Abstract
This paper generalizes the Probabilistic Serial (PS) mechanism of Bogomolnaia and Moulin (2001) to matching markets with arbitrary constraints. The constraints are modeled as a set of permissible ex post allocations. A result of independent interest gives a simple geometric characterization of all lotteries over the set of permissible allocations: they are the ones that satisfy a collection of simple and tractable linear inequalities determined by the constraints. The inequalities correspond to the hyperplanes defining a convex polytope that is intuitively constructed from the given set. When a general version of the PS algorithm is executed under these inequalities, the outcome is an efficient and fair lottery over the set of permissible allocations. The method is general, can be applied to both one-sided and two-sided matching markets, and allows for multi-unit demand.
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