Numerical Representations of Interval Orders

In the framework of the analysis of orderings whose associated indifference relation is not necessarily transitive, we study the structure of an interval order and its representability through a pair of real-valued functions. We obtain a list of characterizations of the existence of a representation, showing that the three main techniques that have been used in the literature to achieve numerical representations of interval orders are indeed equivalent.

[1]  G. Debreu.  Mathematical Economics: Continuity properties of Paretian utility , 1964 .

[2]  G. Cantor,et al.  Beiträge zur Begründung der transfiniten Mengenlehre. (Zweiter Artikel.) , 2022 .

[3]  J. Jaffray.  Existence of a Continuous Utility Function: An Elementary Proof , 1975 .

[4]  P. Fishburn.  Interval representations for interval orders and semiorders , 1973 .

[5]  E. Induráin,et al.  Representability of Interval Orders , 1998 .

[6]  G. Debreu.  Mathematical Economics: Representation of a preference ordering by a numerical function , 1983 .

[7]  Jean-Paul Doignon,et al.  On realizable biorders and the biorder dimension of a relation , 1984 .

[8]  G. Bosi,et al.  Representing preferences with nontransitive indifference by a single real-valued function☆ , 1995 .

[9]  D. Bridges.  Representing interval orders by a single real-valued function , 1985 .

[10]  G. Herden.  On the existence of utility functions ii , 1989 .

[11]  R. Larsen,et al.  The functional equationf(t)f(s)g(ts)=g(t)g(s) , 1970 .

[12]  Esteban Induráin Eraso,et al.  Representación numérica de órdenes totales , 1990 .

[13]  Esteban Induráin Eraso,et al.  Sobre caracterizaciones topológicas de la representabilidad de cadenas mediante funciones de utilidad , 1990 .

[14]  G. Cantor.  Beiträge zur Begründung der transfiniten Mengenlehre , 1897 .

[15]  S. Gensemer.  On relationships between numerical representations of interval orders and semiorders , 1987 .

[16]  P. Fishburn.  Intransitive indifference with unequal indifference intervals , 1970 .

[17]  D. Bridges.  Numerical representation of interval orders on a topological space , 1986 .

[18]  A. Chateauneuf.  Continuous representation of a preference relation on a connected topological space , 1987 .

[19]  Peter C. Fishburn,et al.  Utility theory for decision making , 1970 .

[20]  Peter C. Fishburn,et al.  Intransitive Indifference in Preference Theory: A Survey , 1970, Oper. Res..

[21]  D. Bridges.  Numerical representation of intransitive preferences on a countable set , 1983 .