Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets
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In 1979, Hylland and Zeckhauser <cit.> gave a simple and general scheme for implementing a one-sided matching market using the power of a pricing mechanism. Their method has nice properties – it is incentive compatible in the large and produces an allocation that is Pareto optimal – and hence it provides an attractive, off-the-shelf method for running an application involving such a market. With matching markets becoming ever more prevalant and impactful, it is imperative to finally settle the computational complexity of this scheme. We present the following partial resolution: 1. A combinatorial, strongly polynomial time algorithm for the special case of 0/1 utilities. 2. An example that has only irrational equilibria, hence proving that this problem is not in PPAD. Furthermore, its equilibria are disconnected, hence showing that the problem does not admit a convex programming formulation. 3. A proof of membership of the problem in the class FIXP. We leave open the (difficult) question of determining if the problem is FIXP-hard. Settling the status of the special case when utilities are in the set {0, 1/2, 1 } appears to be even more difficult.
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