Nash-Bargaining-Based Models for Matching Markets: One-Sided and Two-Sided; Fisher and Arrow-Debreu

Authors Mojtaba Hosseini , Vijay V. Vazirani

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Mojtaba Hosseini
  • The Paul Merage School of Business, University of California, Irvine, CA, USA
Vijay V. Vazirani
  • Computer Science Department, University of California, Irvine, CA, USA


The second author would like to thank Richard Zeckhauser for a very enlightening discussion on his "wish list" of models for matching markets. Several of the models considered in this paper have their origins in that discussion.

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Mojtaba Hosseini and Vijay V. Vazirani. Nash-Bargaining-Based Models for Matching Markets: One-Sided and Two-Sided; Fisher and Arrow-Debreu. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 86:1-86:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


This paper addresses two deficiencies of models in the area of matching-based market design. The first arises from the recent realization that the most prominent solution that uses cardinal utilities, namely the Hylland-Zeckhauser (HZ) mechanism [Hylland and Zeckhauser, 1979], is intractable; computation of even an approximate equilibrium is PPAD-complete [Vazirani and Yannakakis, 2021; Chen et al., 2021]. The second is the extreme paucity of models that use cardinal utilities, in sharp contrast with general equilibrium theory. Our paper addresses both these issues by proposing Nash-bargaining-based matching market models. Since the Nash bargaining solution is captured by a convex program, efficiency follow; in addition, it possesses a number of desirable game-theoretic properties. Our approach yields a rich collection of models: for one-sided as well as two-sided markets, for Fisher as well as Arrow-Debreu settings, and for a wide range of utility functions, all the way from linear to Leontief. We also give very fast implementations for these models which solve large instances, with n = 2000, in one hour on a PC, even for a two-sided matching market. A number of new ideas were needed, beyond the standard methods, to obtain these implementations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Theory of computation → Algorithmic mechanism design
  • Mathematics of computing → Convex optimization
  • Matching-based market design
  • Nash bargaining
  • convex optimization
  • Frank-Wolfe algorithm
  • cutting planes
  • general equilibrium theory
  • one-sided markets
  • two-sided markets


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