This talk 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. A motivating application is the study of Cournot-Nash equilibria, when the continuous space of agents has more heterogeneity (that is, is higher dimensional) than the space of strategies. 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. This represents joint work with Luca Nenna, and builds on earlier joint work with Pierre-Andre Chiappori and Robert McCann.