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Geometric division with a fixed point: Not half the cake, but at least 4/9

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Abstract

We study a two-person problem of cutting a homogeneous cake where one player is disadvantaged from the outset: Unlike under the divide-and-choose rule he may only choose a point on the cake through which the other player will then execute a cut and then take the piece that he prefers. We derive the optimal strategy for the disadvantaged player in this game and a lower bound for the share of the cake that he can maximally obtain: It amounts to one third of the cake whenever the cake is bounded. For convex and bounded cakes the minimum share rises to 4/9 of the cake.

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References

  • Berger, Marcel. (1987). Geometry I. Springer: Berlin etc.

    MATH  Google Scholar 

  • Brams, Steven J. and D. Marc Kilgour. (2001). “Competitive Fair Division,” Journal of Political Economy 109, 418–443.

    Article  Google Scholar 

  • Brams, Steven J. and Alan D. Taylor. (1996). Fair Division. From Cake-Cutting to Dispute Resolution. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Brams, Steven J. and Alan D. Taylor. (1995). “An envy-free cake-division protocol,” American Mathematical Monthly 102, 9–18.

    MathSciNet  MATH  Google Scholar 

  • Chawla, Shuchi, Uday Rajan, Ramamoorthi Ravi and Amitabh Sinha. (2003). “Worst-case payoffs of a location game”. Technical report CMU-CS-03-143. Carnegie Mellon University, Pittsburgh.

  • Eiselt, H.A., Gilbert Laporte and Jacques-François Thisse (1993). “Competitive Location Models: A Framework and Bibliography,” Transportation Science 27, 44–54.

    Article  MATH  Google Scholar 

  • Grünbaum, Branko. (1960). “Partitions of Mass-Distributions and of Convex Bodies by Hyperplanes,” Pacific Journal of Mathematics 10, 1257–1261.

    MATH  MathSciNet  Google Scholar 

  • Robertson, Jack and William Webb. (1998). Cake Cutting Algorithms: Be Fair If You Can. Natick: A.K. Peters.

  • Yaglom, Isaac M. and Vladimir G. Boltjanskii. (1961). Convex Figures. New York: Holt, Rinehart & Whinston.

    MATH  Google Scholar 

  • Young, Peyton H. (1994). Equity in Theory and Practice. Princeton: Princeton University Press.

    Google Scholar 

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Authors and Affiliations

  1. Department of Economics, University of Vienna, Hohenstaufengasse 9, 1010, Vienna, Austria

    Andreas Wagener

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  1. Andreas Wagener
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Correspondence to Andreas Wagener.

Additional information

JEL-Classification: D61 D63

The author thanks two anonymous referees for fruitful suggestions and criticism.

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Wagener, A. Geometric division with a fixed point: Not half the cake, but at least 4/9. Group Decis Negot 15, 43–53 (2006). https://doi.org/10.1007/s10726-005-9000-z

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  • DOI: https://doi.org/10.1007/s10726-005-9000-z

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