Before revision:  We introduce a notion of partial homogeneity of preferences to express the degree of alignment in the preferences of colleges over students, and show that the more the preferences are aligned, the more the  matching frequencies tend to concentrate on their expected value given the colleges' types.
After revision:  We introduce a notion of partial homogeneity of preferences to express the degree of alignment in the preferences of colleges over students, and show that the more the preferences are aligned, the more the empirical matching frequencies tend to concentrate on their expected value given the colleges' types.