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Abstract A fundamental result in the theory of stochastic dominance tells that first order dominance between two finite multivariate distributions is equivalent to the property that the one can be obtained from the other by shifting probability mass from one outcome to another that is worse a finite number of times. This paper provides a new and elementary proof of that result by showing that starting with an arbitrary system of mass transfers, whenever the resulting distribution is first order dominated one can gradually rearrange transfers, according to a certain decentralized procedure, and obtain a system of transfers all shifting mass to outcomes that are worse.