Routing Game on Parallel Networks: the Convergence of Atomic to Nonatomic

TL;DR

This paper demonstrates that atomic routing game equilibria can closely approximate nonatomic Wardrop equilibria in parallel networks, providing explicit bounds and a method for precise computation.

Contribution

It establishes a formal link between atomic and nonatomic routing games, offering explicit bounds and a practical approximation method for Wardrop equilibrium.

Findings

Atomic Nash Equilibrium approximates Wardrop Equilibrium

Explicit bounds on equilibrium distance are provided

Method enables arbitrary precision computation of Wardrop equilibrium

Abstract

We consider an instance of a nonatomic routing game. We assume that the network is parallel, that is, constituted of only two nodes, an origin and a destination. We consider infinitesimal players that have a symmetric network cost, but are heterogeneous through their set of feasible strategies and their individual utilities. We show that if an atomic routing game instance is correctly defined to approximate the nonatomic instance, then an atomic Nash Equilibrium will approximate the nonatomic Wardrop Equilibrium. We give explicit bounds on the distance between the equilibria according to the parameters of the atomic instance. This approximation gives a method to compute the Wardrop equilibrium at an arbitrary precision.