Matching Patterns When Group Size Exceeds Two

We study one-sided matching when groups with n > 2 members are being formed. Type-complementarity rules out all but the rank-ordered grouping. Type-substitutability (for example, matching to share risk) rules out much less. It requires that every two groups must be "intertwined," in that each dominates the other at some rank. Intertwined matching is necessary and, in one context, sufficient for any grouping to be a potential equilibrium. But there are many intertwined matching patterns when n > 2. Thus, substitutability can be observationally similar to complementarity; we demonstrate this by showing that dyadic regressions can register intertwined (negative assortative) matching as homogeneous matching.