Consistent solutions to the problem of fair division when preferences are single-peaked. - Thomson, William (Journal of Economic Theory, 1994)
A mathematical model is used to consider the problem of allocating an infinitely divisible commodity fairly among agents with single-peaked preferences. The methods of performing this division, or solutions, sought are ones that satisfy a consistency property. The property is: Any recommendation made for any economy is in agreement with the recommendation made for any “reduced” economy that is obtained by imagining the departure of some of the agents with their allotted consumptions. Essentially, all efficient subsolutions of the no-envy solution that satisfy consistency must contain a particular solution known as the uniform rule. This rule is characterized on the basis of a “converse” of consistency and the distributional requirement of individual rationality from equal division.