On stability of users equilibria in heterogeneous routing games

TL;DR

This paper analyzes the stability of user equilibria in heterogeneous routing games, showing that certain dynamics converge to stable fixed points that approximate Wardrop equilibria as noise diminishes.

Contribution

It proves global stability of fixed points in specific network topologies and links the dynamics to Wardrop equilibria in heterogeneous routing games.

Findings

Dynamics admit a globally asymptotically stable fixed point.

Fixed point approaches Wardrop equilibrium as noise vanishes.

Monotonicity and diagonal dominance are key to stability.

Abstract

The asymptotic behaviour of deterministic logit dynamics in heterogeneous routing games is analyzed. It is proved that in directed multigraphs with parallel routes, and in series composition of such multigraphs, the dynamics admits a globally asymptotically stable fixed point. Moreover, the unique fixed point of the dynamics approaches the set of Wardrop equilibria, as the noise vanishes. The result relies on the fact that the dynamics of aggregate flows is monotone, and its Jacobian is strictly diagonally dominant by columns.