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 quandary. 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 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)? In this paper we show: 1. The problem of finding an EF+PO lottery in a one-sided cardinal-utility matching market is PPAD-complete. 2. A $(2 + \epsilon)$-approximately envy-free and (exactly) Pareto-optimal lottery can be found in polynomial time using Nash bargaining. Moreover, the resulting mechanism is $(2 + \epsilon)$-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.