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.

翻译：与理论和实践均已成熟的序数效用匹配市场不同，从计算机科学视角的最新见解使基数效用匹配市场处于一种不稳定的状态。Hylland和Zeckhauser提出的著名基于定价的单边基数效用匹配市场机制，长期缺乏高效算法，最终被证明是难解的；计算近似均衡的问题是PPAD完全的。这促使我们提出疑问：是否存在一种替代的、多项式时间的机制，用于单边基数效用匹配市场，能够实现HZ机制所期望的性质，即（事前）无嫉妒性（EF）和帕累托最优性（PO）？我们证明了在单边基数效用匹配市场中寻找一个EF+PO抽签方案的问题本身已经是PPAD完全的。然而，利用Hosseini和Vazirani提出的基于纳什议价的机制，可以在多项式时间内找到一个$(2 + \epsilon)$-近似无嫉妒且（完全）帕累托最优的抽签方案。此外，该机制也是$(2 + \epsilon)$-近似激励相容的。我们还提出了关于双边基数效用匹配市场的若干结果，包括EF+PO抽签方案的不存在性，以及合理无嫉妒且弱帕累托最优抽签方案的存在性。