W. Thomson, POPULATION-MONOTONIC SOLUTIONS TO THE PROBLEM OF FAIR DIVISION WHEN PREFERENCES ARE SINGLE-PEAKED, Economic theory, 5(2), 1995, pp. 229-246
We consider the problem of fairly allocating an infinitely divisible c
ommodity among agents with single-peaked preferences. First, we examin
e 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 f
rom the no-envy solution. In the face of this impossibility, we limit
our attention to changes that are not so ''disruptive'', in the follow
ing sense: if initially there is not enough to bring all agents to the
ir 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 t
heir satiation points, then again, this remains the case. The requirem
ent 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 so
lution satisfying this property. It is the solution known as the ''uni
form'' rule.