A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights
This provides a theoretical foundation for understanding convergence vs. recurrence in game dynamics, which is incremental but clarifies behavior in specific game classes.
The paper tackles the complexity of learning dynamics in games by decomposing them into simpler components, establishing that exponential weights dynamics in incompressible (harmonic) games are Poincaré recurrent, meaning almost every trajectory returns arbitrarily close to its starting point infinitely often.
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.