Nonatomic aggregative games with infinitely many types

We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of player types. These equilibria are characterized through an infinite-dimensional variational inequality. We show, under monotonicity conditions, a convergence theorem which enables to compute such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of player types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different player types, with actions subject to coupling constraints.