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Fair and square: Cake-cutting in two dimensions

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  • Segal-Halevi, Erel
  • Nitzan, Shmuel
  • Hassidim, Avinatan
  • Aumann, Yonatan

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.

Suggested Citation

  • Segal-Halevi, Erel & Nitzan, Shmuel & Hassidim, Avinatan & Aumann, Yonatan, 2017. "Fair and square: Cake-cutting in two dimensions," Journal of Mathematical Economics, Elsevier, vol. 70(C), pages 1-28.
  • Handle: RePEc:eee:mateco:v:70:y:2017:i:c:p:1-28
    DOI: 10.1016/j.jmateco.2017.01.007
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    Cited by:

    1. Ortega, Josué & Segal-Halevi, Erel, 2019. "Obvious manipulations in cake-cutting," ZEW Discussion Papers 19-056, ZEW - Leibniz Centre for European Economic Research.
    2. Erel Segal-Halevi & Shmuel Nitzan & Avinatan Hassidim & Yonatan Aumann, 2020. "Envy-Free Division of Land," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 896-922, August.
    3. Josue Ortega & Erel Segal-Halevi, 2019. "Obvious Manipulations in Cake-Cutting," Papers 1908.02988, arXiv.org, revised Oct 2019.
    4. Anna bogomolnaia & Herve Moulin, 2019. "Guarantees in Fair Division: general or monotone preferences," Papers 1911.10009, arXiv.org, revised Sep 2020.
    5. Erel Segal-Halevi & Shmuel Nitzan, 2019. "Fair cake-cutting among families," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(4), pages 709-740, December.
    6. Agnes Cseh & Tamás Fleiner, 2018. "The complexity of cake cutting with unequal shares," CERS-IE WORKING PAPERS 1819, Institute of Economics, Centre for Economic and Regional Studies.
    7. Legut, Jerzy, 2020. "Simple fair division of a square," Journal of Mathematical Economics, Elsevier, vol. 86(C), pages 35-40.

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