Computation of Cournot–Nash Equilibria by Entropic Regularization

Abstract

We consider a class of games with continuum of players where equilibria can be obtained by the minimization of a certain functional related to optimal transport as emphasized in [7]. We then use the powerful entropic regularization technique to approximate the problem and solve it numerically in various cases. We also consider the extension to some models with several populations of players.

On the Wasserstein distance between mutually singular measures

“…This scheme was extended in [5] to unbalanced transport problem and used in [4] to compute Cournot-Nash equilibria. It is well known that the solution of (5.2) is of the form γ i,j = a i (η ε ) i,j b j , where a, b ∈ R N and η ε ∈ R N ×N .…”

Variational problems involving unequal dimensional optimal transport

“…Minimizers of (1) when m = n have been studied extensively; for the subproblems where one marginal is fixed, under mild conditions on the functionals F and G, existence and uniqueness of minimizers has been established, and, depending on the precise forms of F and G, various regularity results and bounds on solutions exist. Solutions can be characterized by partial differential equations, and various numerical schemes for solving them have been proposed (see [24, chapter 6 and section 7.4] and the references therein and [2,4,5,11,22,23]).…”

“…Nestedness is equivalent to this point − k(y) cos(y) of intersection being monotone increasing. Now, the differential equation ( 10) 5 for k(y) reads…”