A generalization of interval orders

Abstract We provide an answer to an open problem concerning the representation of preferences by intervals. Given a finite set of elements and three relations on this set (indifference, weak preference and strict preference), necessary and sufficient conditions are provided for representing the elements of the set by intervals in such a way that 1) two elements are indifferent when the interval associated to one of them is included in the interval associated to the other; 2) an element is weakly preferred to another when the interval of the first is “more to the right” than the interval of the other, but the two intervals have a non empty intersection; 3) an element is strictly preferred to another when the interval of the first is “more to the right” than the interval of the other and their intersection is empty.

[1]  Rian A.W. van Blokland-Vogelesang.  Unfolding and Consensus Ranking: A Prestige Ladder for Technical Occupations , 1989 .

[2]  J. Michell.  An introduction to the logic of psychological measurement , 1995 .

[3]  D. Kendall.  Incidence matrices, interval graphs and seriation in archeology. , 1969 .

[4]  J. Marden,et al.  Use of Nested Orthogonal Contrasts in Analyzing Rank Data , 1992 .

[5]  Rian A. W. van Blokland-Vogelesang,et al.  A Nonparametric Distance Model for Unidimensional Unfolding , 1993 .

[6]  Ayala Cohen,et al.  Analysis of large sets of ranking data , 1982 .

[7]  Adrianus Henricus Gerardus Silverius van der Ven,et al.  Introduction to scaling , 1980 .

[8]  L. A. Goodman.  The Analysis of Cross-Classified Data: Independence, Quasi-Independence, and Interactions in Contingency Tables with or without Missing Entries , 1968 .

[9]  L. Hubert.  SERIATION USING ASYMMETRIC PROXIMITY MEASURES , 1976 .

[10]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[11]  L. Hubert.  Assignment methods in combinatorial data analysis , 1986 .

[12]  J. Marden.  Analyzing and Modeling Rank Data , 1996 .

[13]  C. Coombs.  A theory of data. , 1965, Psychology Review.

[14]  Leo A. Goodman,et al.  Some multiplicative models for the analysis of cross classified data , 1972 .

[15]  Lawrence Hubert,et al.  Problems of seriation using a subject by item response matrix. , 1974 .

[16]  A. Agresti.  An introduction to categorical data analysis , 1997 .

[17]  Leo A. Goodman,et al.  Association Models and Canonical Correlation in the Analysis of Cross-Classifications Having Ordered Categories , 1981 .

[18]  Rian A. W. van Blokland-Vogelesang,et al.  Unfolding and group consensus ranking for individual preferences , 1991 .

[19]  D. Kendall.  Abundance matrices and seriation in archaeology , 1971 .

[20]  C. H. Coombs,et al.  Mathematical psychology : an elementary introduction , 1970 .

[21]  Joseph S. Verducci,et al.  Probability Models and Statistical Analyses for Ranking Data , 1992 .

[22]  Joseph S. Verducci,et al.  Probability models on rankings. , 1991 .

[23]  Harold Gulliksen,et al.  Contributions to mathematical psychology , 1964 .

[24]  C H COOMBS,et al.  Psychological scaling without a unit of measurement. , 1950, Psychological review.