Resource-monotonicity and Population-monotonicity in Cake-cutting

We study the monotonicity properties of solutions in the classic problem of fair cake-cutting — dividing a heterogeneous resource among agents with different preferences. Resourceand 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 all are worse-off (if there is less to share). 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 Paretooptimal, envy-free and satisfying a strong competitive-equilibrium condition. Moreover, we prove that it is the only rule among a natural family of welfaremaximizing rules that is both proportional and resource-monotonic. In contrast, other members of this family, like the utilitarian and leximin rules, can be made either proportional or resource-monotonic, but not both.