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. 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：与从理论和实践视角均已充分发展的序数效用匹配市场不同，近期计算机科学视角的洞见使基数效用匹配市场陷入困境。Hylland 和 Zeckhauser 提出的基于定价的单侧基数效用匹配市场机制，长期以来一直缺乏高效算法，最终被证明是 PPAD-完全的。这引发我们思考：是否存在替代性的、多项式时间的单侧基数效用匹配市场机制，能够实现 HZ 机制所具备的理想性质，即（事前）无嫉妒性（EF）和帕累托最优性（PO）？本文证明：1. 在单侧基数效用匹配市场中寻找 EF+PO 彩票问题是 PPAD-完全的。2. 运用纳什讨价还价可在多项式时间内找到一种 \((2 + \epsilon)\)-近似无嫉妒且（精确）帕累托最优的彩票。我们还提出了关于双侧基数效用匹配市场的若干结果，包括 EF+PO 彩票的不存在性，以及存在合理嫉妒且弱帕累托最优的彩票。