Fair and Square: Cake-Cutting in Two Dimensions

TL;DR

This paper explores fair division of a two-dimensional 'cake' into square or fat rectangular pieces, addressing geometric constraints that challenge traditional fairness guarantees and proposing new division methods.

Contribution

It introduces the problem of fair two-dimensional division with geometric shape restrictions and analyzes the limitations and possibilities for proportional fairness.

Findings

Proportionality cannot always be guaranteed under geometric constraints.

Impossibility results for certain fairness guarantees.

Constructive procedures for fair division with shape restrictions.

Abstract

We consider the classic problem of fairly dividing a heterogeneous good ("cake") among several agents with different valuations. Classic cake-cutting procedures either allocate each agent a collection of disconnected pieces, or assume that the cake is a one-dimensional interval. In practice, however, the two-dimensional shape of the allotted pieces is important. In particular, when building a house or designing an advertisement in printed or electronic media, squares are more usable than long and narrow rectangles. We thus introduce and study the problem of fair two-dimensional division wherein the allotted pieces must be of some restricted two-dimensional geometric shape(s), particularly squares and fat rectangles. Adding such geometric constraints re-opens most questions and challenges related to cake-cutting. Indeed, even the most elementary fairness criterion --- proportionality --- can no longer be guaranteed. In this paper we thus examine the level of proportionality that can be guaranteed, providing both impossibility results and constructive division procedures.