Monotonicity and Competitive Equilibrium in Cake-cutting

TL;DR

This paper investigates monotonicity properties in fair cake-cutting, demonstrating that the Nash-optimal division rule uniquely satisfies resource- and population-monotonicity while maintaining fairness and efficiency.

Contribution

It introduces formal concepts of resource- and population-monotonicity in cake-cutting and proves the Nash-optimal rule uniquely satisfies these properties among welfare-maximizing rules.

Findings

Nash-optimal rule is resource- and population-monotonic.

Nash-optimal rule is Pareto-optimal and envy-free.

It is the only welfare-maximizing rule that is proportional and resource-monotonic.

Abstract

We study the monotonicity properties of solutions in 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 problem and examine whether they are satisfied by various common division rules. We prove that the Nash-optimal rule, which maximizes 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.