Abstract
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.
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Aczél, J. and Dhombres, J. (1991) Functional Equations in Several Variables, Cambridge University Press, Cambridge, UK.
Bosi, G. and Isler, R. (1995) Representing preferences with nontransitive indifference by a single real-valued function, J. Math. Economics 24, 621-631.
Bridges, D. S. (1983) Numerical representation of intransitive preferences on a countable set, J. Economic Theory 30, 213-217.
Bridges, D. S. (1985) Representing interval orders by a single real-valued function, J. Economic Theory 36, 149-155.
Bridges, D. S. (1986) Numerical representation of interval orders on a topological space, J. Economic Theory 38, 160-166.
Bridges, D. S. and Mehta, G. B. (1995) Representations of Preference Orderings, Springer, Berlin.
Candeal, J. C. and Induráin, E. (1990a) Sobre caracterizaciones topológicas de la representabilidad de cadenas mediante funciones de utilidad, Revista Espanõla de Economía 7, 235-244.
Candeal, J. C. and Induráin, E. (1990b) Representación numérica de órdenes totales, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales de Madrid 84, 415-428.
Cantor, G. (1895) Beiträge zur Begründung der transfinite Mengenlehre I, Mathematische Annalen 46, 481-512.
Cantor, G. (1897) Beiträge zur Begründung der transfinite Mengenlehre II, Mathematische Annalen 49, 207-246.
Chateauneuf, A. (1987) Continuous representation of a preference relation on a connected topological space, J. Math. Economics 16, 139-146.
Debreu G. (1954) Representation of a preference ordering by a numerical function, in R. Thrall, C. Coombs and R. Davies (eds), Decision Processes, Wiley, New York.
Debreu, G. (1964) Continuity properties of Paretian utility, Internat. Economic Rev. 5, 285-293.
Doignon, J. P., Ducamp, A. and Falmagne, J. C. (1984) On realizable biorders and the biorder dimension of a relation, J. Math. Psychol. 28, 73-109.
Fishburn, P. C. (1970a) Intransitive indifference with unequal indifference intervals, J. Math. Psychol. 7, 144-149.
Fishburn, P. C. (1970b) Intransitive indifference in preference theory: A survey, Oper. Res. 18(2), 207-228.
Fishburn, P. C. (1970c) Utility Theory for Decision-Making, Wiley, New York.
Fishburn, P. C. (1973) Interval representations for interval orders and semiorders, J. Math. Psychol. 10, 91-105.
Fishburn, P. C. (1985) Interval Orders and Interval Graphs, Wiley, New York.
Gensemer, S. H. (1987) On relationships between numerical representations of interval orders and semiorders, J. Economic Theory 43, 157-169.
Herden, G. (1989) On the existence of utility functions, Math. Social Sci. 17, 297-313.
Jaffray, J. Y. (1975) Existence of a continuous utility function: an elementary proof, Econometrica 43, 981-983.
Oloriz, E., Candeal, J. C. and Induráin, E. (1998) Representability of interval orders, J. Economic Theory 78(1), 219-227.
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Bosi, G., Candeal, J.C., Induráin, E. et al. Numerical Representations of Interval Orders. Order 18, 171–190 (2001). https://doi.org/10.1023/A:1011974420295
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DOI: https://doi.org/10.1023/A:1011974420295