Arrow axiom and full rationality for fuzzy choice functions Irina Georgescu∗ ˙ Akademi University, Institute for Turku Centre for Computer Science, Abo Advanced Management Systems Research, Lemmink¨ aisenkatu 14, FIN-20520 Turku, Finland Abstract A classical result for crisp choice functions shows the equivalence between Arrow axiom and the property of full rationality. In this paper we study a fuzzy form of Arrow axiom formulated in terms of the subsethood degree and of the degree of equality (of fuzzy sets). We prove that a fuzzy choice function verifies Fuzzy Arrow Axiom if and only if it is (fuzzy) full rational. We also show that these conditions are also equivalent with weak and strong fuzzy congruence axioms W F CA and SF CA. It is studied the Arrow index, a new concept that indicates the degree to which a fuzzy choice function verifies the Fuzzy Arrow Axiom. Keywords: Fuzzy choice function; Full rationality; Arrow Axiom
1
Introduction
The rationality of a consumer is a frequent research topic in classical consumer theory. By Uzawa [18] ”the rationality of a consumer may be described by postulating that the consumer has a definite preference over all conceivable bundles and that he chooses those commodity bundles that are optimum with respect to his preference subject to budgetary constraints”. Samuelson’s theory of revealed preference [11] expresses the rationality of a consumer in terms of some preference relations associated with a demand function. Uzawa [17] and Arrow [1] have developed a revealed preference theory in an abstract framework. The work of Uzawa and Arrow was continued by Richter [10], Sen [12, 13, 14], Suzumura [15, 16] and many others. Following [16], a choice space is a pair X, B where X is a non-empty universe of alternatives and B a family of non-empty subsets of X. In the terminology of consumers the pair X, B is called a budget space ; the elements of X ∗ Corresponding author. Tel.: +358-2-2153339; fax: +358-2-215-4809. E-mail address:
[email protected] (I. Georgescu)
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are called bundles and the sets in B are called budgets . Then a set S ∈ B can be taken as an available set of alternatives. A choice function (=consumer ) on a choice space X, B is a function C : B → P(X) which to any S ∈ B assigns a non-empty subset C(S) of S. C(S) is called the choice set of S. By the rationality of C we mean to find a preference relation Q on X such that for any available set S the choice set C(S) coincides with the set of Q-greatest elements of S. Thus C is rational and Q is a rationalization of C. If Q is reflexive, transitive and total then we say that C is full rational . The results in [1, 12, 13] are obtained assuming that B contains the nonempty finite subsets of X. In this framework, by [16], p. 28, the full rationality of a choice function C is equivalent to the following axiom introduced by Arrow in [1]: (AA) For any S1 , S2 ∈ B, if S1 ⊆ S2 then S1 ∩ C(S2 ) = ∅ or S1 ∩ C(S2 ) = C(S1 ). The aim of this paper is to obtain an extension of this result in the context of fuzzy choice functions. In [2] Banerjee introduced a class of fuzzy choice functions and studied their fuzzy revealed preference theory. We work with a more general general definition of fuzzy choice functions. Banerjee fuzzifies only the range of a choice function; in our approach both the domain and the range of a choice function are made of fuzzy subsets of a universe of alternatives X. Papers [6, 7] develop a theory of revealed preference for these fuzzy choice functions. Section 2 contains some preliminary results on the residuated structure of the real interval [0, 1] and some basic things on fuzzy relations. Some basic definitions and results on fuzzy choice functions are included in Section 3. We formulate two hypotheses H1 and H2 as a natural fuzzy extension of the situation in [1, 12, 13, 14, 17]. The revealed preference theory of fuzzy choice functions developed in [6, 7] is based on these hypotheses. Section 4 concentrates the main contributions of this paper. We formulate the Fuzzy Arrow Axiom (F AA) in terms of the subsethood degree I(., .) and the degree of equality E(., .) [4]. We prove that if the hypotheses H1 and H2 are fulfiled then F AA is equivalent to the (fuzzy) full rationality. Then we show that these conditions are also equivalent with weak and strong fuzzy congruence axioms W F CA and SF CA [6]. The main tool in proving this result is the manipulation of the residuum properties. In Section 5 we define the Arrow index of a fuzzy choice function and the similarity of fuzzy choice functions. We study the behaviour of this similarity relation with respect to the Arrow index.
2
Preliminaries
This section contains some preliminary matter with respect to the residuated structure of [0, 1] and some basic notions on fuzzy relations. The background is given by [4, 8, 21].
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For any a, b ∈ [0, 1] we denote a ∨ b =max(a, b) and a ∧ b =min(a, b). More generally, for any {ai }i∈I ⊆ [0, 1] we denote ai =sup{ai |i ∈ I} and
i∈I
ai =inf{ai |i ∈ I}. Then ([0, 1], ∨, ∧, 0, 1) becomes a bounded distributive
i∈I
lattice. Furthermore, [0, 1] is a complete distributive lattice. Then one can define a newbinary operation → on [0, 1], called implication or residuation : a → b = {c ∈ [0, 1]|a ∧ c ≤ b}. A simplecalculation yields 1 if a ≤ b a→b= b if a > b The biresiduum ↔ is a binary operation on [0, 1] defined by a ↔ b = (a → b) ∧ (b → a). The following two lemmas collect some basic properties of the residuum. Lemma 2.1 [4, 8] For any a, b, c ∈ [0, 1] the following properties hold: (1) a ∧ b ≤ c iff a ≤ b → c; (2) a ∧ (a → b) = a ∧ b; (3) a ≤ b iff a → b = 1; (4) a = 1 → a; (5) 1 = a → a; (6) a → (b → c) = (a ∧ b) → c = b → (a → c) Lemma 2.2 [4, 8] For any {ai }i∈I ⊆ [0, 1] and a ∈ [0, 1] the following properties hold: (1) ( ai ) ∧ a = (ai ∧ a); i∈I
(2) a → ( (3)(
i∈I
i∈I
ai ) =
i∈I
ai ) → a =
(a → ai );
i∈I
(ai → a).
i∈I
Let X be a non-empty set. A fuzzy subset of X is a function A : X → [0, 1]. We denote by P(X) the family of crisp subsets of X and by F (X) the family of fuzzy subsets of X. If we identify a crisp subset U of X with its characteristic function χU then we can write P(X) ⊆ F(X). For any A, B ∈ F(X), by A ⊆ B we mean that A(x) ≤ B(x) for each x ∈ X. A fuzzy subset A of X is non-zero if A(x) = 0 for some x ∈ X; A is normal if A(x) = 1 for some x ∈ X. If x1 , . . . , xn ∈ X then [x1 , . . . , xn ] will denote the characteristic function of {x1 , . . . , xn }: 3
1 if y ∈ {x1 , . . . , xn } 0 if y ∈ {x1 , . . . , xn } For A, B ∈ F(X) let us denote I(A, B) = (A(x) → B(x)); E(A, B) = (A(x) ↔ B(x)). [x1 , . . . , xn ](y) =
x∈X
x∈X
It is clear that A ⊆ B iff I(A, B) = 1 and A = B iff E(A, B) = 1. For any x ∈ X we have: I(A, B) ≤ A(x) → B(x) and E(A, B) ≤ A(x) ↔ B(x). I(A, B) is called the subsethood degree of A and B and E(A, B) the degree of equality of A and B. Intuitively I(A, B) expresses the truth value of the statement ”A is included in B” and E(A, B) the truth value of the statement ”A and B contain the same elements” (see [4], p. 82). A fuzzy preference relation R on X is a function R : X 2 → [0, 1]. R is said to be reflexive if R(x, x) = 1 for any x ∈ X; transitive if R(x, y) ∧ R(y, z) ≤ R(x, z) for any x, y, z ∈ X; symmetric if R(x, y) = R(y, x) for any x, y ∈ X; strongly total if R(x, y) = 1 or R(y, x) = 1 for any distinct x, y ∈ X. The notion of similarity relation was introduced by Zadeh [20] as a generalization of the concept of (crisp) equivalence relation. A fuzzy relation R on X is said to be a similarity relation if it is reflexive, symmetric and transitive. An n-ary fuzzy relation P : X n → [0, 1] is extensional w. r. t. a similarity relation R on X if for all x1 , . . . , xn , y1 , . . . , yn ∈ X we have n R(xi , yi ) ≤ P (y1 , . . . , yn ). P (x1 , . . . , xn ) ∧ i=1
By [4], p. 91, Lemma 3.30, the function E(., .) : F (X) × F(X) → [0, 1] defined by the assignment (A, B) → E(A, B) is a similarity relation on F (X).
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Fuzzy choice functions
In this section we recall the definition of the fuzzy choice functions introduced in [6]. We present some fuzzy preference relations associated with a fuzzy choice function and we recall some results from [6]. A fuzzy choice space is a pair X, B where X is a non-empty set of alternatives and B is a non-empty family of non-zero fuzzy subsets of X. A fuzzy choice function (=fuzzy consumer) on X, B is a function C : B → F(X) such that for any S ∈ B, C(S) is a non-zero fuzzy subset of X and C(S) ⊆ S. In terms of fuzzy consumers, X is the set of bundles and B is the family of fuzzy budgets . If x is a bundle and S a fuzzy budget then the real number C(S)(x) can be interpreted as the degree to which the bundle x is chosen subject to the fuzzy budget S. The fuzzy subset S ∈ B offers an availability degree S(x) for each x ∈ X. By identifying a crisp set with its characteristic function, our definition of a fuzzy choice function generalizes Banerjee’s [2]. In [2] the domain of a choice function is made of all non-empty finite subsets and the range is made of fuzzy 4
subsets of X. In our approach, both the domain and the range of a choice function contain fuzzy subsets of X. The results of [6, 7] are proved provided the fuzzy choice function C satisfies the following hypotheses: H1 Every S ∈ B and C(S) are normal fuzzy subsets of X; H2 B includes the fuzzy sets [x1 , . . . , xn ] for any n ≥ 1 and x1 , . . . , xn ∈ X. Since C(S) ⊆ S, in H1 it suffices to assume that C(S) is normal for each S ∈ B. For the crisp choice functions the hypothesis H1 is automatically fulfilled by the definition of such choice function: any S and C(S) are non-empty. In the same case H2 asserts that B includes all non-empty finite subsets of X, hypothesis assumed in [1, 12, 13, 17]. Let X, B be a fuzzy choice space and Q a fuzzy preference relation on X. For any S ∈ B let us define the fuzzy subset G(S, Q) of X: G(S, Q)(x) = S(x) ∧ [S(y) → Q(x, y)] y∈X
for any x ∈ X. In this way we obtain a function G(., Q) : B → F(X). Generally G(., Q) is not a fuzzy choice function: for some S ∈ B, G(S, Q) can be the zero fuzzy set. A fuzzy choice function C : B → F(X) is said to be rational if C = G(., Q) for some fuzzy preference relation Q on X. If Q is reflexive, transitive and strongly total then C is called full rational . Let C : B → F(X) be a fuzzy choice function. We define a fuzzy preference relation Q on X by putting for all x, y ∈ X: [C(S)(x) ∧ S(y)] R(x, y) = S∈B
¯ R:
If C verifies H1 and H2 then we can define another fuzzy preference relation ¯ y) = C([x, y])(x) R(x, ˆ for any x, y ∈ X. Let us denote by C the function G(., R): ˆ C(S)(x) = S(x) ∧ [S(y) → R(x, y)] y∈X
for any S ∈ B and x ∈ X. Cˆ is called the image of C. ˆ Lemma 3.1 [6] C(S)(x) ≤ C(S)(x) for any S ∈ B and x ∈ X. By Lemma 3.1, Cˆ is a fuzzy choice function on X, B. If C = Cˆ then the fuzzy choice function C is called normal . Lemma 3.2 [6] If C satisfies H1 and H2 then ¯ ⊆ R; (i) R ¯ are reflexive and strongly total. (ii) R and R ¯ Lemma 3.3 [6] If C satisfies H1, H2 and C is normal then R = R. Let W be the transitive closure of the fuzzy revealed preference relation R. In [6] the following fuzzy congruence axioms have been introduced:
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W F CA (Weak Fuzzy Congruence Axiom) For any S ∈ B and x, y ∈ X the following inequality holds R(x, y) ∧ C(S)(y) ∧ S(x) ≤ C(S)(x). SF CA (Strong Fuzzy Congruence Axiom) For any S ∈ B and x, y ∈ X the following inequality holds W (x, y) ∧ C(S)(y) ∧ S(x) ≤ C(S)(x). We notice that W F CA (resp. SF CA) is a fuzzy version of the congruence axiom W CA (resp. SCA) (see [10, 12]). The following theorem is a part of a more general result in [6]. It extends to the fuzzy case some equivalences of Arrow-Sen theorem (see [1, 12]). Theorem 3.4 [6] If the fuzzy choice function C verifies H1 and H2 then the following assertions are equivalent: (1) C is normal and R is reflexive, transitive and strongly total; (2) C satisfies W F CA; (3) C satisfies SF CA.
4
Fuzzy Arrow Axiom
In this section we study the Fuzzy Arrow Axiom, a generalization of the Arrow Axiom to fuzzy choice functions. The main result of this section establishes the equivalence of the Fuzzy Arrow Axiom and the full rationality of fuzzy choice functions. Throughout this section we shall assume that hypotheses H1, H2 are verified. Let X, B be a (crisp) choice space. Recall that a choice function C on X, B satisfies the Arrow Axiom (AA) if for any S1 , S2 ∈ B the following implication holds: [S1 ⊆ S2 ] ⇒ [S1 ∩ C(S2 ) = ∅] or [S1 ∩ C(S2 ) = C(S1 )]. In order to obtain the fuzzy form of AA it is convenient to write the above implication in the equivalent form: [S1 ⊆ S2 ] and [S1 ∩ C(S2 ) = ∅] ⇒ [S1 ∩ C(S2 ) = C(S1 )] Definition 4.1 Let C be a fuzzy choice function on a fuzzy choice space X, B. We say that C satisfies the Fuzzy Arrow Axiom F AA if for any S1 , S2 ∈ B and x ∈ X we have I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ E(S1 ∩ C(S2 ), C(S1 )). Remark 4.2 If S1 ⊆ S2 then the previous inequality becomes: S1 (x) ∧ C(S2 )(x) ≤ E(S1 ∩ C(S2 ), C(S1 )) hence it is clear that AA is a particular case of F AA. Remark 4.3 We observe that F AA is formulated in terms of the functions I(., .) and E(., .).
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¯ is transitive. Lemma 4.4 If C satisfies F AA then R Theorem 4.5 If C : B → F(X) is a fuzzy choice function then the following are equivalent: (1) C is full rational; (2) C satisfies F AA. Now we shall connect Theorem 3.4 and Theorem 4.5. Lemma 4.6 If C = G(., Q) and R is the fuzzy revealed preference relation associated to C then R ⊆ Q. Theorem 4.7 If C : B → F(x) is a fuzzy choice function then the following are equivalent: (1) C is full rational; (2) C satisfies F AA; (3) C is normal and R is reflexive, transitive and strongly total; (4) C satisfies W F CA; (5) C satisfies SF CA. We introduce now two more axioms: Fuzzy Chernoff Axiom (F CA) For any S1 , S2 ∈ B and x ∈ X, I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ I(S1 ∩ C(S2 ), C(S1 )) Fuzzy Dual Chernoff Axiom (F DCA) For any S1 , S2 ∈ B and x ∈ X, I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ I(C(S1 ), S1 ∩ C(S2 )). It is obvious that the Fuzzy Arrow Axiom F AA is equivalent to the conjunction of F CA and F DCA. F CA (resp. F DCA) is a fuzzy form of the Chernoff Axiom CA (resp. Dual Chernoff Axiom DCA) (see [16], pp. 31 and 41). Proposition 4.8 For any fuzzy choice function C : B → F(X) the following conditions are equivalent: (1) C satisfies F CA; (2) For any S1 , S2 ∈ B and x ∈ X we have I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x) ≤ C(S1 )(x). Condition (2) from the previous proposition is denoted F α in [7]. Corollary 4.9 Any full rational fuzzy choice function C satisfies condition F α. This corollary improves Proposition 5.1 in [7]. 7
5
Arrow index of a fuzzy choice function
In this section we shall introduce two notions: the Arrow index A(C) of a fuzzy choice function C and the similarity of two fuzzy choice functions. A(C) measures the degree to which C satisfies F AA. We prove an extensionability property of the Arrow index w.r.t. this similarity relation. Definition 5.1 Let C be a fuzzy choice function on X, B. The Arrow index A(C) of C is defined by [(I(S1 , S2 ) ∧ S1 (x) ∧ C(S2 )(x)) → E(S1 ∩ C(S2 ), C(S1 )]. A(C) = S1 ,S2 ∈B x∈X
We remark that A(C) = 1 iff C verifies the Fuzzy Arrow Axiom. Intuitively, the real number A(C) represents the degree of truth of the statement ”The fuzzy choice function C verifies Fuzzy Arrow Axiom”. We introduce next the concept of similarity of two fuzzy choice functions. Definition 5.2 Let C1 , C2 be two fuzzy choice functions on X, B. The degree of similarity of C 1 and C2 is defined by E(C1 , C2 ) = [C1 (S)(x) ↔ C2 (S)(x)]. S∈B x∈X
Lemma 5.3 For any fuzzy choice functions C1 , C2 on X, B we have (i) E(C1 , C2 ) = 1 iff C1 = C2 ; (ii) E(C1 , C2 ) = E(C2 , C1 ); (iii) E(C1 , C2 ) ∧ E(C2 , C3 ) ≤ E(C1 , C3 ).
E(C1 , C2 ) is a By this lemma the function E(., .) defined by (C1 , C2 ) → similarity relation [4] on the set of fuzzy choice functions on X, B. Lemma 5.4 Let C, C be two fuzzy choice functions on X, B. For any S ∈ B and x ∈ X the following inequality holds: E(C, C ) ∧ C(S)(x) ≤ C (S)(x). Lemma 5.5 [4] Let A1 , A2 ∈ F(X) and x ∈ X. Then E(A1 , A2 ) ∧ A1 (x) ≤ A2 (x) The following theorem shows how the similarity relation introduced by Definition 5.2 behaves with respect to the Arrow index. Theorem 5.6 If C, C are two fuzzy choice functions on X, B then A(C) ∧ E(C, C ) ≤ A(C ). Remark 5.7 Let C1 , C2 be two fuzzy choice functions on X, B and 0 ≤ δ ≤ 1. We say that C1 , C2 are δ-equal (C1 =δ C2 in symbols) if E(C1 , C2 ) ≥ δ. In accordance with Theorem 5.6, if C1 satisfies F AA and C1 =δ C2 then C2 satisfies F AA. In other terms, the Arrow index is preserved by the δ-equality. 8
6
Concluding Remarks
In the classic theory of choice functions, the Arrow axiom (AA) is equivalent with each of the following conditions: full rationality (F R), Houthakker’s axiom of revealed preference (HOA), strong axiom of revealed preference (SARP ), weak axiom of revealed preference (W ARP ), strong congruence axiom (SCA) and weak congruence axiom (W CA). These equivalences hold under the hypothesis that the domain of the choice functions contains the finite sets of alternatives (cf. [16], p. 30). The main result of this paper is that the full rationality (F F R) of fuzzy choice functions in the sense of [5, 6, 7] is characterized by a fuzzy form of Arrow Axiom (F AA). Combining this theorem with a result of [6] it follows that F F R and F AA are also equivalent to fuzzy weak congruence axiom W F CA and to fuzzy strong congruence axiom SF CA. These results are summarized in the following diagram.
FAA
FCA & FDCA
FFR
WFCA
SFCA
Figure 1: Hierarchy of axioms Two new concepts are introduced. The first one, the Arrow index, characterizes the degree to which ”the Arrow axiom is verified by a fuzzy choice function”. The second one is the degree of similarity of two fuzzy choice functions. In this way one obtains a similarity relation on the set of fuzzy choice functions defined on a fuzzy choice space. We have proved a preservation theorem of the Arrow index by this similarity relation.
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