An Analysis of Logit Learning with the r-Lambert Function

TL;DR

This paper analyzes the fixed points and stability of logit learning dynamics in two-strategy population games using the r-Lambert function, revealing bifurcation phenomena and convergence to Nash equilibria.

Contribution

It provides an explicit analysis of logit fixed points and their bifurcations across all two-strategy games, extending understanding beyond previous numerical studies.

Findings

Single fixed point in Prisoner's Dilemma and anti-coordination games.

Pitchfork bifurcation in coordination games as rationality increases.

Logit fixed points converge to Nash equilibria at high rationality.

Abstract

The well-known replicator equation in evolutionary game theory describes how population-level behaviors change over time when individuals make decisions using simple imitation learning rules. In this paper, we study evolutionary dynamics based on a fundamentally different class of learning rules known as logit learning. Numerous previous studies on logit dynamics provide numerical evidence of bifurcations of multiple fixed points for several types of games. Our results here provide a more explicit analysis of the logit fixed points and their stability properties for the entire class of two-strategy population games -- by way of the $r$-Lambert function. We find that for Prisoner's Dilemma and anti-coordination games, there is only a single fixed point for all rationality levels. However, coordination games exhibit a pitchfork bifurcation: there is a single fixed point in a low-rationality regime, and three fixed points in a high-rationality regime. We provide an implicit characterization for the level of rationality where this bifurcation occurs. In all cases, the set of logit fixed points converges to the full set of Nash equilibria in the high rationality limit.