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Two-player envy-free multi-cake division

Author

Listed:
  • Cloutier, John
  • Nyman, Kathryn L.
  • Su, Francis Edward

Abstract

We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may not exist for three cakes cut into three pieces each. However, there do exist such divisions for two cakes cut into three pieces each, and for three cakes cut into four pieces each. The resulting allocations of pieces to players are Pareto-optimal with respect to the division. We use a generalization of Sperner's lemma on the polytope of divisions to locate solutions to our generalized cake-cutting problem.

Suggested Citation

  • Cloutier, John & Nyman, Kathryn L. & Su, Francis Edward, 2010. "Two-player envy-free multi-cake division," Mathematical Social Sciences, Elsevier, vol. 59(1), pages 26-37, January.
    • Handle: RePEc:eee:matsoc:v:59:y:2010:i:1:p:26-37
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    References listed on IDEAS

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    1. Simmons, Forest W. & Su, Francis Edward, 2003. "Consensus-halving via theorems of Borsuk-Ulam and Tucker," Mathematical Social Sciences, Elsevier, vol. 45(1), pages 15-25, February.
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    Cited by:

    1. Xiaotie Deng & Qi Qi & Amin Saberi, 2012. "Algorithmic Solutions for Envy-Free Cake Cutting," Operations Research, INFORMS, vol. 60(6), pages 1461-1476, December.
    2. Sagrario Lantarón & Mariló López & Susana Merchán & Javier Rodrigo & José Samuel Rodríguez, 2021. "Envy-Free Allocation by Sperner’s Lemma Adapted to Rotation Shifts in a Company," Mathematics, MDPI, vol. 9(9), pages 1-12, April.
    3. Doğan, Battal, 2016. "Nash-implementation of the no-envy solution on symmetric domains of economies," Games and Economic Behavior, Elsevier, vol. 98(C), pages 165-171.

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