# 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.
• References (49)
• Citations (15)
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References49
#1Salvatore Corrente (University of Catania)H-Index: 17
#2Salvatore Greco (University of Catania)H-Index: 56
Last. Roman Slowifiski (PUT: Poznań University of Technology)H-Index: 69
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Within disaggregation–aggregation approach, ordinal regressionaims at inducing parameters of a preference model, for example, parameters of a value function, which represent some holistic preference comparisons of alternatives given by the Decision Maker (DM). Usually, from among many sets of parameters of a preference model representing the preference information given by the DM, only one specific set is selected and used to work out a recommendation. For example, while there exist many value f...
#1Alfio Giarlotta (University of Catania)H-Index: 11
#2Salvatore Greco (University of Catania)H-Index: 56
A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive and complete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold in many applications, preferences are often modeled by means of weaker binary relations, dropping either completeness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders). We introduce an alternative approach to preference modeling, which uses two binary relations–...
#1Gerhard HerdenH-Index: 11
#2Vladimir L. Levin (Central Economics and Mathematics Institute)H-Index: 5
We prove the existence of arbitrary (resp., semicontinuous, continuous) utility representations for arbitrary (resp., semicontinuous, continuous) preorders satisfying some weakened Debreu order separability conditions. In this way we widely generalize a classical result for total preorders that essentially is due to Debreu.
#1Özgür Evren (New Economic School)H-Index: 6
#2Efe A. Ok (NYU: New York University)H-Index: 29
We develop the ordinal theory of (semi)continuous multi-utility representation for incomplete preference relations. We investigate the cases in which the representing sets of utility functions are either arbitrary or finite, and those cases in which the maps contained in these sets are required to be (semi)continuous. With the exception of the case where the representing set is required to be finite, we find that the requirements of such representations are surprisingly weak, pointing to a wide ...
#1Juan Carlos Candeal (University of Zaragoza)H-Index: 16
#2Esteban Induráin (Universidad Pública de Navarra)H-Index: 17
Abstract We find an internal characterization of the so-called Scott–Suppes representability of a semiorder, defined as the existence of a numerical representation for a semiorder through a real-valued utility function and a positive constant threshold of discrimination.
#1Salvatore GrecoH-Index: 56
#2Roman SlowifiskiH-Index: 69
Last. Vincent MousseauH-Index: 28
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#1Alfio Giarlotta (University of Catania)H-Index: 11
#2Stephen Watson (York University)H-Index: 11
A linear ordering is Debreu (respectively, pointwise Debreu) if each of its suborderings can be mapped into it with an order-preserving function that is both injective (respectively, locally injective) and continuous (respectively, locally continuous) with respect to the order topology on both spaces. Each Debreu linear ordering is pointwise Debreu, but the converse does not hold. In the context of utility representations in mathematical economics, it has been proved that any lexicographic power...
#1Gianni Bosi (UniTS: University of Trieste)H-Index: 12
#2Javier Gutiérrez García (UPV/EHU: University of the Basque Country)H-Index: 10
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We introduce a new approach that deals, jointly and in a unified manner, with the topics of numerical (continuous) representability of total preorders and interval orders. This setting is based on the consideration of increasing scales and the systematic use of a particular kind of codomain, that has a key lattice theoretical structure of a completely distributive lattice and allows us to use a single function (taking values in that codomain) in order to represent both kinds of binary relations.
#1Francisco J. Abrísqueta (Universidad Pública de Navarra)H-Index: 2
#2Juan Carlos Candeal (University of Zaragoza)H-Index: 16
Last. Margarita Zudaire (Universidad Pública de Navarra)H-Index: 6
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We analyze the structure of a semiorder, paying attention to its representability through a real-valued function and a positive constant threshold (the so-called Scott-Suppes representation). We furnish a new set of sufficient conditions for the Scott-Suppes representability of semiorders. Unlike previous characterizations already introduced in the literature, these new conditions can be expressed directly in terms of the given semiordered structure.
#1Agata Caserta (Seconda Università degli Studi di Napoli)H-Index: 6
#2Alfio Giarlotta (University of Catania)H-Index: 11
Last. Stephen Watson (York University)H-Index: 11
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Traditionally the codomain of a utility function is the set of real numbers. This choice has the advantage of ensuring the existence of a continuous representation but does not allow to represent many preference structures that are relevant to utility theory. Recently, some authors have started a systematic study of utility representations that are not real-valued, introducing the notion of a Debreu chain. We continue their analysis defining two Debreu-like properties, which are connected to a l...
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#1Alfio Giarlotta (University of Catania)H-Index: 11
#2Stephen Watson (York University)H-Index: 11
Abstract Consumers’ preferences and choices are traditionally described by appealing to two classical tenets of rationality: transitivity and completeness. In 1971, Schmeidler proved a striking result on the interplay between these properties: On a connected topological space, a nontrivial bi-semicontinuous preorder is complete. Here we reformulate and extend this well-known theorem. First, we show that the topology is not independent of the preorder, contrary to what the original statement sugg...
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On taking a non-trivial and semi-transitive bi-relation constituted by two (hard and soft) binary relations, we report a (i) p-continuity assumption that guarantees the completeness and transitivity of its soft part, and a (ii) characterization of a connected topological space in terms of its attendant properties on the space. Our work generalizes antecedent results in applied mathematics, all following Eilenberg (1941), and now framed in the context of a parametrized-topological space. This re-...
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#1M. Ali Khan (Johns Hopkins University)H-Index: 26
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On taking a non-trivial and semi-transitive bi-relation constituted by two (hard and soft) binary relations, we report a (i) p-continuity assumption that guarantees the completeness and transitivity of its soft part, and a (ii) characterization of a connected topological space in terms of its attendant properties on the space. Our work generalizes antecedent results in applied mathematics, all following Eilenberg (1941), and now framed in the context of a parametrized-topological space. This re-...
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Abstract Under a continuity hypothesis on bi-preferences defined on a topologically-connected choice-set, as formalized by Giarlotta–Greco (2013), this letter reports that a mildly consistent, double-minded decision-maker is single-minded in the sense of being fully consistent and decisive. The results generalize recent work of Giarlotta–Watson (2019), and thereby provide a far-reaching generalization of a result of Schmeidler (1971) that has received considerable recent attention, and extend to...
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A choice space is a finite set of alternatives endowed with a map associating to each menu a nonempty subset of selected items. A congruence on a choice space is an equivalence relation that preserves its structure. Intuitively, two alternatives are congruent if the agent is indifferent between them, and, in addition, her choice is influenced by them in exactly the same way. We give an axiomatic characterization of the notion of congruence in terms of three natural conditions: binary fungibility...
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