In which matching markets does the short side enjoy an advantage?

Abstract

We revisit the popular random matching market model introduced by Knuth (1976) and Pittel (1989), and shown by Ashlagi, Kanoria and Leshno (2013) to exhibit a “stark effect of competition”; in particular, with any difference in the number of agents on the two sides (“imbalance”), the short side agents obtain substantially better outcomes. We generalize the model to allow “partially connected” markets with each agent having an average degree d in a random (undirected) graph. Each agent has a (uniformly random) preference ranking over only their neighbors in the graph. We characterize stable matchings in large markets and find that the short side enjoys a significant advantage only for d exceeding log2 n where n is the number of agents on one side: For moderately connected markets with d = o(log2 n), we find that there is no advantage to being on the short side (for O(n1−ε) market imbalance), with agents on both sides getting a √d-ranked partner on average. Notably, this “mild competition” regime extends far beyond the connectivity threshold of d = Θ(log n). In contrast, for densely connected markets with d = ω(log2 n), we find a strong effect of competition, namely, short side agents get a log n-ranked partner on average, while the long side agents get a partner of (much larger) rank d/log n on average. Our results and analysis suggest that in general matching markets, being on the short side confers an advantage if and only if the number of short-side agents who remain unmatched is small relative to the market imbalance.