Abstract
Consider n players having preferences over the connected pieces of a cake, identified with the interval [0, 1]. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into n connected pieces and assign these pieces to the players in an envy-free manner, i.e., such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when n is a prime number or is equal to 4. When n is at most 3, this has been previously proved by Erel Segal- Halevi, who conjectured that the result holds for any n. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.
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References
R. N. Andersen, M. M. Marjanović and R. M. Schori, Symmetric products and higherdimensional dunce hats, Topology Proceedings 18 (1993), 7–17.
M. Asada, F. Frick, V. Pisharody, M. Polevy, D. Stoner, L. H. Tsang and Z. Wellner, Fair division and generalizations of Sperner- and KKM-type results, SIAM Journal of Discrete Mathematics 32 (2018), 591–610.
V. Bil´o, I. Caragiannis, M. Flammini, A. Igarashi, G. Monaco, D. Peters, C. Vinci and W. S. Zwicker, Almost envy-free allocations with connected bundles, in 10th Innovations in Theoretical Computer Science, Leibniz International Proceedings in Informatics, Vol. 124, Schloss Dagstuhl–Leibniz-Zentrum f¨ur Informatik, Dagstuhl, 2018, pp. 14:1–14:21.
X. Chen and X. Deng, On the complexity of 2D discrete fixed point problem, Theoretical Computer Science 410 (2009), 4448–4456.
J. A. De Loera, X. Goaoc, F. Meunier and N. Mustafa, The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg, Bulletin of the American Mathematical Society 56 (2019), 415–511.
X. Deng, Q. Qi and A. Saberi, Algorithmic solutions for envy-free cake cutting, Operations Research 60 (2012), 1461–1476.
F. Frick, K. Houston-Edwards and F. Meunier, Achieving rental harmony with a secretive roommate, The American Mathematical Monthly 126 (2019), 18–32.
Y. Kannai, Using oriented volume to prove Sperner’s lemma, Economic Theory Bulletin 1 (2013), 11–19.
T. Kir´aly and J. Pap, PPAD-completeness of polyhedral versions of Sperner’s Lemma, Discrete Mathematics 313 (2013), 1594–1599.
D. N. Kozlov, Topology of scrambled simplices, Journal of Homotopy and Related Structures 14 (2019), 371–391.
A. McLennan and R. Tourky, Using volume to prove Sperner’s lemma, Economic Theory 35 (2008), 593–597.
M. Mirzakhani and J. Vondr´ak, Sperner’s colorings, hypergraph labeling problems and fair division, in Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2015, pp. 873–886.
J. R. Munkres, Elements of Algebraic Topology, Vol. 2, Addison-Wesley, Menlo Park, CA, 1984.
E. Segal-Halevi, Fairly dividing a cake after some parts were burnt in the oven, in Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, IFAAMAS, pp. 1276–1284, http://www.ifaamas.org/Proceedings/aamas2018/pdfs/p1276.pdf.
F. W. Simmons and F. E. Su, Consensus-halving via theorems of Borsuk–Ulam and Tucker, Mathematical Social Sciences 45 (2003), 15–25.
W. Stromquist, How to cut a cake fairly, The AmericanMathematical Monthly 87 (1980), 640–644.
F. E. Su, Rental harmony: Sperner’s lemma in fair division, The American Mathematical Monthly 106 (1999), 930–942.
D. R. Woodall, Dividing a cake fairly, Journal of Mathematical Analysis and Applications 78 (1980), 233–247.
Acknowledgments
The authors are grateful to the referee for his thorough reading and his suggestions and questions that helped improve the paper.
This work has been initiated when the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. This material is thus partially based upon work supported by the National Science Foundation under Grant No. DMS-1440140.
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Meunier, F., Zerbib, S. Envy-free cake division without assuming the players prefer nonempty pieces. Isr. J. Math. 234, 907–925 (2019). https://doi.org/10.1007/s11856-019-1939-6
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DOI: https://doi.org/10.1007/s11856-019-1939-6