  This paper is devoted to variational problems on the set of probability
measures which involve optimal transport between unequal dimensional spaces. In
particular, we study the minimization of a functional consisting of the sum of
a term reflecting the cost of (unequal dimensional) optimal transport between
one fixed and one free marginal, and another functional of the free marginal
(of various forms). Motivating applications include Cournot-Nash equilibria
where the strategy space is lower dimensional than the space of agent types.
For a variety of different forms of the term described above, we show that a
nestedness condition, which is known to yield much improved tractability of the
optimal transport problem, holds for any minimizer. Depending on the exact form
of the functional, we exploit this to find local differential equations
characterizing solutions, prove convergence of an iterative scheme to compute
the solution, and prove regularity results.
