The Scenario Approach Meets Uncertain Game Theory and Variational Inequalities

Abstract

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.

Pursuing robust decisions in uncertain traffic equilibrium problems

“…Specifically, it has been identified as one of the weakest conditions allowing proof of consistency of a set S ω K by relying on known results available in the literature. Following the discussion in [31, §8], it appears that the non-degeneracy assumption postulated in similar works (e.g., [19], [31]) is definitely not easier to verify than, for all K ∈ N 0 and p 2…”

“…As a static assignment problem affected by an uncertain traffic demand, it follows from [4,Th. 3.14] that, for the considered model, a network flow vector p ∈ P ω , ω ∈ Ω, satisfies the set of conditions in (4) if and only if it solves the uncertain VI problem associated with the traffic problem (G, L, C, P ω , Ω, P), denoted as VI(P ω , C), ω ∈ Ω, which coincides with the structure of uncertain VIs analysed in [19], [20]. Specifically, a network flow p ∈ P ω , for all ω ∈ Ω, amounts to an agents' traffic equilibrium if it satisfies…”

“…Specifically, we leverage the equivalence between routing games and VIs [4,Th. 3.14] to rely on recent results bridging the realm of the VIs with the scenario approach paradigm [19], [20], with the twofold benefit of enabling for a tractable reformulation of the uncertain routing game, and establishing quantifiable robustness properties for any of the agents' traffic equilibria in a distribution-free fashion. That is, the proposed robustness certificates characterizing randomized optimal routing patterns hold regardless of the probability distribution of the parameter that encodes the uncertainty on the shape of the traffic demand polyhedron.…”

On the robustness of equilibria in generalized aggregative games

“…a recent development of data-driven (or distribution-free) robust approaches, see, e.g., [15], [16], [17].…”

Tight generalization guarantees for the sampling and discarding approach to scenario optimization

“…It is based on representing uncertainty by means of scenarios, accompanying the optimal solution of the resulting scenario program with a priori certificates on the probability of constraint violation. The scenario approach theory has been introduced in [1] and has undergone several developments, including tightness of the bound on the probability of constraint violation [2], the sampling and discarding [4] or constraint removal scheme [5] to trade feasibility to performance, extension to non-convex programs [6], and, more recently, a posteriori assessing the scenario solution [7]- [10] as well as game theoretic extensions [11]- [13].…”