Group Decision and Negotiation

, Volume 15, Issue 1, pp 43–53 | Cite as

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

  • Andreas WagenerEmail author


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.


cake cutting unfair division 


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  1. Berger, Marcel. (1987). Geometry I. Springer: Berlin etc.zbMATHGoogle Scholar
  2. Brams, Steven J. and D. Marc Kilgour. (2001). “Competitive Fair Division,” Journal of Political Economy 109, 418–443.CrossRefGoogle Scholar
  3. Brams, Steven J. and Alan D. Taylor. (1996). Fair Division. From Cake-Cutting to Dispute Resolution. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  4. Brams, Steven J. and Alan D. Taylor. (1995). “An envy-free cake-division protocol,” American Mathematical Monthly 102, 9–18.MathSciNetzbMATHGoogle Scholar
  5. 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.Google Scholar
  6. Eiselt, H.A., Gilbert Laporte and Jacques-François Thisse (1993). “Competitive Location Models: A Framework and Bibliography,” Transportation Science 27, 44–54.CrossRefzbMATHGoogle Scholar
  7. Grünbaum, Branko. (1960). “Partitions of Mass-Distributions and of Convex Bodies by Hyperplanes,” Pacific Journal of Mathematics 10, 1257–1261.zbMATHMathSciNetGoogle Scholar
  8. Robertson, Jack and William Webb. (1998). Cake Cutting Algorithms: Be Fair If You Can. Natick: A.K. Peters.Google Scholar
  9. Yaglom, Isaac M. and Vladimir G. Boltjanskii. (1961). Convex Figures. New York: Holt, Rinehart & Whinston.zbMATHGoogle Scholar
  10. Young, Peyton H. (1994). Equity in Theory and Practice. Princeton: Princeton University Press.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of ViennaViennaAustria

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