Consistent solutions to the problem of fair division when preferences are single-peaked.

*(English)*Zbl 0864.90008Summary: We consider the problem of fairly allocating an infinitely divisible commodity among agents with single-peaked preferences. We search for methods of performing this division, or solutions, that satisfy the following property of consistency: any recommendation made for any economy is in agreement with the recommendation made for any ‘reduced’ economy obtained by imagining the departure of some of the agents with their allotted consumptions. Our main result is that essentially all efficient subsolutions of the no-envy solution satisfying consistency must contain a certain solution known as the uniform rule. We also characterize the uniform rule on the basis of a ‘converse’ of consistency and the distributional requirement of individual rationality from equal divisions.