  Variational inequalities are modelling tools used to capture a variety of
decision-making problems arising in mathematical optimization, operations
research, game theory. The scenario approach is a set of techniques developed
to tackle stochastic optimization problems, take decisions based on historical
data, and quantify their risk. The overarching goal of this manuscript is to
bridge these two areas of research, and thus broaden the class of problems
amenable to be studied under the lens of the scenario approach. First and
foremost, we provide out-of-samples feasibility guarantees for the solution of
variational and quasi variational inequality problems. Second, we apply these
results to two classes of uncertain games. In the first class, the uncertainty
enters in the constraint sets, while in the second class the uncertainty enters
in the cost functions. Finally, we exemplify the quality and relevance of our
bounds through numerical simulations on a demand-response model.
