Abstract
We propose a general framework of mass transport between vectorvalued measures, which will be called simultaneous transport. The new framework is motivated by the need to transport resources of different types simultaneously, i.e., in single trips, from specified origins to destinations. In terms of matching, one needs to couple two groups, e.g., buyers and sellers, by equating supplies and demands of different goods at the same time. The mathematical structure of simultaneous transport is very different from the classic setting of optimal transport, leading to many new challenges. The Monge and Kantorovich formulations are contrasted and connected. Existence and uniqueness of the simultaneous transport and duality formulas are established, and a notion of Wasserstein distance in this setting is introduced. In particular, the duality theorem gives rise to a labour market equilibrium model where each worker has several types of skills and each firm seeks to employ these skills at different levels.
