459/) (2020)
- [On the Last-iterate Convergence in Time-varying Zero-sum Games: Extra Gradient Succeeds where Optimism Fails](https://mlsanity.com/paper/2310.02604/) (2023)
- [Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence](https://mlsanity.com/paper/2211.01681/) (2022)
- [Learning in quantum games](https://mlsanity.com/paper/2302.02333/) (2023)
- [Learning in Quantum Common-Interest Games and the Separability Problem](https://mlsanity.com/paper/2302.04789/) (2023)
- [Last-iterate Convergence Separation between Extra-gradient and Optimism in Constrained Periodic Games](https://mlsanity.com/paper/2406.10605/) (2024)
- [LEAD: Min-Max Optimization from a Physical Perspective](https://mlsanity.com/paper/2010.13846/) (2020)

## Built on by (most cited first, 1 of 1)

- [When and Why is Optimistic Multiplicative Weights Slow? The Geometry of Energy Dissipation](https://mlsanity.com/paper/2605.13242/) (2026)

## Abstract (arXiv)

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.