Transitivity of fuzzy relations and rational choice

Annals of Operations Research 23(1990)265-278 265

TRANSITIVITY OF FUZZY RELATIONS AND RATIONAL CHOICE

Neelam JAIN Department of Economics, University of Minnesota, Minneapolis, MN 55455, USA

Abstract

An attempt is made to explore the implications of weakening the transitivity condition used in defining a fuzzy preference ordering for various results established in the literature.

1. Introduction

The traditional approach to analyzing individual choice behaviour is based on the premise that the individual has preferences which can be represented by an exact binary relation (see [8] for a taxonomy of this approach). This premise has been questioned by a number of economists (Basu [2], Durra [4], Pattanaik [1,4], Salles [1,4], among others). One critique consists of arguing in favour of a framework which allows individuals to express a degree of confidence, ranging between 0 and 1, with which they rank various alternatives. This framework has been developed from concepts in fuzzy set theory. Essentially, a fuzzy set is a set whose boundary is not clearly defined, i.e. it is possible for an element to be a partial member of the set, the degree of membership varying from 0 to 1 [2].

This framework, to be elaborated in the next section, will be referred to as a fuzzy preference framework (FPF) as against the traditional exact preference framework (EPF).

In the existing literature dealing with fuzzy preferences, a number of choice rules, or more appropriately, definitions of rationality, have been discussed as the plausible bases on which exact choice can be generated from fuzzy preferences. (The definitions that we are going to discuss in this paper are the k-dominance rule (Dk rule), pairwise optimality rule (PO rule), the highest scoring rule (the H rule), and the max-rain- difference rule (the max-mD rule).) Consequently, a choice function may be rationalized by different fuzzy binary weak preference relations (henceforth to be referred to as an FWPR) corresponding to different definitions of rationality. The rationalization, satisfying the properties of reflexivity, connectedness and transitivity, imposed generally on a binary weak preference relation, is referred to as a fuzzy preference ordering (FPO). It has been argued that the explanatory potential of these choice rules and hence of the FPF (with the exception of the H rule and the max-mD rule, which are yet to be investigated fully) is not significant [4]. This is so because a choice function rationalizable in terms of an FPO is also rationalizable in terms of an EPO and an Exact

�9 J.C. Baltzer AG, Scientific Publishing Company

266 N. Jain, Transitivity of fuzzy relations

Weak Preference Relation (henceforth EWPR) satisfying the properties of reflexivity, con-nectedness and quasi-transitivity, respectively. Hence, one is not gaining much in terms of explanation of a wider class of choice behaviour by using FPF instead of EPF.

The purpose of this paper is to show that the explanatory content of the FPF can be increased by weakening the transitivity condition required to be satisfied by the underlying FWPR.

Unlike the EPF, the FPF permits us to define transitivity in a number of ways, all of which are equivalent to the standard definition in the EPF. Of the different versions of the standard transitivity condition, the min-transitivity condition has come to be generally accepted as the most reasonable transitivity condition in the FPF. However, since there does not seem to be any a priori, intuitive reason for considering the min-transitivity condition as the most reasonable transitivity condition, it is worth investigating the effects of replacing it by other transitivity conditions [5], discussed in this paper, on the conditions under which the choice function is rationalizable in terms of an FPO and hence on the explanatory content of the FPF.

It is shown in this paper that all major results regarding rationalizability of a choice function change significantly when the min-transitivity condition is replaced by other transitivity conditions. More specifically, we show the following:

(a)

Co)

If the FWPR satisfies min-transitivity, in addition to other regularity conditions, then the max-mD rationalizability of a choice function in terms of an FPO is the same as its PO rationalizability in terms of an FPO.

If we replace the rain-transitivity condition by a weaker but intuitively no less plausible condition of transitivity, then:

(i) if we use Dk rule or PO rule, then the choices that can be rationalized by an FPO can also be rationalized by an EWPR and, therefore, we do not explain any more of choice behaviour in the FPF than in the EPF. However, there is a difference in the nature of rationalization - the same choice behaviour that is explained by an EWPR in the EPF is explained by a reflexive, transitive and connected FWPR in the FPF;

(ii) if we use H rule or max-mD nile, then the choice behaviour that can be rationalized by an EWPR can also be rationalized by an FPO, but the converse is not true. Hence, we do explain a wider class of choice behaviour with a "well-behaved" rationalization in the FPF than in the EPF.

Table 1 contains the relationship between various rationalizability concepts in the EPF and the FPF [4].

In section 2, we present the framework and concepts; section 3 contains the results and, finally, section 4 gives the conclusions.

N. Jain, Transitivity of fuzzy relations 267

Table 1

Rationalizability of a choice function in FPF

Dk PO H Max-mD

Transitive r ~ ~

Quasi-transitive r r :=~

Acyclic r :::O =O

Ordering based on min-transitivity

Ordering based on product transitivity (T4) or T s or T 6

Notes: (1) Both row and column headings refer to different notions of rationality. The symbol "r in a cell indicates that the two notions are equivalent; "~" indicates that the row heading implies the column heading but not vice versa. Similarly, "~" indicates that the column heading implies the row heading but not vice versa. (2) All results related to rnin-~'ansitivity based rationality with the exception of the max-roD rule axe to be found in [4]. All other results are discussed in this paper.

2. F r a m e w o r k and concepts

Let X be the set o f alternatives. Let 3 < I XI < oo. The elements o f X will be denoted by x, y, etc. Let Z = 2x - O. The elements of Z are non-empty, exact subsets of X. A fuzzy subset o f X is a function A: X ~ [0, 1]. Clearly, an exact subset o f X is a special case of fuzzy subsets, where the value assigned to the function is either 0 or 1. An FWPR on X is a function R: X 2 ---> [0, 1], while an EWPR on X is an F W P R such that R(X 2) ---> {0, 1} (if R is an EWPR, then we write xRy instead of R(x, y) = 1).

The condit ions of reflexivity and connectedness, generally imposed on a binary

weak preference relation, are as follows:

DEFINITION 2.1

(a) An FWPR, R, is reflexive iff for all x ~ X, R(x, x) = 1.

(b) An FWPR, R, is connected i ff for all x ~ X, R(x, y) + R(x, y) = 1.

As far as transitivity is concemed, the min-transit ivity condition, which has been preferred to some other condit ions [4], is given below:

DEFINITION 2.2

An F W P R , R, is Min-Transitive or Type-1 transitive (Tctransitive) i ff for all x, y, z in X, R(x, z) > min[R(x, y), R(y, z)].

The fol lowing definit ion gives three "fuzzy versions" of the transit ivity condit ion

used in the EPF, which we are going to discuss in this paper.


DEFINITION 2.3

(a) An FWPR, R, satisfies Product Transitivity T 4 iff for all x,y, z in X, R(x, z) > (R(x, y)) . (R(y, z)).

(b) An FWPR, R, satisfies T 5 iff for all x , y , z in X, R (x , z )>R(x , y ) + R(y, z) - 1.

(c) An FWPR, R, satisfies T 6 iff for all x,y, z in X, if R(x ,y )> 0 and R(y, z) >_ O, then R(x, z) > O.

Also, it easy to show that all these conditions boil down to the standard transitivity condition in the EPF.

Notation: An FWPR, R, satisfying reflexivity, connectedness and transitivity condition Tj will be denoted by FPO(T i) ( j = 1 . . . . . 6).

The transitivity conditions T 2 and "1~3, discussed in [4], will not be taken up here because of their uninteresting implications. The exact counterpart of an FPO is an EPO (Exact Preference Ordering).

The next step is to see how an agent makes an exact choice from given sets of alternatives when the underlying preferences are fuzzy. We summarize these choice rules below, after defining the concept of choice function:

DEFINITION 2.4

A Choice Function is a function C such that for all A ~ Z, C(A) c_ A.

DEFINITION 2.5

L e t R be an FWPR.

(1) (k-dominance rule or Dk rule). An element x E A, k-dominates y ~ A iff there exists k ~ [0, 1] such that R(x, y) > k and the choice set is given by BDk(A,R) - {x ~ A [R(x, y) > k for all y ~ A}.

(2) (Pairwise Optimality rule or PO rule). An element x ~ A is pairwise optimal vis-a-vis another element y ~ A iff R(x, y) >_ R(y, x) and the choice set is given by Bpo(A,R) -- {x ~ A IR(x,y) > R(y ,x) for all y ~ A}.

(3) (Max-mD rule)*. Let x e A and define

mD(x, A, R) =-y e~'m{x} (R(x, y) - R(y, x)).

Then the choice set is given by BmD(A, R) = {x E A I mD (x, A, R) > mD(x, A, R) for all y in A).

(4) (Highest Scoring rule or H rule or H criterion). For all A ~ Z, and for some FWPR, R, define the Fuzzy R-Greatest set G*(A, R) as a function, G*: X ---> [0,1] such that for all x ~ X - A, G*(A, R)(x) = 0 and for all x ~ A,

*See [1].


G*(A, R)(x) -y r~in{x }(R(x, y)).

Then the choice set is given by BH(A, R) - {x e A IG*(A, R)(x) > G* (A, R)( y) for all y in A }.

In the exact counterpart, while the choice function is defined in an identical manner, the generation of choice from preference is much more clearcut. The following approach is commonly used:

DEFINITION 2.6

For all A ~ 3(, for some EWPR, R, x ~ A is the R-greatest element of A iff xRy for all y ~ A. Then, the R-greatest set is G(A, R) = {x ~ A [xRy for all y in A}.

Clearly, G ~ defined with respect to an FWPR corresponds to the set G. Another concept from EPF that will be used later is that of the base relation defined below:

DEFINITION 2.7

For all x, y ~ X, Reis a Base-Relation if xRey e-~ [x ~ C({x, y})].

Next, one may like to know whether the choice behaviour of the agent can be explained by an FWPR. Intuitively, this is the problem of rationalizability. The formal definition follows:

DEFINITION 2.8

Let C be an exact choice function and let R be an FWPR. Then, C is, respectively,

(1) Dk-rationalizable,

(2) PO-rationalizable,

(3) H-rationalizable,

(4) Max-mD-rationalizable

in terms of R iff

(1') C(A) = BDk(A, R),

(2') C(A) = Br,o(A, R),

(3') C(A) = BH(A, R), and

(4') C(A) = BmD(A, R),

for all A e Z, respectively.

In the EPF, rationalizability of the choice function can be similarly defined. Further, it has been shown that rationalizability of a choice function in terms of any EWPR, R, is equivalent to its rationalizability in terms of the base relation [9].


3. Rationalizability of choice functions in FPF

To begin with, we look into the max-mD criterion. This criterion has been proposed by Barrett et al. [1], where they evaluate a number of choice rules in terms of certain desirable rationality conditions. They have shown that the max-mD rule satisfies all of these conditions, thereby suggesting that this rule can possibly be considered more satisfactory than others. At the same time, however, they discard the PO rule as uninteresting since choice sets generated by this rule can also be generated by an EWPR, satisfying reflexivity, connectedness and quasi-transitivity. However, our investigation shows that when the underlying FWPR is an FPO, then the Bpo(A, R) and BraD(A, R) sets are identical* The proof of this claim requires the following lemma.

LEMMA 3.1

For all x ~ X, for any FWPR, R, x ~ BIn(A, R) iff mD(x, A, R) > 0.

The proof is simple and therefore omitted.

PROPOSITION 3.1

For all A ~ X and for all FPO(T1), R, Bpo(A, R) = BmD(A, R).

Proof

Let A ~ Z and let R be an FPO(Tx). The proof will be complete by showing the following:

(a) x ~ BraD(A, R) --4 x E Bpo(A, R) for all A E Z, and

(b) x ~ BIn(A, R) ---> x ~ BraD(A, R) for all A ~ X.

To prove (a), consider x ~ BmD(A, R). Then, by definition, mD(x, A, R) > mD(y, A, R) for all y inA - {x}. Le tx ~ Bpo(A, R). Then there exists y ~ A - {x} such that R( y, x) > R(x, y), which implies R(x, y) - R( y, x) < 0, or mD(x, A, R) < 0. However, since Bpo(A, R) is non-empty when R is an FPO(T1) [4], there exists z in A such that z ~ Bpo(A,R ). By lemma 3.1, mD(z ,A ,R) > O. It follows from mD(x ,A ,R) < 0 and mD(z, A, R) > 0 that mD(z, A, R) > mD(x, A, R), which implies that x ~ BmD(A, R), a contradiction.

To prove (b), let A ~ Z and let R be an FPOff~). Let x ~ BIn(A, R). Then, by lemma 3.1, mD(x, A, R) > 0. Let x ~ BraD(A, R). Then let y in A be such that mD( y, A, R) > mD(x, A, R) > 0. This together with lemma 3.1 implies y ~ BIn(A, R).

*This result has been developed independently on the basis of the preliminary draft of the Barrett, Pattanaik and Salles [1] paper. Their revised version contains the necessary part of the proof; we provide both the necessity and sufficiency parts of the proof. Furthermore, the necessity part of our proof is different from theirs.


Now, since x, y are in Bpo(A, R) and mD(x, A, R), mD(y, A, R) > O, R(x, y) = R(y, x). Hence, mD(x, A, R) = mD(y, A, R) = 0, which contradicts mD(y, A, R) > mD(x, A, R). Therefore, x is in BraD(A, R).

Thus, from (a) and (b) we obtain Bpo(A, R) = BraD(A, R) for all A in Z where R is an FPO(T1).

As far as the H rule is concemed, the existing literature [4] shows that there are some necessary conditions for H-rationalizability of a choice function. Our investigation shows that, with a slight modification of one of the conditions, the H rule can be completely characterized if X contains only three alternatives, but not in general. The proof is tedious and hence not presented here.

Next, we explore the consequences of replacing T~ by weaker transitivity conditions. The relationships among various transitivity conditions are summarized below:

T 1 D+ T4 a~

T 5

Consequently,

FPOCr~) ~ FPO(T 4)

FPO(T s)

FPOCO

It can be easily checked that T 5 and T 6 are independent. We begin with Dk-rationalizability.

Dk-rationalizability. On using T 4 instead of T 1, we find that rationalizability of a choice function in terms of an EPO is no longer necessary, although it remains sufficient for its Dk-rationalizability. More importantly, one can establish that if the confidence threshold k lies strictly between 0.5 and 1, then all choice functions rationalizable in terms of an FPO(T4) must also be rationalizable in terms of an EWPR and vice versa. Intuitively speaking, it appears reasonable to restrict k in this manner since an agent can be reasonably expected to choose an alternative only if he is at least 50% sure about his preferences (in a weak sense) over all other alternatives. Similarly, exclusion of 1 from the domain of k only amounts to requiring R to be strictly non-exact, thus increasing the possibility of a non-empty choice set, induced by the k-dominance approach.

We prove this result in the following proposition:


PROPOSITION 3.2

For k ~ (0.5,1), a choice function is Dk-rationalizable in terms of an FPO(T 4) iff it is rationalizable in terms of an EWPR.

Proof

Necessity: Let k ~ (0.5,1) and let a choice function be Dk-rationalizable in terms of an FPO(T4), R, i.e. C(A) = BDk(A, R) for all A in Z. Let C not be rationalizable in terms of an EWPR. Then, for some A in Z, C(A) ~ G(A, Re). Then there are two possibilities:

(a) there exists x in A such that x is in C(A) but not in G(A, Re), or

(b) there exists x in A such that x is in G(A, R_ c) but not in C(A).

To prove (a), we note that, since x is not in G(A, RE), there exists y in A such that yP.x. Then, by definition of R e, C({x, y}) = {y} and by the fact that C is Dk-rationalizable, R(x, y) < k. However, since x is in C(A) and C(A) = Bog(A, R), we know that R(x, y) > k for all y in A. Thus, we have a contradiction.

To prove (b), notice that, since x is in G(A, Re), xR~y for all y in A. Hence, by def'mition, x ~ C({x, y}) for all y in A. By Dk-rationalizability of C, we know that there exists FPO(T4), R, such that R(x, y) > k for all y in A. Hence, x ~ BDk(A, R), and given that C(A) = BDk(A, R), x is in C(A), leading to a contradiction.

Sufficiency: Let C be rationalizable in terms of an EWPR, R. Then, for all

A ~ Z,

(1) C ( A ) = G ( A , R ) = { x ~ A I x R y f o r a U y ~ A } .

Let k e (0.5, 1) and let e > 0 be infinitesimally small. Then define an FWPR, R ~ as follows: for all x, y in X,

(2) [x = y] ~ [R*(x, y) = 1];

(3) [x ~ y and C({x, y}) = {x}] +-> [R'(x, y) = k, R*(y ,x) = k - e];

(4) [x r y and C({x, y }) = {x, y } ] ~ [R *(x, y) = R*(y, x) = k].

It is easy to verify that R*, so defined, satisfies reflexivity and connectedness. As for T 4, we note that min[k, k - e] > m a x [ ~ , k(k - e)] for all k ~ (0.5,1). Thus, R* satisfies T 4. Hence, R* is an FPO(T4).

Now, from (1) to (4) it follows that for all x, y ~ X,

(5) xRy iff R*(x, y) > k.

From (1) to (5) it follows that C(A) = {x ~ A iR*(x,y) > k for all y ~ A} for all A ~ Z- Hence, C is Dk-rationalizable in terms of R*, which was shown to be an FPO(T4).


Thus, we find that one consequence of using T 4 is to render Dk-rationalizability and rationalizability in terms of an EWPR in the EPF equivalent.

PO-rationalizability.

PROPOSITION 3.3

A choice function is PO-rationalizable in terms of an FPO(T4) iff it is rationalizable in terms of an EWPR.

The proof is similar to that of proposition 3.2 with the following FWPR being used in the sufficiency part: Let g, h ~ (0, 1) be such that g > h > g2 > 0.5. Then define an FWPR, R*, such that for all x, y in X,

Ix = y ] <---> [R*(x,y) = R*(y ,x) = 1];

[x ~ y and C({x, y}) = {x}] ~ [R*(x, y) : g, R*(y ,x) = hi;

[x ~:y and C({x, y}) = {x,y}) ~ [R*(x, y) = R*(y,x) = h].

It is easy to verify that R* is an FPO(T4).

Remark 3.1. Propositions 3.2 and 3.3 indicate that replacing TI with T 4 renders Dk- rationalizability and PO-rationalizability concepts equivalent when k ~ (0.5,1).

Remark 3.2. While just as with T 1 equivalence has been shown between Dk- and PO- rationalizability on the one hand, and rationalizability in the EPF on the other, it is worth noting that the nature of rationalization is different. While in EPF it is any EWPR, in the FPF it is a reflexive, transitive and connected FWPR that rationalizes the choice behaviour.

H-rationalizability. On replacing T~ with T 4, we find that while all choice functions rationalizable in terms of an EWPR are also rationalizable in terms of an FPO(T4), the converse is not true.* This result is proved in the following proposition:

PROPOSITION 3.4

If a choice function is rationalizable in terms of an EWPR, it is also H-rationalizable in terms of an FPO(T4).

Proof

Let the choice function C C(A) = G(A, R) for all A ~ Z, i.e.

be rationalizable in terms of an EWPR, R. Then

(1) C(A) = {x ~ A IxRy for all y ~ A} for all A ~ Z-

*See [4] for a counter-example.


Construct an FWPR, R*, as follows: let g, h �9 (0,1) be such that g > h >_ g2 > 0.5. For allx, y � 9

(2) [x = y] ~ [R*(x, y) = R*(y, x) = 11;

(3) [x ~ y and C({x, y}) = {x}] <--) [R*(x, y) = g and R*(y,x) = h];

(4) [x ~- y and C({x, y}) = {x, y}] ~ [R'(x, y) = R * ( y , x ) = g].

It can be easily verified that this FWPR is an FPO(T4). The proof will be complete by showing that for all A �9 Z, x ~ C(A ) ~ [G (A, R )(x) _ G (A, R *)( y) for all y ~ A]. Since C is rationalizable in terms of an EWPR, R, it follows that xRy for all y ~ A and hence x �9 C({x, y}) for all y �9 A. From (2) to (4), we obtain R*(x, y) = g for all y �9 A. Thus,

(5) G'(A, n *)(x) = g.

Consider any y ~ x in A. Then there are two possibilities:

(a) y �9 C(A), or

Co) y ~ C(A).

If (a) holds then, by the above argument,

(6) G*(A, R')(y) = g.

If (b) holds then, given rationalizability of C, there exists z e A such that zPey. Hence, C({z,y}) = {z}. Then from (3), R*(y, z) = h. Thus,

(7) G*(A, e*)( y) = h.

Combining (5), (6) and (7), we obtain

G~ R ~ > G*(A, R *)( y) for all y in A.

Next, let x in A be such that G*(A, R*)(x) > G*(A, R*)(y) for all y in A and let x ~ C(A). Since C is rationalizable in terms of an EWPR, it follows that x ~ G(A, Re). Then there exists y in A such that yP~x. By definition, therefore, x ~ C({x, y}). By (3), R*(x, y) = h and thus

(8) G*(A, R *)(x) = h.

However, given our assumption of G*(A, R*)(x) > G*(A, R)(y) for all y in A and (8), it follows that


(9) G*(A, R*)(y) = h for all y in A.

In (a) above, we have shown that i fy is in C(A) then G*(A, R*)(y) = g. Hence, (9) clearly implies that C(A) = 0 , which cannot be true. Therefere, a contradiction is established.

PROPOSITION 3.5

There exist choice functions not H-rationalizable in terms of any FWPR.

Proof

The proof consists of the following example:

Let X = {x,y,z} and let C({x,y})= {x,y}, C({y,z})= {z}, C({x,z})= {z}, C({x,y, z}) = {x,y, z}.

For H-rationalizability in terms of an FWPR, R, the following inequalities must hold:

(1) R(x, y) = R(y,x),

(2) R(y, z) < R(z, y),

(3) R(x, z) < R(z, x), and

(4) G*(A, R)(x) = G*(A, R)(y) = G*(A, R)(z).

However, this is impossible for all possible values of R. Hence, the given choice behaviour is not H-rationalizable in terms of any FWPR.

Max-mD-rationalizability. Earlier in this section, it was shown that this concept was equivalent to PO-rationalizability. Here, we explore the status of this result as T 1 is replaced by T 4.

PROPOSITION 3.6

For all A ~ Z and for all FPO(T4), R, Br, o(A, R) c_ BraD(A, R), but the converse is not true in general.

Proof

In proposition 3.2, we showed that if x is in Bpo(A, R) then x is in BraD(A, R). AS can be checked easily, this result remains valid for FPO(T 4) as well. Therefore, to prove the proposition, we need to show that for some A ~ X and for some FPO(T4), R,

x ~ BraD(A, R) ~ x ~ Bpo(A, R).

For this, an example is sufficient: let X = {x, y, z}. Consider an FWPR, R, onX defined as follows: R(x, y) = R(y, z) = R(z, x) = 0.9 and R( y, x) = R(z, y) = R(x, z) = 0.5. It is


clear that this R is an FPO(T4). However, it can be easily checked that Br,o(X, R) = and BraD(X, R) = {x, y, z}.

Hence the result.

Remark 3.3. It needs to be emphasized that the above result is essentially due to the fact that an FPO(T 4) (and hence an FPO(T 5) and FPO(T6)) does not ensure a non-empty Bpo set, whereas an FPO(T~) does.

Remark 3.4. We give an example below to show that max-mD-rationalizability of a choice function in terms of an FPO(T 4) does not imply its PO-rationalizability in terms of an FPO(T4):

EXAMPLE

Let X = {x,y, z} and let" C({x ,y} )= {x}, C({y, z})= {y}, C({x , z} )= {z} and C({x,y ,z}) = {x,y,z} .

It can be checked that the FPO(T 4) used in proving proposition 3.6 above max- mD-rationalizes this choice function and also that there does not exist any F I ~ g r 4) which will PO-rationalize it. However, the converse is true since the latter implies non- emptiness of the Bpo set, which is sufficient for equality between the Br, 0 set and the BmD set, as shown above. Further, since PO-rationalizability of a choice function in terms of an FPO(T4) is equivalent to its rationalizability in terms of an EWPR (see proposition 3.4), the latter is also sufficient for max-mD-rationalizability in terms of an FPO(T4), but not necessary.

Remark 3.5. We also note that there are choice functions which are not max-mD- rationalizable in terms of any FWPR. This can be seen from the following example, used in proposition 3.5, as follows: m D ( x , X , R ) < O, m D ( y , X , R ) < 0 and mD(z, x, R) > 0 for all FPO(T4), R. Therefore, only z ~ BmD(X, R), which contradicts our assumption.

Summing up our results related to the condition T 4, we observe that while whatever choice behaviour is explained by a fuzzy preference framework by using the k-dominance and pairwise optimality rules can also be explained in the exact preference framework, the explanations are different and the differences are significant. However, if one uses the highest-scoring rule or the max-mD rule, then one can find choice functions which may be considered irrational in terms of the standard criteria of the exact preference framework but are fuzzy-reflexive-transitive-connected-rational - a result which is intuitively justified given the stringent requirement of exactness in preferences in the exact preference framework.

As far as the transitivity conditions T 5 and T 6 are concerned, we note that since T 4 is a stronger condition than either of T 5 and T 6, all conditions not necessary for H- rationalizability or max-mD-rationalizability of a choice function in terms of an FPO(T 4)


must not be necessary for T 5- or T6-based rationalizability as well, and the conditions which ensure the former must also ensure the latter. Hence, rationalizability of a choice function in terms of an EWPR is not necessary but is sufficient for H- or max-mD- rationalizability in terms of an FPO(Ts) or FPO(T6) as well.

Further, propositions 3.2 and 3.3 also remain valid when T 4 is replaced by T s or T 6. While the sufficiency part is obvious, we note that in proving the necessity part of these propositions, we have not used the transitivity property of the FWPR at all. Hence, it does not matter whether the choice function is Dk- or PO-rationalizable in terms of an FPO(T4), an FPO(T 5) or an FPO(T6) in this particular context.

These propositions can be stated in a more general form, as follows:

PROPOSITION 3.7

A choice function is Dk- or PO-rationalizable in terms of an FPO(Tj), j = 4,5,6 iff it is rationalizable in terms of an EWPR.

4. Conclusions

In this paper, an attempt has been made to explore the implications of weakening the transitivity condition used in defining a fuzzy preference ordering for various results established in the literature using the transitivity condition T r

The upshot of various propositions proved here is that the fuzzy preference framework does help in explaining choice behaviour which is not explained at all in the traditional framework if we use the highest-scoring approach or the max-mD approach and replace the min-transitivity condition by weaker but no less plausible transitivity conditions. Even if we use the k-dominance or the pairwise optimality rules, the fuzzy rationalization of a choice function, explained in the EPF in terms of an EWPR, is reflexive, transitive (T 4, T 5 or T6) and connected.

Acknowledgement

Acknowledgements are due to Santosh Panda and Kaushik Basu who supervised the author's dissertation at the Delhi School of Economics, from which this paper is an extract. The author.is grateful to Maurice Salles, P.K. Pattanaik and M.K. Richter for extremely valuable comments, and benefited from comments by the referees. Finally, editorial assistance from Samiran Banerjee is gratefully acknowledged.

References

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[3] H. Chernoff, Rational selection of decision functions, Economica 22(1954). [4] B. Duma, S.C. Panda and P.K. Pattanaik, Exact choice and fuzzy preferences, Math. and Social Sci.

11(1986). [5] A. Kaufmarm, Introduction to the Theory of Fuzzy Sets, Vol. 1 (Academic Press, New York, 1975). [6] A.K. Sen, Quasi-transitivity, rational choice and collective decisions, Rev. Econ. Studies 36(1960). [7] A.K. Sen, Collective Choice and Social Welfare (Holden-Day, San Francisco, 1970). [8] A.K. Sen, Choice functions and revealed preferences, Rev. Econ. Studies 38(1971). [9] A.K. Sen, Social choice theory: A re-examination, Econometrica 45(1977). [10] A.K. Sen, Choice, Welfare and Measurement (Blackwell, Oxford, 1982). [11] K. Suzumura, Rational choice and revealed preferences, Rev. Econ. Studies 41(1973).

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Transitivity of fuzzy relations and rational choice