Probabilistic sensitivity of Nash equilibria in multi-agent games: a wait-and-judge approach

Abstract

Motivated by electric vehicle charging control problems, we consider multi-agent noncooperative games where, following a data driven paradigm, unmodeled externalities acting on the players' objective functions are represented by means of scenarios. Building upon recent developments in scenario-based optimization, based on the evaluation of the computed solution, we accompany the Nash equilibria of the uncertain game with an a posteriori probabilistic robustness certificate, providing confidence on the probability that the computed solution remains unaffected when a new uncertainty realisation is encountered. The latter constitutes, to the best of our knowledge, the first application of the so-called scenario approach to multi-agent Nash equilibrium problems. The efficacy of our approach is demonstrated in simulation for the charging coordination of an electric vehicle fleet.

Agent independent probabilistic robustness certificates for robust optimization programs with uncertain quadratic cost

“…This is of crucial importance in large-scale applications, where a large number of agents is present. A similar claim that agent independent bounds could be derived for a class of games with structure similar to that of our optimization problem, was conjectured in [20], [21]. Therein, by adopting a variational inequalities approach, the authors provide probabilistic guarantees for the Nash equilibria of a game affected by uncertainty.…”

Probabilistic feasibility guarantees for solution sets to uncertain variational inequalities

“…• We consider a broad family of uncertain VIs in (3) rather than just VI problems arising in computing variational generalized Nash equilibria (v-GNE), a popular subclass of generalized Nash equilibria (GNE) in GNEPs [19,20,36,16], thus complementing [36], where GNEPs were not considered ( §2); • By focusing on the entire set of solutions, we are able to bypass the uniqueness and nondegeneracy assumptions postulated in [34] ( §3); • Along the direction of [16], we provide a-posteriori robustness certificates for the entire set of solutions rather than for a single one [34] or for the feasible set only [36]. Compared to latter ones, we show that the resulting bounds are less conservative ( §3); • The proposed robustness certificates strongly depend on the number of support subsamples characterizing the set of solutions to the uncertain VI ( §2, §3).…”

“…Remarkably, the family of VIs we consider is quite large, encompassing practical applications in several domains. Indeed, Nash equilibrium problems (NEPs) and generalized Nash equilibrium problems (GNEPs) can be modelled as VIs [18,17], whose popular robust version falls in the class investigated in this paper [1,19,36,16]. As a static assignment problem, an optimal network flow in traffic networks can also be computed via solution to an associated VI [37,Th.…”

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].…”

“…• Along the direction of [17], we focus on the entire set of equilibria, implicitly relaxing the assumption on the uniqueness of the equilibrium postulated in [16]; • We extend the results in [15], [17] providing a-posteriori robustness certificates for the set of GNE rather than for the feasible set or Nash equilibria without coupling constraints. We also show that the resulting bounds are less conservative ( §III); • The obtained probabilistic guarantees rely on the notion of support subsample, a key concept of the scenario approach theory.…”