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), 20612070
MSC:
Primary 90A06; Secondary 28A60
MathSciNet review:
1233964
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Abstract: One of the most wellknown 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 conditionan Archimedean property that obviously holds if the relation is induced by a measureand 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 envyfree cake division protocol, Amer. Math.
Monthly 102 (1995), no. 1, 9–18. MR
1321451, http://dx.doi.org/10.2307/2974850
 [D]
G. Debreu, Continuity properties of Paretian utility, Internat. Econom. Rev. 5 (1964), 285293.
 [DS]
L.
E. Dubins and E.
H. Spanier, How to cut a cake fairly, Amer. Math. Monthly
68 (1961), 1–17. MR 0129031
(23 #B2068)
 [G]
David
Gale, Mathematical entertainments, Math. Intelligencer
16 (1994), no. 2, 25–31. MR 1270837
(95i:90130), http://dx.doi.org/10.1007/BF03024280
 [K]
B.
O. Koopman, The axioms and algebra of intuitive probability,
Ann. of Math. (2) 41 (1940), 269–292. MR 0001474
(1,245a)
 [P]
Bezalel
Peleg, Utility functions for partially ordered topological
spaces, Econometrica 38 (1970), 93–96. MR 0281166
(43 #6885)
 [S]
Leonard
J. Savage, The foundations of statistics, Second revised
edition, Dover Publications, Inc., New York, 1972. MR 0348870
(50 #1364)
 [V]
C.
Villegas, On qualitative probability 𝜎algebras, Ann.
Math. Statist. 35 (1964), 1787–1796. MR 0167588
(29 #4860)
 [BT]
 S. J. Brams and A. D. Taylor, An envyfree cake division protocol, Amer. Math. Monthly 102 (1995), 918. MR 1321451
 [D]
 G. Debreu, Continuity properties of Paretian utility, Internat. Econom. Rev. 5 (1964), 285293.
 [DS]
 L. E. Dubins and E. H. Spanier, How to cut a cake fairly, Amer. Math. Monthly 68 (1961), 117. MR 0129031 (23:B2068)
 [G]
 D. Gale, Mathematical entertainments, Math. Intelligencer 15 (1993), 4852. MR 1270837 (95i:90130)
 [K]
 B. O. Koopman, The axioms and algebra of intuitive probability, Ann. of Math. (2) 41 (1940), 269292. MR 0001474 (1:245a)
 [P]
 B. Peleg, Utility functions for partially ordered topological spaces, Econometrica 38 (1970), 9396. MR 0281166 (43:6885)
 [S]
 L. J. Savage, The foundations of statistics, Dover, New York, 1972. MR 0348870 (50:1364)
 [V]
 C. Villegas, On qualitative probability algebras, Ann. of Math. Statist. 35 (1964), 17871796. MR 0167588 (29:4860)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512339645
PII:
S 00029939(1995)12339645
Article copyright:
© Copyright 1995
American Mathematical Society
