Probably Approximately Correct Nash Equilibrium Learning

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.…”

“…Each A(θ m ) ∈ R n×n , is considered to be a diagonal matrix, whose elements {a(θ m )}, m ∈ M are extracted according to a lognormal distribution, normalised with respect to the number of agents. The elements of b(θ m ) ∈ R n follow a uniform distribution, as in [21]. For each agent i ∈ N , the upper bound x i takes a random integer value in the set [15,25], the lower bound x i is set to zero and the final energy to be achieved by the end of the charging cycle is chosen as reported in [21].…”

“…The elements of b(θ m ) ∈ R n follow a uniform distribution, as in [21]. For each agent i ∈ N , the upper bound x i takes a random integer value in the set [15,25], the lower bound x i is set to zero and the final energy to be achieved by the end of the charging cycle is chosen as reported in [21]. The diagonal entries of A 0 = diag({a t } n t=1 ) ∈ R n×n and b 0 are derived by rescaling a winter weekday demand profile in the UK.…”

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).…”

“…Future research directions involve synthesizing algorithms to enumerate the number of support subsamples in a general convex setting, as well as investigating extensions of the developed approach to quasi-variational inequalities. This enables us to incorporate the uncertainty within the mapping defining the VI, thus extending the results of [19,20] to the entire set of solutions to VIs.…”