A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights

TL;DR

This paper introduces a geometric decomposition of finite games using a Riemannian framework, revealing that exponential weights dynamics exhibit recurrence in harmonic and incompressible games, thus clarifying long-term behavior.

Contribution

It develops a novel geometric decomposition based on the Shahshahani metric, linking incompressible and harmonic games to recurrence properties of exponential weights dynamics.

Findings

Exponential weights dynamics are volume-preserving and recurrent in incompressible games.

Incompressible games are characterized as harmonic, connecting game decomposition with dynamic behavior.

Recurrence is established for harmonic games under exponential weights dynamics.

Abstract

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 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.