Unbalanced Random Matching Markets: The Stark Effect of Competition

Abstract

We study competition in matching markets with random heterogeneous preferences and an unequal number of agents on either side. First, we show that even the slightest imbalance yields an essentially unique stable matching. Second, we give a tight description of stable outcomes, showing that matching markets are extremely competitive. Each agent on the short side of the market is matched with one of his top choices, and each agent on the long side either is unmatched or does almost no better than being matched with a random partner. Our results suggest that any matching market is likely to have a small core, explaining why small cores are empirically ubiquitous.

Large Matching Markets as Two-Sided Demand Systems

“…an arbitrarily large share of all the men in the market under the stable matching most preferred by the women (W-optimal matching), whereas under the stable matching that is most preferred by all men (M-optimal matching), their match is ranked above a share that is arbitrarily close to zero. In more recent work, Ashlagi, Kanoria, and Leshno (2013) showed that this finding depends crucially on both sides of the market having a (close to equal) number of participants. Our results concern only the observable implications of pairwise stability, where we find that, for large markets, the differences in distributions of matched characteristics resulting for different stable matchings become irrelevant with no regard to the relative number of agents on either side of the market.…”

Marketplaces, Markets, and Market Design

“…54 53 See Dubins and Freedman (1981) and Roth (1982a) on incentives in the one-to-one marriage problem, and Roth (1985) when there is many-to-one matching as in the medical match. Roth and Peranson (1999) observed the small size of the set of stable matchings, and Ashlagi, Kanoria, and Leshno (2017) have given a compelling theoretical account of why we should expect the set of stable matchings to be small in these kinds of matching models. In simple models (e.g., of one-to-one matching) this in turn implies that profitably manipulating preferences will not be a viable option for virtually all participants, because manipulation of preferences cannot move outcomes outside of the set of stable matchings (see Demange, Gale, and Sotomayor 1987;Roth 2015c).…”

Can Everyone Benefit from Social Integration?

“…All results in Section 3 extend to the case of societies unbalanced in their gender ratio. Regarding the expected gains from integration, Ashlagi et al (2017) show that the spouse rank for men and women in the MOSM roughly reverses if there are more men than women. Therefore, if we merge several societies, all in which there are more men than women, the expected gains from integration roughly reverse as well.…”

“…This result extends to manyto-one SMPs (Theorem 5 in Kelso and Crawford, 1982, Theorems 1 and 2 in Crawford, 1991, and Theorems 2.25 and 2.26 in Roth and Sotomayor, 1992). Ashlagi et al (2017) describe the exact magnitude of these welfare changes in random MPs. Toda (2006) uses this monotonicity property to characterize the set of stable outcomes.…”

“…Because this analysis will use existing results, henceforth I will assume that each community has n men and n women. Albeit restrictive, this assumption allows me to isolate the effects of integration from those of having societies that are unbalanced in their gender ratio, which are discussed in detail by Ashlagi et al (2017). This assumption implies that all agents are married in any stable matching.…”