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After defining a pure-action profile in a nonatomic aggregative game, whereplayers have specific compact convex pure-action sets and nonsmooth convex costfunctions, as a square-integrable function, we characterize a Wardropequilibrium as a solution to an infinite-dimensional generalized variationalinequality. We show the existence of Wardrop equilibrium and variationalWardrop equilibrium, a concept of equilibrium adapted to the presence ofcoupling constraints, in monotone nonatomic aggregative games. The uniquenessof (variational) Wardrop equilibrium is proved for strictly or aggregativelystrictly monotone nonatomic aggregative games. We then show that, for asequence of finite-player aggregative games with aggregative constraints, ifthe players' pure-action sets converge to those of a strongly (resp.aggregatively strongly) monotone nonatomic aggregative game, and theaggregative constraints in the finite-player games converge to the aggregativeconstraint of the nonatomic game, then a sequence of so-called variational Nashequilibria in these finite-player games converge to the variational Wardropequilibrium in pure-action profile (resp. aggregate-action profile). Inparticular, it allows the construction of an auxiliary sequence of games withfinite-dimensional equilibria to approximate the infinite-dimensionalequilibrium in such a nonatomic game. Finally, we show how to constructauxiliary finite-player games for two general classes of nonatomic games.

Nonsmooth Aggregative Games with Coupling Constraints and Infinitely Many Classes of Players