*Published Paper*

**Inserted:** 5 nov 2010

**Last Updated:** 28 oct 2011

**Journal:** Записки научных семинаров ПОМИ

**Volume:** 390

**Pages:** 69-91

**Year:** 2011

**Notes:**

This paper is a survey (no new contribution) of recent results that the authors obtained with many collaborators, inserted in a more general framework involving game theory, optimal transport and PDEs. It has been written for possible publication in the proceedings of a conference organized in St Petersburg in May 2010

**Abstract:**

In the classical Monge-Kantorovich problem, the transportation cost only depends on the amount of mass sent from sources to destinations and not on the paths followed by each particle forming this mass. Thus, it does not allow for congestion effects, which depend instead on the proportion of mass passing through a same point or on a same path. Usually the travelling cost (or time) of a path depends on ``how crowded'' this path is. Starting from a simple network model, we shall define equilibria in the presence of congestion. We will then extend this theory to the continuous setting mainly following some recent papers. After an introduction with almost no mathematical details, we will give a survey of the main features of this theory.

**Keywords:**
Degenerate elliptic equations, Wardrop equilibria, Fast Marching Method, Convex optimization

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