# Fair and square: Cake-cutting in two dimensions

*Erel Segal-Halevi*,
*Shmuel Nitzan*,
*Avinatan Hassidim* and
*Yonatan Aumann*

*Journal of Mathematical Economics*, 2017, vol. 70, issue C, 1-28

**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.

**Keywords:** Cake cutting; Fair division; Land economics; Geometry (search for similar items in EconPapers)

**Date:** 2017

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:mateco:v:70:y:2017:i:c:p:1-28

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