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.

本文致力于概率度量集上的变分问题，这些度量涉及不等维空间之间的最优输运。特别地，我们研究了一个函数的最小化，该函数的总和反映一个固定和一个自由边际之间的（不等维）最优运输成本以及一个自由边际（各种形式）之间的最优运输成本。激励应用包括古诺-纳什均衡，其中策略空间的维数小于代理类型的空间。对于上述术语的各种不同形式，我们显示出嵌套条件对于任何最小化条件都成立，已知该嵌套条件可产生最佳运输问题的易处理性。根据功能的确切形式，