Riemannian game dynamics

TL;DR

This paper introduces a broad class of Riemannian game dynamics that generalize known evolutionary dynamics, ensuring desirable properties like convergence and positive correlation, and explores their connection to reinforcement learning.

Contribution

It defines Riemannian game dynamics, shows they encompass replicator and projection dynamics, and links them to reinforcement learning under certain conditions.

Findings

All Riemannian game dynamics satisfy positive correlation.

They exhibit global convergence in potential games.

Connections to reinforcement learning are established.

Abstract

We study a class of evolutionary game dynamics defined by balancing a gain determined by the game's payoffs against a cost of motion that captures the difficulty with which the population moves between states. Costs of motion are represented by a Riemannian metric, i.e., a state-dependent inner product on the set of population states. The replicator dynamics and the (Euclidean) projection dynamics are the archetypal examples of the class we study. Like these representative dynamics, all Riemannian game dynamics satisfy certain basic desiderata, including positive correlation and global convergence in potential games. Moreover, when the underlying Riemannian metric satisfies a Hessian integrability condition, the resulting dynamics preserve many further properties of the replicator and projection dynamics. We examine the close connections between Hessian game dynamics and reinforcement learning in normal form games, extending and elucidating a well-known link between the replicator dynamics and exponential reinforcement learning.