On Gradient like Properties of Population games, Learning models and Self Reinforced Processes

TL;DR

This paper studies differential equations in population games and learning models, establishing conditions under which these dynamics are gradient-like, leading to convergence results and insights into equilibrium stability.

Contribution

It generalizes the concept of gradient-like properties for population game dynamics and provides conditions for convergence and stability in these systems.

Findings

Omega limit sets consist of equilibria

Trajectories converge to equilibria in the real analytic case

Reversible dynamics are close to gradient vector fields

Abstract

We consider ordinary differential equations on the unit simplex of $\RR^n$ that naturally occur in population games, models of learning and self reinforced random processes. Generalizing and relying on an idea introduced in \cite{DF11}, we provide conditions ensuring that these dynamics are gradient like and satisfy a suitable "angle condition". This is used to prove that omega limit sets and chain transitive sets (under certain smoothness assumptions) consist of equilibria; and that, in the real analytic case, every trajectory converges toward an equilibrium. In the reversible case, the dynamics are shown to be $C^1$ close to a gradient vector field. Properties of equilibria -with a special emphasis on potential games - and structural stability questions are also considered.