  This paper considers variational inequalities (VI) defined by the conditional
value-at-risk (CVaR) of uncertain functions and provides three stochastic
approximation schemes to solve them. All methods use an empirical estimate of
the CVaR at each iteration. The first algorithm constrains the iterates to the
feasible set using projection. To overcome the computational burden of
projections, the second one handles inequality and equality constraints
defining the feasible set differently. Particularly, projection onto to the
affine subspace defined by the equality constraints is achieved by matrix
multiplication and inequalities are handled by using penalty functions.
Finally, the third algorithm discards projections altogether by introducing
multiplier updates. We establish asymptotic convergence of all our schemes to
any arbitrary neighborhood of the solution of the VI. A simulation example
concerning a network routing game illustrates our theoretical findings.
