Cutting a Cake Is Not Always a 'Piece of Cake': A Closer Look at the Foundations of Cake-Cutting Through the Lens of Measure Theory

TL;DR

This paper critically examines the foundational assumptions in cake-cutting literature, analyzing how different model choices impact fairness and proposing measure-theoretic perspectives for more consistent definitions.

Contribution

It provides a comprehensive survey of cake-cutting models, highlighting their strengths and weaknesses, and offers recommendations for measure-theoretic approaches to improve foundational clarity.

Findings

Analysis of common model assumptions in cake-cutting

Identification of strengths and weaknesses in existing definitions

Recommendations for measure-theoretic frameworks

Abstract

Cake-cutting is a playful name for the fair division of a heterogeneous, divisible good among agents, a well-studied problem at the intersection of mathematics, economics, and artificial intelligence. The cake-cutting literature is rich and edifying. However, different model assumptions are made in its many papers, in particular regarding the set of allowed pieces of cake that are to be distributed among the agents and regarding the agents' valuation functions by which they measure these pieces. We survey the commonly used definitions in the cake-cutting literature, highlight their strengths and weaknesses, and make some recommendations on what definitions could be most reasonably used when looking through the lens of measure theory.