N-Person cake-cutting: there may be no perfect division
AbstractA cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability but not both). We prove that no perfect division exists for an extension of the example for three or more players.
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Bibliographic InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 34264.
Date of creation: 22 Oct 2011
Date of revision:
Cake-cutting; fair division; efficiency; envy-freeness; equitability; heterogeneous good;
Find related papers by JEL classification:
- D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
- D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
- D30 - Microeconomics - - Distribution - - - General
- D74 - Microeconomics - - Analysis of Collective Decision-Making - - - Conflict; Conflict Resolution; Alliances
- D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
- C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
- C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
This paper has been announced in the following NEP Reports:
- NEP-ALL-2011-10-22 (All new papers)
- NEP-CIS-2011-10-22 (Confederation of Independent States)
- NEP-GTH-2011-10-22 (Game Theory)
- NEP-HPE-2011-10-22 (History & Philosophy of Economics)
- NEP-MIC-2011-10-22 (Microeconomics)
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Barbanel, Julius B. & Brams, Steven J., 2011. "Two-person cake-cutting: the optimal number of cuts," MPRA Paper 34263, University Library of Munich, Germany.
- Barbanel, Julius B. & Brams, Steven J., 2010. "Two-person pie-cutting: The fairest cuts," MPRA Paper 22703, University Library of Munich, Germany.
- I. D. Hill, 2008. "Mathematics and Democracy: Designing Better Voting and Fair-division Procedures," Journal of the Royal Statistical Society Series A, Royal Statistical Society, Royal Statistical Society, vol. 171(4), pages 1032-1033.
- Brams,Steven J. & Taylor,Alan D., 1996. "Fair Division," Cambridge Books, Cambridge University Press, Cambridge University Press, number 9780521556446.
- Nurmi, Hannu, 1996. "Fair division: From cake-cutting to dispute resolution : Steven J. Brams and Alan D. Taylor, (Cambridge University Press, Cambridge, 1995) pp. xiv + 272, US$ 54.95 (hardcover), US$ 18.95 (paper)," European Journal of Political Economy, Elsevier, Elsevier, vol. 12(1), pages 169-172, April.
- Barbanel, Julius B. & Brams, Steven J., 2004. "Cake division with minimal cuts: envy-free procedures for three persons, four persons, and beyond," Mathematical Social Sciences, Elsevier, Elsevier, vol. 48(3), pages 251-269, November.
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