  This paper focuses on a class of variational inequalities (VIs), where the
map defining the VI is given by the component-wise conditional value-at-risk
(CVaR) of a random function. We focus on solving the VI using sample average
approximation, where solutions of the VI are estimated with solutions of a
sample average VI that uses empirical estimates of the CVaRs. We establish two
properties for this scheme. First, under continuity of the random map and the
uncertainty taking values in a bounded set, we prove asymptotic consistency,
establishing almost sure convergence of the solution of the sample average
problem to the true solution. Second, under the additional assumption of random
functions being Lipschitz, we prove exponential convergence where the
probability of the distance between an approximate solution and the true
solution being smaller than any constant approaches unity exponentially fast.
The exponential decay bound is refined for the case where random functions have
a specific separable form in the decision variable and uncertainty. We adapt
these results to the case of uncertain routing games and derive explicit sample
guarantees for obtaining a CVaR-based Wardrop equilibria using the sample
average procedure. We illustrate our theoretical findings by approximating the
CVaR-based Wardrop equilibria for a modified Sioux Falls network.
