On the stability of the logit dynamics in population games

TL;DR

This work analyzes the asymptotic stability of the logit evolutionary dynamics in population games, addressing how stability changes with the noise level $\eta$ and the implications of multiple strict Nash equilibria. It proves that strict Nash equilibria are locally asymptotically stable for low noise, while there exists a globally exponentially stable logit equilibrium for sufficiently high noise; in addition, it identifies a broad class of monotone separable games that admit global asymptotic stability for every $\eta>0$, with a continuous curve of logit equilibria connecting Nash equilibria in the vanishing-noise limit to a uniform distribution in the high-noise limit. The monotone separable condition is shown to include heterogeneous routing scenarios, particularly on series compositions of parallel networks, yielding global stability even when no potential function exists. Collectively, the results reveal noise-induced bifurcations in population games with multiple strict equilibria and broaden the spectrum of games where logit dynamics are globally stable, providing theoretical justifications for observed routing and coordination phenomena under stochastic decision-making.

Abstract

We study the asymptotic stability of the logit evolutionary dynamics in population games, possibly with multiple heterogenous populations. For general population games, we prove that, on the one hand, strict Nash equilibria are asymptotically stable under the logit dynamics for low enough noise levels, on the other hand, a globally exponentially stable logit equilibrium exists for sufficiently large noise levels. This suggests the emergence of bifurcations in population games admitting multiple strict Nash equilibria, as observed in numerous examples. We then provide sufficient conditions on the population game structure for the existence of globally asymptotically stable logit equilibria for every noise level. The considered class of monotone separable games finds applications, e.g., in routing games on series compositions of networks with parallel routes when there are multiple populations of users that differ in the reward functions.

Figures (5)

• Figure 1: Bifurcation diagrams of the logit dynamics in the population games of Example \ref{['ex:coordination']} in the special cases $\gamma=1$ (top) and $\gamma=2$ (bottom).
• Figure 2: Solutions of the logit dynamics in the games of Example \ref{['ex:RSP']} for different values of $\gamma$ and noise levels $\eta$. The dots indicate the initial configurations.
• Figure 3: Top: Solutions of the logit dynamics with noise level $\eta = 0.01$ in the game of Example \ref{['ex:2pop2link']} for different initial configurations, all converging to a globally asymptotically stable logit equilibrium. Bottom: The curve of logit equilibria as the noise level $\eta$ varies in the interval $[0.01,10]$. In both the plots the variables $z_i(x) = \sum_{p \in \mathcal{P}}x_{ip}$ are reported.
• Figure 4: The multigraph and the link cost functions of the routing game in Example \ref{['ex:toso']}. The bottom panels report two solutions of the logit dynamic with noise level $\eta = 0.05$ corresponding to different initial configurations. The curves represent the entries of the link flow $\textcolor{black}{y(x(t))}$.
• Figure 5: Top: The multigraph of the routing game of Example \ref{['ex:3']}. Bottom: Two components of the curve of globally asymptotically stable logit equilibria as a function of the noise level $\eta$.

Theorems & Definitions (23)

• Proposition 1
• Example 1: Binary coordination
• Example 2: Generalized Rock-Scissor-Paper
• Example 3
• Lemma 1
• Lemma 2
• Example 1: continued
• Example 2: continued
• Example 3: continued
• Lemma 3