POPULATION-MONOTONIC SOLUTIONS TO THE PROBLEM OF FAIR DIVISION WHEN PREFERENCES ARE SINGLE-PEAKED

We consider the problem of fairly allocating an infinitely divisible commodity among agents with single-peaked preferences. First, we examine the implications of the requirement that a change in the population affect all agents that are present before and after the change in the same direction. We show that this requirement is met by no selection from the no-envy solution. In the face of this impossibility, we limit our attention to changes that are not so ''disruptive'', in the following sense: if initially there is not enough to bring all agents to their satiation points, then this still is the case after the change, and if initially there is so much that agents have to be brought beyond their satiation points, then again, this remains the case. The requirement is that such changes affect all agents that are present before and after the change in the same direction. Our main result is that there is essentially only one selection from the envy-free and efficient solution satisfying this property. It is the solution known as the ''uniform'' rule.