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 single heterogeneous resource among agents with subjective utilities. 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 all are worse-off (if there is less to share). We formally introduce these concepts to the cakecutting setting and present a meticulous axiomatic analysis. We show that classical cakecutting protocols, like the Cut and Choose, Banach-Knaster, Dubins-Spanier and many other fail to be monotonic. We also show that, when the allotted pieces must be contiguous, proportionality and Pareto-optimality are incompatible with each of the monotonicity axioms. We provide a resource-monotonic protocol for two players and show the existence of rules that satisfy various combinations of contiguousness, proportionality, Pareto-optimality and the two monotonicity axioms.