Cardinal-Utility Matching Markets: The Quest for Envy-Freeness, Pareto-Optimality, and Efficient Computability

TL;DR

This paper investigates the computational complexity of achieving envy-freeness and Pareto-optimality in cardinal-utility matching markets, proposing approximation algorithms and analyzing their properties.

Contribution

It establishes the PPAD-completeness of finding exact EF+PO lotteries and presents a polynomial-time approximation mechanism with provable guarantees.

Findings

Exact EF+PO lotteries are PPAD-complete to compute.

A polynomial-time $(2 + oldsymbol{ extepsilon})$-approximate mechanism exists for EF and PO.

Results include non-existence of EF+PO lotteries in two-sided markets and existence of weaker fairness notions.

Abstract

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, which had long eluded efficient algorithms, was finally shown to be intractable; the problem of computing an approximate equilibrium is PPAD-complete. 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) envy-freeness (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 $(2 + \epsilon)$-approximately envy-free and (exactly) Pareto-optimal lottery can be found in polynomial time using the Nash-bargaining-based mechanism of Hosseini and Vazirani. Moreover, the mechanism is also $(2 + \epsilon)$-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.