A geometric decomposition of finite games: Convergence vs. recurrence under no-regret learning

In view of the complexity of the dynamics of no-regret learning in games, we seek to decompose a finite game into simpler components where the day-to-day behavior of the dynamics is well understood. A natural starting point for this is Helmholtz's theorem, which resolves a vector field into a potential and an incompressible component. However, the geometry of no-regret dynamics - and, in particular, the dynamics of exponential / multiplicative weights (EW) schemes - is not compatible with the Euclidean underpinnings of Helmholtz's theorem, leading us to consider a Riemannian framework based on the Shahshahani metric. Using this geometric construction, we introduce the class of incompressible games, and we prove 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 Poincar\'e 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 Poincar\'e recurrence in harmonic games.