Sample average approximation of CVaR-based Wardrop equilibrium in routing under uncertain costs

TL;DR

This paper introduces a method to compute CVaR-based Wardrop equilibria in routing games with uncertain costs using sample average approximation, proving its consistency and exponential convergence.

Contribution

It develops a novel sample average approximation scheme for CVaR-based Wardrop equilibrium under unknown distributions, with proven convergence properties.

Findings

Almost sure convergence of approximate equilibria to CWE

Exponential convergence rate under Lipschitz costs

Simulation validates theoretical results

Abstract

This paper focuses on the class of routing games that have uncertain costs. Assuming that agents are risk-averse and select paths with minimum conditional value-at-risk (CVaR) associated to them, we define the notion of CVaR-based Wardrop equilibrium (CWE). We focus on computing this equilibrium under the condition that the distribution of the uncertainty is unknown and a set of independent and identically distributed samples is available. To this end, we define the sample average approximation scheme where CWE is estimated with solutions of a variational inequality problem involving sample average approximations of the CVaR. We establish two properties for this scheme. First, under continuity of costs and boundedness of uncertainty, we prove asymptotic consistency, establishing almost sure convergence of approximate equilibria to CWE as the sample size grows. Second, under the additional assumption of Lipschitz cost, we prove exponential convergence where the probability of the distance between an approximate solution and the CWE being smaller than any constant approaches unity exponentially fast. Simulation example validates our theoretical findings.