Entropy, desegregation, and proportional rationing

An assignment of students to schools displays zero-segregation if all schools have the same distribution of the different types of students (ethnic, geographical, gender). We axiomatize the choice of an optimally desegregated assignment under arbitrary capacity constraints. The celebrated Consistency axiom, together with standard rational choice requirements, identify the choice rule minimizing a canonical index of proportional fairness: the entropy of the assignment matrix. This is an alternative vindication of the Mutual Information index of segregation recently characterized in Frankel and Volij (2011). A similar result holds in the capacity-constrained extension of the bipartite rationing model: there we must minimize the entropy of the rationing matrix augmented by individual deficits.