A NaP-preference (necessary and possible preference) is a pair of nested reflexive relations on a set such that the smaller (necessary) is transitive, the larger (possible) is complete, and the two components jointly satisfy natural forms of mixed completeness and transitive coherence . A NaP-preference is normalized if the necessary component is a partial order. NaP-preferences were originally introduced in Multiple Criteria Decision Aid within the framework of Robust Ordinal Regression (ROR) . In ROR, all information provided by an economic agent on a set of multi-dimensional alternatives is used to build a compatible family of additive value functions. Then, a (usually normalized) NaP-preference arises from this family by universal and existential quantification. Recent research on NaP-preferences is devoted to the study of their properties, as well as to the analysis of related preference structures (see, e.g., [2, 3, 4]). In fact, NaP-preferences are linked to interval orders and semiorders. The collection of all normalized NaP-preferences on a set can be arranged into a poset, whose ordering relation is semantically connected to the informative content of their two components: “positive” for the necessary preference, and “negative” for the possible one. Upon taking into account both types of information (and allowing no compensation between them), normalized NaP-preferences are partially ordered according to their stability . We show in  that the poset of normalized NaP-preferences on a finite set of alternatives is well-graded in the sense of Doignon and Falmagne . This means that this poset is rich enough to ensure that we can efficiently “migrate” from a NaP-preference to a different one by modifying (i.e., adding or eliminating) one connection (i.e., one edge) at a time. The well-gradedness of this poset paves the way toward developing a stochastic theory of NaP-preferences, where a smooth transition of states is done via the random introduction of quantum tokens of information, similarly to what has been done for semiorders. Doignon, J.-P., Falmagne, J.-C. (1997). Well-graded families of relations. Disc. Math. 173, 35-44. Giarlotta, A. (2014). A genesis of interval orders and semiorders: Transitive NaP-preferences. Order 31, 239-258. Giarlotta, A., Greco, S. (2013). Necessary and possible preference structures. J. Math. Econ. 42/1, 163-172. Giarlotta, A., Watson, S. (2017). Well-graded families of NaP-preferences. J. Math. Psychol. 77, 21-28 Greco, S., Mousseau, V., Slowinski R. (2008). Ordinal regression revisited: Multiple criteria ranking with a set of additive value functions. Eur. J. Oper. Res. 191, 415-435.
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