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

We study individually rational rules to be used toallot, among a group of agents, a perfectly divisiblegood that is freely available only in whole units. Arule is individually rational if, at each preferenceprofile, each agent finds that her allotment is at leastas good as any whole unit of the good. We study andcharacterize two individually rational and efficientfamilies of rules, whenever agents' preferences aresymmetric single‐peaked on the set of possibleallotments. Rules in the two families are in additionenvy‐free, but they differ on whether envy‐freeness isconsidered on losses or on awards. Our main resultstates that (a) the family of constrained equal lossesrules coincides with the class of all individuallyrational and efficient rules that satisfy justified envy‐freeness on losses and (b) the family of constrainedequal awards rules coincides with the class of all in-dividually rational and efficient rules that satisfyenvy‐freeness on awards.