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

Authors Vijay V. Vazirani, Mihalis Yannakakis

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Vijay V. Vazirani
  • Department of Computer Science, University of California, Irvine, CA, USA
Mihalis Yannakakis
  • Department of Computer Science, Columbia University, New York, NY, USA


We wish to thank Federico Echenique, Jugal Garg, Tung Mai and Thorben Trobst for valuable discussions and Richard Zeckhauser for providing us with the Appendix to his paper [Hylland and Zeckhauser, 1979]. In addition, the first author wishes to thank Simons Institute for running a program on matching markets in Fall 2019; this provided valuable exposure to the topic.

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Vijay V. Vazirani and Mihalis Yannakakis. Computational Complexity of the Hylland-Zeckhauser Scheme for One-Sided Matching Markets. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 59:1-59:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In 1979, Hylland and Zeckhauser [Hylland and Zeckhauser, 1979] 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 prevalent 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 dichotomous case, i.e., 0/1 utilities, and more generally, when each agent’s utilities come from a bi-valued set. 2) An example that has only irrational equilibria, hence proving that this problem is not in PPAD. 3) A proof of membership of the problem in the class FIXP. 4) A proof of membership of the problem of computing an approximate HZ equilibrium in the class PPAD. We leave open the (difficult) questions of determining if computing an exact HZ equilibrium is FIXP-hard and an approximate HZ equilibrium is PPAD-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Hyland-Zeckhauser scheme
  • one-sided matching markets
  • mechanism design
  • dichotomous utilities
  • PPAD
  • FIXP


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