Preference relations and measures in the context of fair division

Authors:
Julius B. Barbanel and Alan D. Taylor

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2061-2070

MSC:
Primary 90A06; Secondary 28A60

MathSciNet review:
1233964

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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.

**[BT]**Steven J. Brams and Alan D. Taylor,*An envy-free cake division protocol*, Amer. Math. Monthly**102**(1995), no. 1, 9–18. MR**1321451**, 10.2307/2974850**[D]**G. Debreu,*Continuity properties of Paretian utility*, Internat. Econom. Rev.**5**(1964), 285-293.**[DS]**L. E. Dubins and E. H. Spanier,*How to cut a cake fairly*, Amer. Math. Monthly**68**(1961), 1–17. MR**0129031****[G]**David Gale,*Mathematical entertainments*, Math. Intelligencer**16**(1994), no. 2, 25–31. MR**1270837**, 10.1007/BF03024280**[K]**B. O. Koopman,*The axioms and algebra of intuitive probability*, Ann. of Math. (2)**41**(1940), 269–292. MR**0001474****[P]**Bezalel Peleg,*Utility functions for partially ordered topological spaces*, Econometrica**38**(1970), 93–96. MR**0281166****[S]**Leonard J. Savage,*The foundations of statistics*, Second revised edition, Dover Publications, Inc., New York, 1972. MR**0348870****[V]**C. Villegas,*On qualitative probability 𝜎-algebras*, Ann. Math. Statist.**35**(1964), 1787–1796. MR**0167588**

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1233964-5

Article copyright:
© Copyright 1995
American Mathematical Society