The Price of Indivisibility in Cake Cutting

TL;DR

This paper investigates how limiting the number of disjoint pieces in envy-free cake cutting affects social welfare, providing bounds on the 'Price of Indivisibility' for various welfare measures and fairness constraints.

Contribution

It introduces the concept of the Price of Indivisibility, quantifies its bounds, and explores the trade-offs between continuous and disjoint piece allocations in cake cutting.

Findings

Bounds for the Price of Indivisibility in social welfare.

Analysis of the impact of disjoint pieces on envy-free allocations.

Comparison of welfare gains with and without fairness constraints.

Abstract

We consider the problem of envy-free cake cutting, which is the distribution of a continuous heterogeneous resource among self interested players such that nobody prefers what somebody else receives to what they get. Existing work has focused on two distinct classes of solutions to this problem - allocations which give each player a continuous piece of cake and allocations which give each player arbitrarily many disjoint pieces of cake. Our aim is to investigate allocations between these two extremes by parameterizing the maximum number of disjoint pieces each player may receive. We characterize the Price of Indivisibility (POI) as the gain achieved in social welfare (utilitarian and egalitarian), by moving from allocations which give each player a continuous piece of cake to allocations that may give each player up to $k$ disjoint pieces of cake. Our results contain bounds for the Price of Indivisibility for utilitarian as well as egalitarian social welfare, and for envy-free cake cutting as well as cake cutting without any fairness constraints.