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In 1979, Hylland and Zeckhauser [26] gave a simple and general mechanism for a 6 one-sided matching market, given cardinal utilities of agents over goods. They use the power of a 7 pricing mechanism, which endows their mechanism with several desirable properties – it produces an 8 allocation that is Pareto optimal and envy-free, and the mechanism is incentive compatible in the 9 large. It therefore provides an attractive, off-the-shelf method for running an application involving 10 such a market. With matching markets becoming ever more prevalent and impactful, it is imperative 11 to characterize the computational complexity of this mechanism. 12 We present the following results: 13 1. A combinatorial, strongly polynomial time algorithm for the dichotomous case, i.e., 0/1 14 utilities, and more generally, when each agent’s utilities come from a bi-valued set. 15 2. An example that has only irrational equilibria; hence this problem is not in PPAD. 16 3. A proof of membership of the problem in the class FIXP; as a corollary we get that an 17 HZ equilibrium can always be expressed via algebraic numbers. For this purpose, we give 18 a new proof of the existence of an HZ equilibrium using Brouwer’s fixed point theorem; 19 the proof of Hylland and Zeckhauser used Kakutani’s fixed point theorem, which is more 20 involved. 21 4. A proof of membership of the problem of computing an approximate HZ equilibrium in the 22 class PPAD.
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Cardinal-Utility Matching Markets: The Quest for Envy-Freeness, Pareto-Optimality, and Efficient Computability
Unlike ordinal-utility matching markets, which are well-developed from the viewpoint of both theory and practice, recent insights from a computer science perspective have left cardinal-utility matching markets in a state of flux. The celebrated pricing-based mechanism for one-sided cardinal-utility matching markets due to Hylland and Zeckhauser [1979](HZ), which had long eluded efficient algorithms, was finally shown to be intractable; the problem of computing an approximate equilibrium is PPAD-complete (Vazirani and Yannakakis [2021], Chen et al. [2022]). This led us to ask the question: is there an alternative, polynomial time, mechanism for one-sided cardinal-utility matching markets which achieves the desirable properties of HZ, i.e. (ex-ante) envyfreeness (EF) and Pareto-optimality (PO)? We show that the problem of finding an EF+PO lottery in a one-sided cardinal-utility matching market is by itself already PPAD-complete. However, a (2 + ǫ)-approximately envy-free and (exactly) Pareto-optimal lottery can be found in polynomial time using the Nash-bargaining-based mechanism of Hosseini and Vazirani [2022]. Moreover, the mechanism is also (2+ǫ)-approximately incentive compatible. We also present several results on two-sided cardinal-utility matching markets, including non-existence of EF+PO lotteries as well as existence of justified-envy-free and weak Pareto-optimal lotteries.
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- Award ID(s):
- 2230414
- PAR ID:
- 10627962
- Publisher / Repository:
- ACM EC
- Date Published:
- Format(s):
- Medium: X
- Location:
- New Haven, Connecticut
- Sponsoring Org:
- National Science Foundation
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