Fuzzy Choice Functions, Consistency, and Sequential Fuzzy Choice
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16th World Congress of the International Fuzzy Systems Association (IFSA) 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT) Fuzzy choice functions, consistency, and sequential fuzzy choice José Carlos R. Alcantud,1 S. Díaz2 1Facultad de Economía y Empresa and IME, University of Salamanca, 37007 Salamanca, Spain 2Dept. of Statistics and O. R., University of Oviedo, Calvo Sotelo s/n, 33007 Oviedo, Spain Abstract by this sequential procedure. This implies that the same is true for full-rationality (i.e., for the In the setting of fuzzy choice functions existence of a well-behaved fuzzy relation that G- (Georgescu [6]), we explore the relationships rationalizes the fuzzy choice). among known and new consistency axioms. Then Secondly, when the sequential application of we define the notion of sequential application of G-rational fuzzy choice functions produces a G- fuzzy choice functions, and investigate its nor- rational fuzzy choice function, the fuzzy preference mative implications. The Fuzzy Arrow Axiom that G-rationalizes their sequential application is is preserved by this sequential procedure, which identified. ensures that full-rationality is preserved too. This paper is structured as follows. In Section 2 we recall some preliminary definitions and concepts Keywords: Choice function, Fuzzy choice function, from the crisp literature. In Section 3 we introduce Fuzzy Arrow Axiom. the fuzzy set theory and discuss the connection be- tween some consistency conditions in this context. 1. Introduction In Section 4 we present the results obtained for se- quential choices. We conclude in Section 5. In the act of choice, rationality is a central issue that has raised many controversial analysis. There 2. The Crisp Case is a trend in the literature that considers choice as rational only when derives from well-behaved pref- The remote antecedent of our investigation is the erences. Transitivity of preference is the paradig- following model of abstract choice. matic rational principle to obey, and it is assumed The collection of all situations that a decision- by many descriptive theories of decision making. maker can conceivably face are represented by B, a However experimental evidence contradicts this po- nonempty domain of nonempty subsets of X. That sition, as recognized as soon as in May [9] and Tver- is, if P(X) denotes the set of all subsets of X then sky [12]. we require ∅ 6= B ⊆ P(X) and S 6= ∅ for all S ∈ B. Therefore other authors attempted to study the For convenience sometimes we drop brackets, e.g., object of choice –formally, choice functions– and dis- the set {x} can be represented as x. cuss which possible rationality properties could be Choices (among the subsets in B) are captured by tested without any reference to an underlying pref- the following notion: erence. As soon as in Uzawa [13] the term “rational choice function” means a choice function satisfying Definition 1 A decisive choice correspondence on axioms of rational behavior. Nevertheless the idea (X, B) is a map C : B → P(X) such that C(S) ⊆ S that these axioms can be used to test the model and C(S) 6= ∅ for all S ∈ B. by optimization of nice preferences with the observ- ables (the actual choices) is central in the analysis Henceforth we simply refer to decisive choice cor- of economic or abstract decisions (cf., Arrow [1] and respondences as choice functions. They attach with Sen [10] among others). each feasible set in the domain B, the set of alterna- If we recognize that social phenomena involve in- tives that are either potentially or actually chosen trinsically vague concepts then we can perform more by the decision-maker. extensive studies in the lines highlighted above. In Particular specifications of this abstract model this work we are interested in fuzzy choice functions call for suitably adapted terms. For example in as defined in Georgescu [6]. Economics, demand theory is concerned with de- We explore the relationships among known and mand functions or demand correspondences, whose new consistency axioms for this setting. Then we domain is constituted by budget sets. define the new concept of sequential fuzzy choice functions, and explore its normative implications. 2.1. Consistency properties Our main findings are two when the minimum t- norm is assumed. Firstly, a foremost consistency A good deal of the literature on choice functions is axiom known as Fuzzy Arrow Axiom is preserved devoted to study properties of choice that embody © 2015. The authors - Published by Atlantis Press 564 rationality in various forms and their relationships. the choice function can be rationalized through We proceed to list some rationality or consistency some specific procedure constitute another impor- properties of choice that we are interested in. tant branch in choice theory, especially in revealed The strongest version of the consistency axioms preference theory. These relationships and char- that we define is the crisp Arrow Axiom (AA), acterizations rely on the structure of the domain which demands that for any S, T ∈ B, the following of subsets for which choices are defined or known. implication holds: Particularly, we may be bound by the following con- straints: S ⊆ T ⇒ either S ∩ C(T ) = ∅ or S ∩ C(T ) = C(S) . Definition 3 Condition H is satisfied if B contains Other weaker properties used to capture rationality all non-empty finite subset of X. Condition WH in the context of crisp choice functions follow: is satisfied if B contains at least all the pairs and Definition 2 The choice function C satisfies: triplets of alternatives. • The Chernoff condition, also CH, if for any To grasp the importance of assumption H we re- S, T ∈ B such that S ⊆ T we have call that already Arrow [1] emphasized in his in- sightful analysis on the theory of consumer’s de- C(T ) ∩ S ⊆ C(S). mand: “the demand-function point of view would • The Superset property, also SUP, if for all be greatly simplified if the range over which the S, T ∈ B such that S ⊆ T it is the case that choice functions are considered to be determined is broadened to include all finite sets” and also: “re- C(T ) ⊆ C(S) ⇒ C(T ) = C(S) . quiring the choice functions to be defined for finite sets is thoroughly consistent with the intuitive ar- • Property γ (cf., Sen [10]), also γ, if for any guments underlying revealed preference. It should collection {M } of subsets of B the following i i∈I also be observed that any hope of using experimen- holds true: tal methods for studying preference will require in- x ∈ C(Mi) for all i ∈ I entails x ∈ C(∪i∈I Mi) . ferring from choices on finite sets to choices on in- finite ones.” Even under assumption WH, he shows This property is stronger than the Concor- among other issues that Samuelson’s weak axiom of dance property which only establishes that revealed preference is equivalent to the strong ax- C(S) ∩ C(T ) ⊆ C(S ∪ T ) throughout. iom of revealed preference and to Arrow’s Axiom. • The Binariness property, also B, if for any And that they are equivalent to the property that S ∈ B we have: x ∈ S and x ∈ C(x, y) for all the choice function is rationalizable in the most sat- y ∈ S implies x ∈ C(S). isfactory manner: there exists a complete and tran- sitive relation Q such that for each S ∈ B, C(S) is 2.2. Relationships among consistency the set of maximizers of R in S, this is, properties C(S) = {x ∈ S | (x, y) ∈ R for all y ∈ S} . In the crisp case there are implications among the properties in Subsection 2.1, which are represented 3. The Fuzzy Case: a Study in Consistency by Figure 1. Neither of them can be reversed. More- over, there is no connection among the properties on We now move forward to the fuzzy context. We the right, except for the ones shown between Prop- begin with the analysis of the issues that Section 2 erty γ and Binariness and Property γ and Concor- motivates. The main concept that we need to es- dance. tablish is the following (cf., Georgescu [6, Definition 5.13]): Chernoff condition Definition 4 Let X be a non-empty set and B a non empty set of non-zero fuzzy subsets of X. Superset property A fuzzy choice function on (X, B) is a mapping Arrow Axiom ⇒ Property γ ⇒ Binariness C : B → F(X) such that for each S ∈ B, C(S) is non-zero and C(S) ⊆ S, i.e., C(S)(x) ≤ S(x) for ⇓ all x ∈ X. Concordance S(x) is called the availability degree of the alter- native x in S or if S represents a vague criterion, Figure 1: Implications among consistency proper- it captures the degree to which x verifies the crite- ties of crisp choice functions. rion. Definition 4 includes the class of fuzzy choice functions originally studied by Banerjee [2]. The relationships among the consistency prop- In order to define consistency properties of fuzzy erties mentioned above and the possibility that choice functions, we first need to recall the concepts 565 that permit to use intersection, union, implications 3.1. Consistency of fuzzy choice functions: and inclusions in the fuzzy context. some new axioms The intersection of fuzzy sets is usually defined by Let X be a non-empty set and B a non empty set of the minimum t-norm, and we follow this tradition. non-zero fuzzy subsets of X. We proceed to define A t-norm is a binary operation ∗ on [0, 1] that is some known and new axioms of consistency for C, commutative, associative, monotone and has 1 as a fuzzy choice function on (X, B).