Preference relations and measures in the context of fair division
HTML articles powered by AMS MathViewer
- by Julius B. Barbanel and Alan D. Taylor
- Proc. Amer. Math. Soc. 123 (1995), 2061-2070
- DOI: https://doi.org/10.1090/S0002-9939-1995-1233964-5
- PDF | Request permission
Abstract:
One of the most well-known metaphors in the mathematical theory of fair division concerns the problem of dividing a cake among n people in such a way that each person is satisfied with the piece he or she receives, even though different people value different parts of the cake differently. Our concern here is with how an individual’s preferences are formalized. David Gale has pointed out that although most of the deeper results in the field assume that preferences are given by an additive measure, the fundamental algorithms in the field require only that preferences be given by a binary relation satisfying a few natural properties. We introduce here one additional condition—an Archimedean property that obviously holds if the relation is induced by a measure—and we show that a preference relation satisfying Gale’s conditions is induced by a finitely additive measure if and only if it satisfies this Archimedean property.References
- Steven J. Brams and Alan D. Taylor, An envy-free cake division protocol, Amer. Math. Monthly 102 (1995), no. 1, 9–18. MR 1321451, DOI 10.2307/2974850 G. Debreu, Continuity properties of Paretian utility, Internat. Econom. Rev. 5 (1964), 285-293.
- L. E. Dubins and E. H. Spanier, How to cut a cake fairly, Amer. Math. Monthly 68 (1961), 1–17. MR 129031, DOI 10.2307/2311357
- David Gale, Mathematical entertainments, Math. Intelligencer 16 (1994), no. 2, 25–31. MR 1270837, DOI 10.1007/BF03024280
- B. O. Koopman, The axioms and algebra of intuitive probability, Ann. of Math. (2) 41 (1940), 269–292. MR 1474, DOI 10.2307/1969003
- Bezalel Peleg, Utility functions for partially ordered topological spaces, Econometrica 38 (1970), 93–96. MR 281166, DOI 10.2307/1909243
- Leonard J. Savage, The foundations of statistics, Second revised edition, Dover Publications, Inc., New York, 1972. MR 0348870
- C. Villegas, On qualitative probability $\sigma$-algebras, Ann. Math. Statist. 35 (1964), 1787–1796. MR 167588, DOI 10.1214/aoms/1177700400
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2061-2070
- MSC: Primary 90A06; Secondary 28A60
- DOI: https://doi.org/10.1090/S0002-9939-1995-1233964-5
- MathSciNet review: 1233964