Group Formation under Transferable Utility with Group Size greater than Two: Complementarity vs Substitutability

One-sided matching under transferable utility is well understood when groups of size two are being formed. In this setting, complementarity or substitutability of types in the group payoff function completely pins down key features of the equilibrium matching pattern, without it being necessary to know the distribution of types or specifics of the payoff function. This note explores equilibrium matching patterns when groups of fixed size n > 2 are being formed. Complementarity of types continues to pin down a unique equilibrium grouping. Substitutability of types rules out much less. We show that it rules out groupings in one which one group rank-wise dominates the other. Conversely, we show in a simple setting that any grouping not exhibiting rankwise dominance can be the unique equilibrium candidate. Thus, “no rank-wise dominance” is all that can be said generically about matching under substitutability. In the case of 2n agents, “no rank-wise dominance” leaves a unique grouping when n = 2 but rules out a vanishingly small fraction of groupings ( 2 n+1) as n increases. In general, the results suggest that substitutability by itself has much less predictive power than complementarity, when group size is bigger than two. ∗I thank Ashley Ahlin, Pierre-Andre Chiappori, Vivek Dhand, Patrick Legros, Andrew Newman, Debraj Ray, Katerina Sherstyuk, and Myrna Wooders for valuable input. All errors are mine.