Individually rational rules for the division problem when the number of units to be allotted is endogenous

Abstract

We study individually rational rules to be used to allot, among a group of agents, a perfectly divisible good that is freely available only in whole units. A rule is individually rational if, at each preference pro…le, each agent …nds that her allotment is at least as good as any whole unit of the good. We study and characterize two individually rational and e¢cient rules, whenever agents' preferences are symmetric single-peaked on the set of possible allotments. The two rules are in addition envy-free, but they di¤er on wether envy-freeness is considered on losses or on awards. Our main result states that (i) the constrained equal losses rule is the unique individually rational and e¢cient rule that satis…es justi…ed envy-freeness on losses and (ii) the constrained equal awards rule is the unique individually rational and e¢cient rule that satis…es envy-freeness on awards.