Representation of preference relations on σ -algebras of nonatomic measure spaces: Convexity and continuity

Authors:
Nobusumi Sagara;Milan Vlach
Affiliations:
Faculty of Economics, Hosei University, 4342, Aihara, Machida, Tokyo 194--0298, Japan;School of Mathematics and Physics, Charles University, Malostranské náměstí 25, 118 00 Praha 1, Czech Republic
Venue:
Fuzzy Sets and Systems
Year:
2009

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Abstract

The purpose of this paper is to show how preference relations on @s-algebras can be represented by means of nonadditive set functions that satisfy appropriate requirements of convexity and continuity on @s-algebras. To this end, we introduce convex combinations of measurable sets, and quasiconcave and concave functions on @s-algebras, which conform with the standard results in convex analysis. We formulate the convexity and the continuity axioms for preference relations on @s-algebras with a metric topology and show the existence of utility functions for convex continuous preference relations. We also show that monotone continuous preference relations are representable by fuzzy measures.