Continuity of utility functions representing fuzzy preferences

In a previous paper, we established necessary and sufficient conditions for a given binary fuzzy relation to be representable by a utility function. In this article, we construct a crisp order topology associated to a given weakly complete fuzzy pre-order and introduce the notion of “continuous fuzzy pre-order.” We show that this new condition and the conditions introduced in the previous paper are together necessary and sufficient for a numerical representation of a given weakly complete fuzzy pre-order by a continuous utility function.

[1]  Nicolas Gabriel Andjiga,et al.  Utility function of fuzzy preferences on a countable set under max-*-transitivity , 2007, Soc. Choice Welf..

[2]  Henri Gwet,et al.  Fuzzy utility and non cardinal : Representation of preferences , 2000 .

[3]  Nicolas Gabriel Andjiga,et al.  Fuzzy strict preference and social choice , 2005, Fuzzy Sets Syst..

[4]  Magalì E. Zuanon,et al.  Representation of preference orderings on totally ordered semigroups , 2000 .

[5]  H. Wold.  A synthesis of pure demand analysis: Part II , 1943 .

[6]  C. R. Barrett,et al.  On choosing rationally when preferences are fuzzy , 1990 .

[7]  Gregory S. Richardson,et al.  The structure of fuzzy preferences: Social choice implications , 1998 .

[8]  Sergei Ovchinnikov,et al.  Numerical representation of transitive fuzzy relations , 2002, Fuzzy Sets Syst..

[9]  Antoine Billot,et al.  An existence theorem for fuzzy utility functions: A new elementary proof , 1995, Fuzzy Sets Syst..

[10]  Louis Aimé Fono,et al.  On strict lower and upper sections of weakly complete fuzzy pre-orders based on co-implication , 2008, Fuzzy Sets Syst..

[11]  Bernard De Baets,et al.  Additive decomposition of fuzzy pre-orders , 2007, Fuzzy Sets Syst..

[12]  Bhaskan Dutta,et al.  Fuzzy preferences and social choice , 1987 .

[13]  Bernard De Baets,et al.  A compendium of fuzzy weak orders: Representations and constructions , 2007, Fuzzy Sets Syst..

[14]  Bernard De Baets.  Coimplicators, the forgotten connectives. , 1997 .

[15]  Herman Wold,et al.  A synthesis of pure demand analysis , 1943 .

[16]  Louis Aimé Fono,et al.  On strict lower and upper sections of fuzzy orderings , 2003, Fuzzy Sets Syst..

[17]  Ashley Piggins,et al.  Instances of Indeterminacy , 2007 .