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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. 

The Micali–Vazirani (MV) algorithm for finding a maximum cardinality matching in general graphs, which was published in 1980, remains to this day the most efficient known algorithm for the problem. The current paper gives the first complete and correct proof of this algorithm. The MV algorithm resorts to finding minimum-length augmenting paths. However, such paths fail to satisfy an elementary property, called breadth first search honesty in this paper. In the absence of this property, an exponential time algorithm appears to be called for—just for finding one such path. On the other hand, the MV algorithm accomplishes this and additional tasks in linear time. The saving grace is the various “footholds” offered by the underlying structure, which the algorithm uses in order to perform its key tasks efficiently. The theory expounded in this paper elucidates this rich structure and yields a proof of correctness of the algorithm. It may also be of independent interest as a set of well-knit graph-theoretic facts. 

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.