# Nonatomic Aggregative Games with Infinitely Many Types

## TL;DR

This paper introduces a framework for analyzing nonatomic aggregative games with infinitely many player types, characterizing equilibria via infinite-dimensional variational inequalities and demonstrating convergence of finite approximations.

## Contribution

It defines variational Wardrop equilibrium for infinite-type games, proves existence and convergence of finite-type approximations, and applies the model to smart grid electricity consumption.

## Key findings

- Existence of variational Wardrop equilibrium under monotonicity.
- Finite-type symmetric equilibria converge to the infinite-type equilibrium.
- Equilibrium computation reduces to finite-dimensional variational inequalities.

## Abstract

We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of players types. These equilibria are characterized through an infinite-dimensional variational inequality. We show, under monotonicity conditions, a convergence theorem enables to approximate such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of players 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 players types, with actions subject to coupling constraints.