Matching of like rank and the size of the core in the marriage problem

- We study Gale-Shapley stable matchings when rankings are correlated.
- The size of a ranking set measures the displacement of individuals across rankings.
- The rank gaps measure how close a matching is to being assortative.
- The size of the core is a measure of the maximal distance between stable matchings.
- We bound the rank gaps and the size of the core, given the size of the ranking sets.

When men and women are objectively ranked in a marriage problem, say by beauty, then pairing individuals of equal rank is the only stable matching. We generalize this observation by providing bounds on the size of the rank gap between mates in a stable matching in terms of the size of the ranking sets. Using a metric on the set of matchings, we provide bounds on the diameter of the core - the set of stable matchings - in terms of the size of the ranking sets and in terms of the size of the rank gap. We conclude that when the set of rankings is small, so are the core and the rank gap in stable matchings. We construct examples showing that our bounds are essentially tight, and that certain natural variants of the bounds fail to hold.