Collective Behavior: Macroscopic versus Kinetic Descriptions

Cournot-Nash equilibria

Adrien Blanchet

Université Toulouse 1 Capitole

We consider a non-atomic anonymous game with a continuum of players, or mean-field game. The agents are heterogeneous and have a pay-off which depends on congestion and interaction effects. We prove that a Cournot-Nash equilibrium is the limit when the number of agents goes to infinity of a Nash equilibrium. We also give some properties of existence, uniqueness of the equilibria in the potential game case using optimal transport techniques. We characterise the equilibrium and give different numerical methods to numerically compute them in the potential and non-potential case. These results are obtained in collaboration with P. Mossay, F. Santambrogio and G. Carlier.