We study monotonicity properties of solutions to the classic problem of fair cake-cutting—dividing a heterogeneous resource among agents with different preferences. Resource- and population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all of them are better-off (if there is more to share or fewer to share among) or all are worse-off (if there is less to share or more to share among). We formally introduce these concepts to the cake-cutting setting and show that they are violated by common division rules. In contrast, we prove that the Nash-optimal rule—maximizing the product of utilities—is resource-monotonic and population-monotonic, in addition to being Pareto-optimal, envy-free and satisfying a strong competitive-equilibrium condition. Moreover, we prove that it is the only rule among a natural family of welfare-maximizing rules that is both proportional and resource-monotonic.
The consistency axiom (cf. @cite_20 or @cite_2 ) is related to population-monotonicity, but it is fundamentally different as in that case the leaving agents take their fair shares with them.
@cite_20 @cite_2
2094005555 2116299292
A basic principle of distributive justice states that if an allocation among a group of individuals is fair, then it should be perceived as fair when restricted to each subgroup of individuals. This 'consistency' principle applies in particular to methods for allocating taxes among citizens according to their ability to pay. Every consistent, continuous taxation method optimizes an additively separable objective function. Optimal allocations may be computed by a simple Lagrange multiplier technique. Similar results hold for allocating assets among creditors according to their claims. The techniques are illustrated by constructing an objective function for a bankruptcy method from the Babylonian Talmud. An allocation rule is ‘consistent’ if the recommendation it makes for each problem ‘agrees’ with the recommendation it makes for each associated reduced problem, obtained by imagining some agents leaving with their assignments. Some authors have described the consistency principle as a ‘fairness principle’. Others have written that it is not about fairness, that it should be seen as an ‘operational principle’. We dispute the particular fairness interpretations that have been offered for consistency, but develop a different and important fairness foundation for the principle, arguing that it can be seen as the result of adding ‘some’ efficiency to a ‘post-application’ and efficiency-free expression of solidarity in response to population changes. We also challenge the interpretations of consistency as an operational principle that have been given, and here identify a sense in which such an interpretation can be supported. We review and assess the other interpretations of the principle, as ‘robustness’, ‘coherence’ and ‘reinforcement’.