Fair and Square: Cake-cutting in Two Dimensions

TL;DR

This paper explores fair division of two-dimensional resources with shape constraints, such as squares or rectangles, revealing new challenges and solutions in ensuring proportional fairness under geometric restrictions.

Contribution

It introduces the problem of geometric constrained cake-cutting, analyzes the limits of proportional fairness, and provides algorithms for guaranteed fair divisions with shape restrictions.

Findings

Proportionality cannot always be guaranteed under shape constraints.

Algorithms are developed for fair division with rectangular and square plots.

Impossibility results highlight the limitations of fair division in geometric settings.

Abstract

We consider the problem of fairly dividing a two dimensional heterogeneous good among multiple players. Applications include division of land as well as ad space in print and electronic media. Classical cake cutting protocols primarily consider a one-dimensional resource, or allocate each player multiple infinitesimally small "pieces". In practice, however, the two dimensional \emph{shape} of the allotted piece is of crucial importance in many applications (e.g. squares or bounded aspect-ratio rectangles are most useful for building houses, as well as advertisements). We thus introduce and study the problem of fair two-dimensional division wherein the allotted plots must be of some restricted two-dimensional geometric shape(s). Adding this geometric constraint re-opens most questions and challenges related to cake-cutting. Indeed, even the elementary \emph{proportionality} fairness criteria can no longer be guaranteed in all cases. In this paper we thus examine the \emph{level} of proportionality that \emph{can} be guaranteed, providing both impossibility results (for proportionality that cannot be guaranteed), and algorithmic constructions (for proportionality that can be guaranteed). We focus primarily on the case when the cake is a rectilinear polygon and the allotted plots must be squares or bounded aspect-ratio rectangles.