Constrained optimal transport with applications to large matching markets
We establish a variant of Monge--Kantorovich duality for matching problems with a continuum of agents, a finite set of alternatives, and general linear constraints. Under primal feasibility alone, both the primal and dual problems attain their optimal values, and the values coincide. We develop two applications. First, in a large market with indivisible goods, equilibrium prices are characterized as minimizers of a finite-dimensional convex potential function, and a t{â}tonnement-style subgradient process converges to equilibrium prices even in the presence of complementarities. Second, in a transferable-utility matching market with couples, where the core may be empty in finite markets, dual optimizers support a nonempty core in the continuum economy. These results extend optimal-transport methods to a broader class of constrained large matching markets.