An Analysis of Logit Learning with the r-Lambert Function

TL;DR

The paper analyzes logit learning in two-strategy population games by linking logit fixed points to the $r$-Lambert function, yielding explicit expressions for fixed points across all $\beta \ge 0$. It shows that coordination games exhibit a pitchfork bifurcation (1 fixed point at low rationality to 3 at high rationality), while Prisoner’s Dilemma and anti-coordination have a single fixed point for all $\beta$, and that as $\beta\to\infty$ these fixed points converge to Nash equilibria. Stability analyses provide a universal condition $\dfrac{1}{k}W_r^2(kr) - W_r(kr) - 1 \le 0$ governing finite-$\beta$ stability, with high-$\beta$ dynamics matching best-response behavior. The results offer precise, actionable insights into controlling logit dynamics via the rationality parameter and have implications for engineered systems and socio-technical applications.

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.

Figures (1)

• Figure 1: Plots of the fixed points $x^*$ of \ref{['eq:logit_dyn']} as a function of $\beta$ for points in the parameter space $(\delta_{SP}, \delta_{RT}) = (\pm 1, \pm 2)$, $(\pm 2, \pm 1)$, and $(\pm 2, \pm 2)$. Dashed lines denote mixed Nash equilibria. This diagram plots stable values of $x^*$ in green and unstable ones in red, as calculated numerically using \ref{['eq:genFPstab']}. Gaps in the plots indicate the limits of the numerical methods used to compute the solutions to \ref{['eq:logit_dyn_sol']}.

Theorems & Definitions (19)

• Definition 1
• Theorem 4.1: Fixed Point Values as $\beta \to \infty$
• proof
• Theorem 4.2: The Number of Fixed Points
• proof
• Theorem 4.3: Fixed Point Stability as $\beta \to \infty$
• proof
• Theorem 4.4: Fixed Point Stability for Finite $\beta$
• proof
• Lemma 1.1