Matching Patterns when Group Size Exceeds Two

Matching under transferable utility is well understood when groups of fixed size n = 2 are being formed: Complementarity or substitutability of types in the group payoff function pins down the matching pattern, whatever the distribution of types or specifics of the payoff function. But little is known about one-sided matching in the case of groups with fixed size n > 2. This subject is taken up here. Type-complementarity continues to rule out all but one matching pattern. Type-substitutability rules out much less. It requires that in equilibrium, every two groups must be “intertwined”, in that each dominates the other at some rank. Intertwined matching is necessary and, in at least one context, sufficient for any grouping to be an equilibrium for some set of types; thus intertwined matching is all that substitutability generically predicts. But, the number of intertwined matching patterns increases rapidly in n. Thus, substitutability by itself has much less predictive power than complementarity when n > 2. One implication is that substitutability can be observationally similar to complementarity using common empirical techniques that detect homogeneity/heterogeneity of matching. This is demonstrated through dyadic regressions on simulated data, which show that statistically homogeneous groups are observable under intertwined (negative assortative) matching. ∗Department of Economics, Michigan State University; +1 517 3558306; [email protected]. I thank Ashley Ahlin, Alfredo Burlando, Pierre-Andre Chiappori, Vivek Dhand, Jan Eeckhout, Sergei Izmalkov, Patrick Legros, Andrew Newman, Debraj Ray, Katerina Sherstyuk, Myrna Wooders, and workshop/seminar participants at ThReD-Oslo, Universitat Pompeu Fabra, New Economic School, and University of Oregon for valuable input. All errors are mine.