  Nash`s classical bargaining solution suggests that n players in a
non-cooperative bargaining situation should find a solution that maximizes the
product of each player's utility functions. We consider a special case: Suppose
that the players are chosen from a continuum distribution $\mu$ and suppose
they are to divide up a resource $\nu$ that is also on a continuum. The utility
to each player is determined by the exponential of a distance type function.
The maximization problem becomes an optimal transport type problem, where the
target density is the minimizer to the functional \[
F(\beta)=H_{\nu}(\beta)+W^{2}(\mu,\beta) \] where $H_{\nu}(\beta)$ is the
entropy and $W^{2}$ is the 2-Wasserstein distance. This minimization problem is
also solved in the Jordan-Kinderlehrer-Otto scheme. Thanks to optimal transport
theory, the solution may be described by a potential that solves a fourth order
nonlinear elliptic PDE, similar to Abreu's equation. Using the PDE, we prove
solutions are smooth when the measures have smooth positive densities.
