The uniform rule in economies with single-peaked preferences, endowments and population-monotonicity

Abstract

We consider the problem of fair allocate an infinitely divisible commodity among agents with single-•peaked preferences and private endowments. First, we show that the adaptation of the uniform rule to this context is the only selection from the no-envy in redistributions and Pareto efficient solution to depend only on peaks and endowments. Second, we examine the implications of the requirement that a change in the population affect all agents that are present befare and after the change in the same direction. We show there is no selection from the noenvy and Pareto efficient solution satisfying this requirement. However, if a mild additional restriction on the domain is imposed, there are selections from the individually rational and Pareto efficient solution satisfying it. Finally, we relax the population-monotonicity requirement by applying it only when the direction of the inequality between the sum of the endowments and the sum of the peaks does not change. Our main result is that there is only one selection from the no-emy and Pareto efficient solution satisfying this property. It is the uniform rule.Single-peaked preferences, endowments, population-monotonicity, uniform rule