Fair Reallocation in Economies with Single-Peaked Preferences

We consider the problem of fairly reallocating the individual endowments of a perfectly divisible good among agents with single-peaked preferences. We provide a new concept of fairness, called position-wise envy-freeness, that is compatible with individual rationality. This new concept requires that each demander (i.e., agent whose most preferred amount is strictly greater than his endowment) should not envy another demander who does not receive his endowment and that each supplier (i.e., agent whose most preferred amount is strictly less than his endowment) should not envy another supplier who does not receive his endowment. We establish that a rule is efficient, individually rational, strategy-proof, and position-wise envy-free if and only if it is the gradual uniform rule, which is an extension of the well-known uniform rule.