Mathematical Social Sciences 38 (1999) 21–33 Variable intervals model * Marc-Arthur Diaye ´ Universite de Paris 1, LAMIA, Paris, France Received 1 June 1997; received in revised form 1 July 1998; accepted 1 August 1998 Abstract The variable intervals model is a generalization of Fishburn’s Intervals model. It fully characterizes the complete acyclic relation when the alternatives set is countable. In the uncountable case, a perfect separability condition has to be added.  1999 Elsevier Science B.V. All rights reserved. Keywords: Acyclicity; Representation; Underlying preference JEL classification: C00; D00 1. Introduction A number of models are now in use for the modeling of preferences. The most famous are the semiorder model and the interval order model (a review of the variants of the semiorder model can be found in Fishburn, 1997). However, a common point in all of these models is that their asymmetric parts are transitive. The purpose of this paper is to study the relationship between the variable intervals model (first formulated by Abbas and Vincke, 1993), which is a generalization of the intervals model (Fishburn, 1970), and the complete acyclic binary relations. The paper is organized as follows. Section 2 reviews some basic definitions in ordered sets theory. Section 3 is devoted to the variable intervals model. This model characterizes the complete acyclic relations when the set of alternatives is countable. In the uncountable case, a particular separability condition has to be added. According to the variable intervals model, an agent having a complete acyclic preference has an underlying true preorder (complete and transitive) preference. Section 4 deals with the issue of the best approximation of this underlying preference. Section 5 presents some conclusions. * E-mail address: diaye@univ-paris1.fr (M.-A. Diaye) 0165-4896 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S0165-4896(98)00034-1