IMPLEMENTING RANDOM ASSIGNMENTS : A GENERALIZATION OF THE BIRKHOFF-VON NEUMANN THEOREM

The literature on random mechanisms often describes outcomes incompletely as “random assignments” — expressing the expected number of objects of each type assigned to different agents — and a set of feasibility constraints that a pure assignment must satisfy. We provide a necessary and sufficient condition (the “bihierarchy” condition) for the set of constraints to have the property that if the random assignment satisfies them, then it is implementable by a lottery over feasible pure assignments. Our theorem maximally generalizes the celebrated Birkhoff-von Neumann theorem. We also provide an algorithm to implement any such random assignment. Several applications are described, including (i) single-unit random assignment, such as school choice; (ii) multi-unit random assignment, such as course allocation and fair division; and (iii) twosided matching problems, such as the scheduling of inter-league sports matchups. The same method also finds applications outside economics, generalizing previous results on the minimize makespan problem in the computer science literature.

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