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Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets

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  • Vijay V. Vazirani
  • Mihalis Yannakakis

Abstract

In 1979, Hylland and Zeckhauser \cite{hylland} 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 $\{0, {\frac 1 2}, 1 \}$ appears to be even more difficult.

Suggested Citation

  • Vijay V. Vazirani & Mihalis Yannakakis, 2020. "Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets," Papers 2004.01348, arXiv.org, revised Apr 2020.
    • Handle: RePEc:arx:papers:2004.01348
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    References listed on IDEAS

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    1. Yinghua He & Antonio Miralles & Marek Pycia & Jianye Yan, 2018. "A Pseudo-Market Approach to Allocation with Priorities," American Economic Journal: Microeconomics, American Economic Association, vol. 10(3), pages 272-314, August.
    2. Federico Echenique & Antonio Miralles & Jun Zhang, 2019. "Fairness and efficiency for probabilistic allocations with participation constraints," Papers 1908.04336, arXiv.org, revised May 2020.
    3. Eric Budish, 2011. "The Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes," Journal of Political Economy, University of Chicago Press, vol. 119(6), pages 1061-1103.
    4. Hylland, Aanund & Zeckhauser, Richard, 1979. "The Efficient Allocation of Individuals to Positions," Journal of Political Economy, University of Chicago Press, vol. 87(2), pages 293-314, April.
    5. Phuong Le, 2017. "Competitive equilibrium in the random assignment problem," International Journal of Economic Theory, The International Society for Economic Theory, vol. 13(4), pages 369-385, December.
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    Cited by:

    1. Jugal Garg & Thorben Trobst & Vijay V. Vazirani, 2020. "One-Sided Matching Markets with Endowments: Equilibria and Algorithms," Papers 2009.10320, arXiv.org, revised Jul 2021.
    2. Miralles, Antonio & Pycia, Marek, 2021. "Foundations of pseudomarkets: Walrasian equilibria for discrete resources," Journal of Economic Theory, Elsevier, vol. 196(C).

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