# On the Decomposition of Differential Game

arXiv: https://arxiv.org/abs/2411.03802 · HTML: https://mlsanity.com/paper/2411.03802/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 33 references that are in the map.
- First posted on arXiv 6 Nov 2024.

## Builds on (most cited first, 14 of 14)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Multi-Agent Actor-Critic for Mixed Cooperative-Competitive Environments](https://mlsanity.com/paper/1706.02275/) (2017)
- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [Differentiable Game Mechanics](https://mlsanity.com/paper/1905.04926/) (2019)
- [Optimality and Stability in Federated Learning: A Game-theoretic Approach](https://mlsanity.com/paper/2106.09580/) (2021)
- [On Last-Iterate Convergence Beyond Zero-Sum Games](https://mlsanity.com/paper/2203.12056/) (2022)
- [Multiplicative Weights Update with Constant Step-Size in Congestion Games: Convergence, Limit Cycles and Chaos](https://mlsanity.com/paper/1703.01138/) (2017)
- [Learning in Congestion Games with Bandit Feedback](https://mlsanity.com/paper/2206.01880/) (2022)
- [Generalized Mirror Descents in Congestion Games](https://mlsanity.com/paper/1605.07774/) (2016)
- [Decomposition of games: some strategic considerations](https://mlsanity.com/paper/1901.06048/) (2019)
- [Semi Bandit Dynamics in Congestion Games: Convergence to Nash Equilibrium and No-Regret Guarantees](https://mlsanity.com/paper/2306.15543/) (2023)
- [A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights](https://mlsanity.com/paper/2405.07224/) (2024)
- [A Note On Orthogonal Decomposition of Finite Games](https://mlsanity.com/paper/1905.08112/) (2019)

## Abstract (arXiv)

To understand the complexity of the dynamic of learning in differential games, we decompose the game into components where the dynamic is well understood. One of the possible tools is Helmholtz's theorem, which can decompose a vector field into a potential and a harmonic component. This has been shown to be effective in finite and normal-form games. However, applying Helmholtz's theorem by connecting it with the Hodge theorem on ℝn (which is the strategy space of differential game) is non-trivial due to the non-compactness of ℝn. Bridging the dynamic-strategic disconnect through Hodge/Helmoltz's theorem in differential games is then left as an open problem citeletcher2019differentiable. In this work, we provide two decompositions of differential games to answer this question: the first as an exact scalar potential part, a near vector potential part, and a non-strategic part; the second as a near scalar potential part, an exact vector potential part, and a non-strategic part. We show that scalar potential games coincide with potential games proposed by citemonderer1996potential, where the gradient descent dynamic can successfully find the Nash equilibrium. For the vector potential game, we show that the individual gradient field is divergence-free, in which case the gradient descent dynamic may either be divergent or recurrent.