Continuity Assumptions in Cake-Cutting

TL;DR

This paper clarifies the measure-theoretic foundations of cake-cutting, proposing the Borel σ-algebra and sliceable measures as the appropriate framework, and analyzes the relationships between divisibility and absolute continuity properties.

Contribution

It identifies the suitable measure-theoretic setting for cake-cutting and clarifies the relationship between continuity properties like divisibility and absolute continuity.

Findings

Borel σ-algebra is the appropriate family of sets for cake pieces.

Sliceable Borel measures are suitable for evaluation.

Divisibility is equivalent to being atom-free, while absolute continuity is a stronger condition.

Abstract

In important papers on cake-cutting -- one of the key areas in fair division and resource allocation -- the measure-theoretical fundamentals are not fully correctly given. It is not clear (i) which family of sets should be taken for the pieces of cake, (ii) which set-functions should be used for evaluating the pieces, and (iii) which is the relationship between various continuity properties appearing in cake-cutting. We show that probably the best choice for the familiy of subsets of $[0,1]$ is the Borel $\sigma$-algebra and for the set-function any `sliceable' Borel measure. At least in dimension one it does not make sense to work with only finitely additive contents on finite unions of intervals. For the continuity property we see two possibilities. The weaker is the traditional divisibility property, which is equivalent to being atom-free. The stronger is simply absolute continuity with respect to Lebesgue measure. We also consider the case of a base set (cake or pie) more general than $[0,1]$.