Le Chatelier principle in replicator dynamics

TL;DR

This paper explores how the Le Chatelier principle, which describes stability and resistance to perturbations in physical systems, applies to evolutionary game theory and replicator dynamics, revealing new stability criteria and insights.

Contribution

It reformulates the Le Chatelier principle as a majorization relation within replicator dynamics, establishing a generalized stability concept for Nash equilibria.

Findings

Le Chatelier principle can be expressed as a majorization relation in replicator dynamics.

Mutualistic interactions can lead to more stable Nash equilibria.

Some globally stable equilibria violate the Le Chatelier principle, amplifying perturbations.

Abstract

The Le Chatelier principle states that physical equilibria are not only stable, but they also resist external perturbations via short-time negative-feedback mechanisms: a perturbation induces processes tending to diminish its results. The principle has deep roots, e.g., in thermodynamics it is closely related to the second law and the positivity of the entropy production. Here we study the applicability of the Le Chatelier principle to evolutionary game theory, i.e., to perturbations of a Nash equilibrium within the replicator dynamics. We show that the principle can be reformulated as a majorization relation. This defines a stability notion that generalizes the concept of evolutionary stability. We determine criteria for a Nash equilibrium to satisfy the Le Chatelier principle and relate them to mutualistic interactions (game-theoretical anticoordination) showing in which sense mutualistic replicators can be more stable than (say) competing ones. There are globally stable Nash equilibria, where the Le Chatelier principle is violated even locally: in contrast to the thermodynamic equilibrium a Nash equilibrium can amplify small perturbations, though both this type of equilibria satisfy the detailed balance condition.