Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Fair Division*

  • Steven J. Brams
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_198

Definition of the Subject

Cutting a cake, dividing up the property in an estate, determining the borders in an international dispute – such allocation problems are ubiquitous. Fair division treats all these problems and many more through a rigorous analysis of procedures for allocating goods, or deciding who wins on what issues, in a dispute.


The literature on fair division has burgeoned in recent years, with five academic books [1,13,23,28,32] and one popular book [15] providing overviews. In this review, I will give a brief survey of three different literatures: (i) the division of a single heterogeneous good (e. g., a cake with different flavors or toppings); (ii) the division, in whole or part, of several divisible goods; and (iii) the allocation of several indivisible goods. In each case, I assume the different people, called players , may have different preferences for the items being divided.

For (i) and (ii), I will describe and illustrate procedures for dividing...

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  1. 1.
    Barbanel JB (2005) The Geometry of Efficient Fair Division. Cambridge University Press, New YorkzbMATHGoogle Scholar
  2. 2.
    Barbanel JB, Brams SJ (2004) Cake Division with Minimal Cuts: Envy-Free Procedures for 3 Persons, 4 Persons, and Beyond. Math Soc Sci 48(3):251–269MathSciNetzbMATHGoogle Scholar
  3. 3.
    Barbanel JB, Brams SJ (2007) Cutting a Pie Is Not a Piece of Cake. Am Math Month (forthcoming)Google Scholar
  4. 4.
    Brams SJ, Edelman PH, Fishburn PC (2001) Paradoxes of Fair Division. J Philos 98(6):300–314MathSciNetGoogle Scholar
  5. 5.
    Brams SJ, Edelman PH, Fishburn PC (2004) Fair Division of Indivisible Items. Theory Decis 55(2):147–180MathSciNetGoogle Scholar
  6. 6.
    Brams SJ, Fishburn PC (2000) Fair Division of Indivisible Items Between Two People with Identical Preferences: Envy-Freeness, Pareto‐Optimality, and Equity. Soc Choice Welf 17(2):247–267MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brams SJ, Jones MA, Klamler C (2006) Better Ways to Cut a Cake. Not AMS 35(11):1314–1321MathSciNetGoogle Scholar
  8. 8.
    Brams SJ, Jones MA, Klamler C (2007) Proportional Pie Cutting. Int J Game Theory 36(3–4):353–367MathSciNetGoogle Scholar
  9. 9.
    Brams SJ, Kaplan TR (2004) Dividing the Indivisible: Procedures for Allocating Cabinet Ministries in a Parliamentary System. J Theor Politics 16(2):143–173Google Scholar
  10. 10.
    Brams SJ, Kilgour MD (2001) Competitive Fair Division. J Political Econ 109(2):418–443Google Scholar
  11. 11.
    Brams SJ, King DR (2004) Efficient Fair Division: Help the Worst Off or Avoid Envy? Ration Soc 17(4):387–421Google Scholar
  12. 12.
    Brams SJ, Taylor AD (1995) An Envy-Free Cake Division Protocol. Am Math Month 102(1):9–18MathSciNetzbMATHGoogle Scholar
  13. 13.
    Brams SJ, Taylor AD (1996) Fair Division: From Cake‐Cutting to Dispute Resolution. Cambridge University Press, New YorkzbMATHGoogle Scholar
  14. 14.
    Brams SJ, Taylor AD (1999a) Calculating Consensus. Corp Couns 9(16):47–50Google Scholar
  15. 15.
    Brams SJ, Taylor AD (1999b) The Win-Win Solution: Guaranteeing Fair Shares to Everybody. W.W. Norton, New YorkGoogle Scholar
  16. 16.
    Brams SJ, Taylor AD, Zwicker SW (1995) Old and New Moving‐Knife Schemes. Math Intell 17(4):30–35MathSciNetzbMATHGoogle Scholar
  17. 17.
    Brams SJ, Taylor AD, Zwicker WS (1997) A Moving‐Knife Solution to the Four‐Person Envy-Free Cake Division Problem. Proc Am Math Soc 125(2):547–554MathSciNetzbMATHGoogle Scholar
  18. 18.
    Edelman PH, Fishburn PC (2001) Fair Division of Indivisible Items Among People with Similar Preferences. Math Soc Sci 41(3):327–347MathSciNetzbMATHGoogle Scholar
  19. 19.
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  20. 20.
    Haake CJ, Raith MG, Su FE (2002) Bidding for Envy-Freeness: A Procedural Approach to n‑Player Fair Division Problems. Soc Choice Welf 19(4):723–749MathSciNetzbMATHGoogle Scholar
  21. 21.
    Herreiner D, Puppe C (2002) A Simple Procedure for Finding Equitable Allocations of Indivisible Goods. Soc Choice Welf 19(2):415–430MathSciNetzbMATHGoogle Scholar
  22. 22.
    Jones MA (2002) Equitable, Envy-Free, and Efficient Cake Cutting for Two People and Its Application to Divisible Goods. Math Mag 75(4):275–283MathSciNetzbMATHGoogle Scholar
  23. 23.
    Moulin HJ (2003) Fair Division and Collective Welfare. MIT Press, CambridgeGoogle Scholar
  24. 24.
    Peterson E, Su FE (2000) Four‐Person Envy-Free Chore Division. Math Mag 75(2):117–122MathSciNetGoogle Scholar
  25. 25.
    Pikhurko O (2000)On Envy-Free Cake Division. Am Math Month 107(8):736–738Google Scholar
  26. 26.
    Potthoff RF (2002) Use of Linear Programming to Find an Envy-Free Solution Closest to the Brams‐Kilgour Gap Solution for the Housemates Problem. Group Decis Negot 11(5):405–414Google Scholar
  27. 27.
    Robertson JM, Webb WA (1997) Near Exact and Envy-Free Cake Division. Ars Comb 45:97–108MathSciNetzbMATHGoogle Scholar
  28. 28.
    Robertson J, Webb W (1998) Cake-Cutting Algorithms: Be Fair If You Can. AK Peters, NatickzbMATHGoogle Scholar
  29. 29.
    Shishido H, Zeng DZ (1999) Mark‐Choose‐Cut Algorithms for Fair and Strongly Fair Division. Group Decis Negot 8(2):125–137Google Scholar
  30. 30.
    Stromquist W (1980) How to Cut a Cake Fairly. Am Math Month 87(8):640–644MathSciNetzbMATHGoogle Scholar
  31. 31.
    Su FE (1999) Rental Harmony: Sperner's Lemma in Fair Division. Am Math Month 106:922–934Google Scholar
  32. 32.
    Young HP (1994) Equity in Theory and Practice. Princeton University Press, PrincetonGoogle Scholar
  33. 33.
    Zeng DZ (2000) Approximate Envy-Free Procedures. Game Practice: Contributions from Applied Game Theory. Kluwer Academic Publishers, Dordrecht, pp. 259–271Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Steven J. Brams
    • 1
  1. 1.Department of PoliticsNew York UniversityNew YorkUSA