D. Legacci, P. Mertikopoulos, and B. S. R. Pradelski. In ICML '24: Proceedings of the 41st International Conference on Machine Learning, 2024.

In view of the complexity of the dynamics of learning in games, we seek to decompose a game into simpler components where the dynamics' long-run behavior is well understood. A natural starting point for this is Helmholtz’s theorem, which decomposes a vector field into a potential and an incompressible component. However, the geometry of game dynamics - and, in particular, the dynamics of exponential / multiplicative weights (EW) schemes - is not compatible with the Euclidean underpinnings of Helmholtz’s theorem. This leads us to consider a specific Riemannian framework based on the so-called *Shahshahani metric*, and introduce the class of *incompressible games*, for which we establish the following results: First, in addition to being volume-preserving, the continuous-time EW dynamics in incompressible games admit a constant of motion and are *Poincaré recurrent* - i.e., almost every trajectory of play comes arbitrarily close to its starting point infinitely often. Second, we establish a deep connection with a well-known decomposition of games into a potential and harmonic component (where the players' objectives are aligned and anti-aligned respectively): a game is incompressible if and only if it is harmonic, implying in turn that the EW dynamics lead to Poincaré recurrence in harmonic games.

arXiv link: https://arxiv.org/abs/2405.07224

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