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A Genesis of Interval Orders and Semiorders: Transitive NaP-preferences

Published on Jul 1, 2014in Order0.424
· DOI :10.1007/s11083-013-9298-0
Alfio Giarlotta11
Estimated H-index: 11
(University of Catania)
Abstract
A NaP-preference (necessary and possible preference) on a set A is a pair \({\left(\succsim^{^{_N}}\!,\,\succsim^{^{_P}}\!\right)}\) of binary relations on A such that its necessary component \({\succsim^{^{_N}} \!\!}\) is a partial preorder, its possible component \({\succsim^{^{_P}} \!\!}\) is a completion of \({\succsim^{^{_N}} \!\!}\), and the two components jointly satisfy natural forms of mixed completeness and mixed transitivity. We study additional mixed transitivity properties of a NaP-preference \({\left(\succsim^{^{_N}}\!,\,\succsim^{^{_P}}\!\right)}\), which culminate in the full transitivity of its possible component \({\succsim^{^{_P}} \!\!}\). Interval orders and semiorders are strictly related to these properties, since they are the possible components of suitably transitive NaP-preferences. Further, we introduce strong versions of interval orders and semiorders, which are characterized by enhanced forms of mixed transitivity, and use a geometric approach to compare them to other well known preference relations.
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