The Proportional Random Allocation of Indivisible Units

Indivisible units are randomly allocated among agents with a claim/demand on the resources. The available resources fall short of the sum of individual claims. The proportional method distributes units sequentially, and the probability of receiving a unit at any step is proportional to the unsatisfied claims. We characterize the family of probabilistic rationing methods meeting the three axioms Consistency, Lower and Upper Composition. It contains the proportional method, all deterministic fixed priority methods, and the priority compositions of proportional methods. The proportional method is the only fair method in the family.