We discuss equations associated to Cournot–Nash Equilibria as put forward recently
by Blanchet and Carlier. These equations are related to an optimal transport
problem in which the source measure is known, but the target measure is part of the
problem. The resulting equation is of Monge–Ampère type with possible
nonlocal terms. If the cost function is of a particular form, the equation is
vulnerable to standard optimal transportation PDE techniques, with some
modifications to deal with the new terms. We give some sufficient conditions for
the problem on the sphere from which we can conclude that solutions are
smooth.
