Journal of Mathematical Psychology 81 (2017) 98–109 Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp Necessary and possible indifferences Alfio Giarlotta a, *, Stephen Watson b a Department of Economics and Business, University of Catania, Italy b Department of Mathematics and Statistics, York University, Toronto, Canada highlights • A necessary and possible indifference is a suitable pair of nested symmetric relations on a set of alternatives. • The symmetric relations induced by a NaP-preference form a necessary and possible indifference. • Necessary and possible indifferences are characterized by the existence of a family of equivalence relations. • Necessary and possible indifference naturally arise in applications, for instance in the field of choice theory. • We classify necessary and possible indifferences in two types: derived (from a NaP-preference) and primitive. article info Article history: Received 9 December 2016 Received in revised form 30 August 2017 Available online 31 October 2017 Keywords: Preference modeling NaP-preference NaP-indifference Resolution Equivalence relation Similarity relation Revealed preference Primitive similarity Comparability graph abstract A NaP-preference (necessary and possible preference) is a pair of nested reflexive relations on a set such that the smaller is transitive, the larger is complete, and the two relations jointly satisfy properties of transitive coherence and mixed completeness. It is known that a NaP-preference is characterized by the existence of a set of total preorders whose intersection and union give its two components. We introduce the symmetric counterpart of a NaP-preference, called a NaP-indifference: this is a pair of nested symmetric relations on a set such the smaller is an equivalence relation, and the larger is a transitively coherent extension of the first. A NaP-indifference can be characterized by the existence of a set of equivalence relations whose intersection and union give its two components. NaP-indifferences naturally arise in applications: for instance, in the field of individual choice theory, suitable pairs of similarity relations revealed by a choice correspondence yield a NaP-indifference. We classify NaP-indifferences in two categories, according to their genesis: (i) derived, which are canonically obtained by taking the symmetric part of a NaP-preference; (ii) primitive, which arise independently of the existence of an underlying NaP-preference. This partition into two classes turns out to be related to the notion of incomparability graph. © 2017 Elsevier Inc. All rights reserved. 1. Introduction The classical way to represent the (non-stochastic) preference structure of an economic agent on a set of alternatives is by means of a binary relation satisfying suitable order properties, which are usually forms of transitivity and/or completeness. Preorders, semiorders (Luce, 1956; Pirlot & P.Vincke, 1997), and interval or- ders (Fishburn, 1970, 1985) are the binary relations that are often used for the modelization of preference structures, due to their intrinsic properties: see (Aleskerov, Bouyssou, & Monjardet, 2007; Pirlot & P.Vincke, 1997) and references therein. * Corresponding author. E-mail addresses: giarlott@unict.it (A. Giarlotta), watson@mathstat.yorku.ca (S. Watson). A very recent approach to preference modeling employs instead a pair of interconnected binary relations on the same set of alterna- tives. This bi-preference approach has the advantage of allowing a more flexible modelization of an economic agent’s (or a set of eco- nomic agents’) preference structure in several scenarios. The two preference relations are nested into each other, and are connected by (economically and psychologically) meaningful properties. The main feature of a bi-preference structure is that the two ‘‘core properties’’ of transitivity and completeness are not required to fully hold for both relations, instead they are suitably spread over the combination of the two relations. NaP-preferences (necessary and possible preferences) belong to the family of bi-preferences. Originally, NaP-preferences were introduced in the field of Multiple Criteria Decision Aid, in the process of constructing a new methodology called Robust Ordi- nal Regression (Greco, Mousseau, & Słowiński, 2008). However, https://doi.org/10.1016/j.jmp.2017.09.006 0022-2496/© 2017 Elsevier Inc. All rights reserved.