Choice rules with fuzzy preferences: Some characterizations

Choice rules with fuzzy preferences: Some characterizationsAuthor(s): Kunal SenguptaSource: Social Choice and Welfare, Vol. 16, No. 2 (February 1999), pp. 259-272Published by: SpringerStable URL: http://www.jstor.org/stable/41106303 .

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Soc Choice Welfare (1999) 16: 259-272 bOCÌfll VlMMCC mtWOm

© Springer- Verlag 1999

Choice rules with fuzzy preferences: Some characterizations Kunal Sengupta

CESP, Jawaharlal Nehru University, New Delhi, India

Received: 3 August 1995/ Accepted: 19 November 1997

Abstract. Consider an agent with fuzzy preferences. This agent, however, has to make exact choices when faced with different feasible sets of alternatives. What rule does he follow in making such choices? This paper provides an axiomatic characterization of a class of binary choice rules called the a satisfying rule. When a = 1, this rule is the Orlovsky choice rule. On the other hand, for a < 1/2, the rule coincides with the Ma rule that has been extensively analyzed in the literature on fuzzy preferences.

1. Introduction

Consider an agent with fuzzy preferences who has to make exact choices when faced with different feasible sets of alternatives. What rule does he follow in making such choices? Unlike in the case of exact preference ordering where the choice set of an agent coincides with the most preferred elements, there does not seem to be any natural way of defining the choice rule of an agent when his preferences are fuzzy [see Orlovsky (1978); Basu (1984); Barrett and Pattanaik (1985); Dutta et al. (1986), Barrett et al. (1990)]. This paper tries to provide an answer to this question.

Broadly, one can classify choice rules on the basis of fuzzy preferences into two categories. The first type of rules are binary. In such a rule, choice from a feasible set of alternatives is essentially based on looking at the outcome in the pair wise comparisons of the elements in the set. * In non binary rules, how-

I am extremely grateful to Asis Banerjee, Rajat Deb and Prasanta Pattanaik and for many helpful discussions. I am also grateful to two anonymous referees for their helpful comments.

Orlovsky's rule is an example of a binary choice rule.



260 K. Sengupta

ever, it is possible that choices could depend on the overall preferences and not only on the pair wise preferences. (See Barrett et al. (1990) for a fairly comprehensive list of binary and non binary rules.)

In the class of all such possible rules (binary or otherwise), this paper characterizes (axiomatically) a family of binary choice rules that is called the a satisfying rule. For any a e [0,1], under this rule, x is chosen in a pair wise choice {x,y} whenever the extent of preference for x to y is no less than a. If a < 1/2, this rule is known in the literature as the M% rule (see Dutta et al. (1986) and Dasupta and Deb (1995)]. On the other hand, for a = 1, this is the Orlovsky rule. Since different values of a are consistent with a given prefernce ordering of an individual, it is possible that two agents with identical preferences can end up making different choices.

The approach taken in this paper should be contrasted with the literature that looks at the question of 'rationalizing' choices through fuzzy preferences (Basu 1984; Dutta et al. 1986). In this literature, one starts with a given choice rule for the individual and a set of observed choices and then asks the question whether there exists a set of preferences (possibly fuzzy) such that the predicted choice outcomes (according to the given choice rule) will match the actual choices?2 In this context, Dutta et al. proves an interesting result. Assuming that the decision maker uses the Orlovsky choice rule, they show that if the agent's behaviour is rationalized by a fuzzy preference, then it is also rationalized by an exact quasi-ordering.3 More interestingly, this exact quasi ordering can be constructed directly from the fuzzy preference of the individual and without the knowledge of his actual choices. Their result thus points to the redundancy of use of fuzzy preferences in explaining individual choices.

In our approach, however, the choice rule of the agent is not fixed at the outset. Indeed, the purpose of this paper is to characterize choice rules that satisfy a set of plausible axioms. Since these set of axioms typically characterize a whole family of choice rules, it is not possible to replace one's fuzzy preferences by an exact ordering and make the predicted choice sets identical.4

The rest of the paper is organized as follows: preliminaries are set up in the next section. In Section 3, we discuss three different notions of transitivity of fuzzy preferences. Section 4 formally defines the notion of a satisfying rule and discusses its relationship to Orlovsky and the Ma choice rule. Section 5 pres- ents and discusses the axioms that a decision maker is supposed to satisfy. Section 6 characterizes the class of choice rules under each of the there notion of transitivity that was introduced in Sect. 3. Section 7 contains the proofs while Sect. 8 concludes.

2 It is possibly not surprising that results in this literature depends critically on the choice rule that is assumed for the decision maker (Basu 1984).

In a quasi ordering, the strict preference relation is transitive. Indifference relation need not be.

Although, if the choices made by the individual (who is following one of these choice rules) were known, it would have been possible to rationalize his choices by an exact ordering.



Choice rules with fuzzy preferences 261

2. Preliminaries

Let X be a nonempty set that stands for a fixed finite number of alternatives. Throughout, we assume that 'X' > 3. Also let x be the set of all possible nonempty subsets of X.

Definition 1. A fuzzy binary preference relation (FBPR) is a function R : X2 -> [0, 1]. The relationship R is said to be exact iff for all x,y, R(x,y) is either Oori. Ris said to be reflexive iff for all x e X, R(x, x) = 1. R is said to be connected iff for all x,yeX, R(x,y) + R{y, x)>l.

Denote by SR, the set of all reflexive and connected FBPR.

Definition 2. A preference based choice function (PCF) on domain Q^Mis a function C : x x fi -► X- A (PCF) is said to be well defined if for all Reí! and Aex,0*C(A,R)czA.

3. Transitivity of R

When preferences are fuzzy, there are many ways of defining the notion of transitivity (see Dasgupta and Deb 1996). In this paper, I restrict my attention to the following. Definition 3.1 R e 9Ì is said to be max-min transitive if and only if for all xyy,z e X, we have R(x,z) > min[R(x,y)iR(y,z)].

Denote byffc«, the set of all reflexive and connected FBPR satisfying max-min transitivity.

Max-min transitivity is by far the most commonly used notion in the analysis of fuzzy preferences. However, Dasgupta and Deb (1996) have argued that the use of max-min transitivity could be somewhat restrictive. In particular, they show that under max-min transitivity and in a three element set if the relationship is exact over at least one pair, then the relationship is exact for at least two pairs. To get away from such restrictiveness, they pre- scribe using a weaker notion of max-min transitivity. Definition 3.2 R e <R is said to be weak max-min transitive if and only if for all x,y,zeX, R(x,y) > R(y,x), R(y,z) > R(z,y) implies R{x,z)> min[R(x,y)ìR(yìz)}.

Denote by Hw c % the set of all reflexive and connected FBPR satisfying weak max-min transitivity. Clearly, H a Hw

The following Lemma gives us an useful property of weak max-min transitive preferences. Lemma 1. Suppose ReHw and let x,y,ze X with R{x,y) = R(y,x) = n and R(y,z) = R(z,y) = m, then, R(x,z) = R(z,x), whenever, n ¿ m.

Proof Assume without loss of generality that n> m. If the Lemma is false, then R(x, z) # R(z, x). We will consider two cases.



262 K. Sengupta

Case 1. R{x,z) > R{z,x).

R(y,x) - R(x,y) and R(x,z) > R(z,x) implies by weak max-min transitivity that m = R(y,z) > min[R(y,x),R(x,z)]. Since R{y,x) = n, and n> m, we have *0>,z) >*(*,*).

Now R{z,y) =R(y,z) and R(y,x) = R(x,y) implies by weak max-min transitivity that R(z, x) > min[jR(z, y) , R(y, x)] = R{z, y) since R(zy y) = m < R(y, x) = n.

Thus, R(y,z) > R(x,z) > R(z,x) > R{z,y). But this contradicts our supposition that R(y,z) = R(z,y), Case 2. R(z,x) > R(x,z). Since R(z,x) > R{x,z) and R(x,y) = R(y,x), by weak max-min transitivity we have m = R(z,y) > min[R(z,x),R(x,y)]. Since R(x,y) = n > m, we thus have/?(z,>>) > R(z,x).

Also, ä(jc,^) = R(y,x) and ̂ (^,2) = R(z,y) implies by weak max-min transitivity that R(x, z) > mm[R{x, y), R{y, z)] = R{y, r), since, R(x, y) = w > /?(y, z) = m.

Thus, #(z,j) > R(z,x) > R{x,z) > R(jy,z). But this contradicts our assumption that R(z,y) = R(y,z). ■

Remark 1. Since H a Hw, clearly any Re H satisfies Lemma 1.

Interestingly, however, Lemma 1 fails to be true when n = m. Consider this example.

Example 1. Let X = {x,y,z} with R(x,y) = R(y,x) = R(y,z) = R(zyy) = .5 and R(x,z) = .6 and R(z,x) = .5.

It is easy to check that Re H and thus in Hw. In the above example, we have R{z,y) > R{y,z) and R{y,x) > R{x,y)

while R(z,x) is strictly less than R(x,z). This is somewhat at odds with one's notion of transitivity. To rule out such possibilities, one can impose the following condition:

Condition T. ReW is said to satisfy condition T if for all x, y, z s X, R{x,y) = R(y,x) = R{y,z) = R(z,y) implies R(x,z) = R(z,x). Definition 3.3 R e 9Î is said to be T-transitìve if Re H and satisfies condition T.5

Denote by H cz H, the set of all reflexive and connected FBPR satisfying T-transitivity. Definition 3.4 R e 91 is said to be weakly T-transitive if Re Hw and satisfies condition T.

5 It is possible to check that if Re H and R satisfies condition Tf then, R(x,y) > R(y,x), R{y,z) > R(z,y) implies that R(x,z) > R(z,x). This notion of transitivity has been used in Banerjee (1993).




Denote by Hw the set of all reflexive and connected FBPR satisfying weak T-transitivity.

Note that H a H and H a Hw H and Hw, however, are unrelated. In the rest of the paper, my characterization results will use H (Theorem

1), H (Theorem 2) and Hw (Theorem 3) as the respective domains where the preferences of the agent are defined.

4. A binary choice rule 5a

Definition 4.1 Given R and for any a g [0, 1], we say x ^-dominates y, written x>*y if and only if R(x,y) > R(y,x) and R(y,x) < ot.

We now define the following choice rule.

C(A, R) = Sa(A, R) = {xeA: for no y e A, y >a x}

Orlovsky (1978) proposed the following choice rule.

C{A,R) = OV(A,R) = {xeA: R(x,y) > R(y,x) for all y e A}.

The following result is immediate.

Lemma 2. For all a e [0, 1],

(a) OV(A1R)ŒSa{A,R) (b) OV(A,R) = S{(A,R).

Consider now the following rule that has been extensively analyzed in the literature. For any R and A e x and a e (0, 1/2], define

C(A, R) = MÂ, R) = {xeA: R(x,y) > a for all y e A}.

Since a < 1/2, and R is connected, it is clear that Ma procedure generates a well defined PCF.

Lemma 3. If a < 1/2, then, Ma{A,R) = Sa(A,R).

Proof It is clear that Ma(A, R) cSÂ.R). To see the reverse implication, let x e Sa. If x is not in Ma, then, we must have R(x,y) < a for some y e A. Since xeSa, R(x,y)> R(y,x). Thus, R(yix)<<x. This implies that R(x,y) + R(y,x) < 1, violating the connectedness of R. ■

We end this section by noting that for any a, the Sa procedure generates a well defined PCF.

Lemma 4. Suppose R e Hw, then, for any A ex, Sa(A,R) # 0.

Proof In Lemma (1.2) of Dasgupta and Deb (1996), the following result is proved: if ReHWi then, R(xiy)>R(y,x) and R{y,z) >R{z,y) implies R{x,z) > R(z,x). Thus, from Kolodziejczyk (1986), it follows that OV{A,R) is nonempty for all Aex whenever ReHw- Since OV(A,R) c: SÂ^R) (Lemma 2 (part (a)), it follows that Sa is nonempty whenever R e Hw ■



264 K. Sengupta

5. Some properties of a PCF

Definition 5.1 A PCF, C satisfies independence / iff for any Aexand R, R! e 9?, [R(x,y) = R!{x,y)'for all x,y e A implies C{A,R) = C{A,R'). Remark 2. If a PCF satisfies I, then given any R e 9? and A ex, the choice from A is independent of the agent's preference with respect to any option not any A. It seems to be a desirable feature of any PCF.

Definition 5.2 Let C be a PCF satisfying I. C satisfies (5.2. 1 ) Neutrality (N) iff for all R,R! e<H and all jc, y, z, w e X, (not necessarily distinct), [R(x,y) = R'(z,w) and R(y,x) = R'(w,z)], implies [x e C({x, y},R) iffzeC({z,w},R'))and[yeC({x,y},R) iff w e C({z, w}, *')]• (5.2.2) Monotonicity (M) iff for all R,R' eW and all x,yeX, [R(x,y) < R'(x,y) and R(y,x) > R'(y,x)] implies [if {x} = C({jc,y},Ä), then {x} = C({x, >>},*')] and [ifxeCfaylR), then,xe C{{x,y},R!)). Remark 3. Note that the above two axioms are imposed only for PCFs satisfying independence. Neutrality says that as long as the preference between two pairs of alternatives (x,y) and (z,w) are symmetric across two different preference orderings, the choice in one situation should be sufficient to deter- mine the choice in the other situation. In other words, in determining choices, only the preference is important and not the identity of the alternatives.

Condition M, on the other hand, requires that if R(x, y) does not go down and R(y, x) does not go up, then x does not get treated less favorably (or y more favorably) by the choice rule as compared to the situation before the change in R. M seems to be an extremely plausible property of any choice function.

Definition 5.3 Let C be a PCF. C satisfies (5.3.1) Chernoff Condition (CC) iff for all xje X and Ae/M^ {x,y}, [x i C{{x,y},R)] implies [x i C{A,R)]. (5.3.2) Condition Beta iß) iff for all x,yeX and A e /, if [{x,y} = C({x, >>},/?)], then,xeC{A,R) if and only ifyeC(A,R). Remark 4. The conditions, Chernoff and ß are well known. These two together are necessary and sufficient to generate transitive rationalization of a choice function when preferences are exact (see Sen 1970). While conditions (specially) the Chernoff condition seem to be intuitively compelling, they are not entirely innocuous. For instance, the presence of Chernoff condition in effect rules out the possibility of non-binary choice rules.6

Finally consider the following continuity property of the choice rule.

6 I am indebted to a referee for pointing this out.




Definition 5.4 A PCF C satisfies Continuity (CON) iff for all RUR2... e ft and x, y e X,xe C({x,y) , R¡) for all R¡ and the sequence ((Ri(x,y),Ri(y,x)) converges to some (/£(x,j),/Ü(j>,jc)), then, x e C({x, >>},/£).

6. The characterization results

Theorem 1. Consider C : x x H -* X- Then, C satisfies (I, CC, ß,_N, M, CON) if and only if there exists a g [0, 1] such that for all Re H and A e x> C(A,R) = Sa(A,R). Theorem 2. Suppose C : x x H -> X- Then, C satisfies (I, CQ ß, N, M, CON) if and only if there exists a < 1/2 such that for all Re H and A ex, C{A,R) = Sa(A,R) = Ma{A,R).

Remark 5. The difference in the two theorems results because of axiom /?. For instance, consider the Orlovsky choice rule (Si). Since Re H, need not satisfy condition T (see Example 1), it is not possible to satisfy ß for all R e H using the Orlovsky procedure.7

We now state our final result that characterizes uniquely the Orlovsky choice rule.8

We will need the following definition.

Definition 6.1 A PCF C on domain Q is said to be non trivial if there exists some A ex and some Red such that C(A,R) ^ A.

Theorem 3. Consider C : x x Hw -> X that is nontrivial Then, C satisfies (I, CC, ß, N, M, CON) if and only if for all ReHw and A ex, C(A,R) =

Sl(A,R) = OV(A,R). Remark 6. Note that in the above theorem, the condition that C be non trivial is important since the trivial PCF, where a = 0 also satisfies all the axioms. Also note that if we enlarge our domain to H¡y, one gets an impossibility result. This is because the Orlovsky prpcedure does not satisfy condition ß if R fails to satisfy condition T.

One might wonder whether our characterization results are tight, i.e whether the axioms in each of the above theorems are independent or not?9 This is stated in the following theorem.

7 This is precisely why Dutta et al. obtained the equivalence between a fuzzy preferences and that of an exact quasi ordering and not exact ordering. Recall that their domain of preference was H. 0 Banerjee (1993) provides an alternative characterization ot the Orlovsky procedure. His axioms, however, are somewhat restrictive (see Sengupta 1998).

Independence means that if one drops any of the axioms, then, it is possible to get choice rules (not covered by the theorems) that satisfy the rest of the axioms.



266 K. Sengupta

Theorem 4. The axioms (I, CC, ß, N, M, CON) used in Theorem 1-3 are independent

7. Proofs

Proof of Theorem 1

Sufficiency. First note that since H cz Hw, by Lemma 4, Sa is well defined for any Re H and Aex- Now, the Sa procedure clearly satisfies the axioms I, CC, N and M. We now show that it satisfies condition ß. To show this, let {x,y} = Sa({x, y},R) and suppose in some A => {x,y}, y e Sa(A,R). We will show that xeSa(A,R). If x^ Sa(AyR)9 then, there exists zeA such that z >a x. We consider two cases.

Case 1. R(x,y) > R(y,x). Since z>,x, we have, R(z,x) > R(x,z). Since ReH, we then have R(z,y) > R{y,z) (with strict in equality if R(x, y) > R(y,x). (Dasgupta and Deb 1996).

Now R(xìz)>rmn[R(xìy),R(yìz)]. But R(x,y) > R{y,x) and thus R(x,z) >min[R(y,x),R(y,z)) =nán[min[R{yiz),R(z,x)],R{y,z)]. Therefore, R{x,z) > R(y,z) (because R{z,x) > ä(a',z)). Now if R(y,z) > a, then, we contradict our hypothesis that z >0Lx. Thus, R{y,z)oL. Since y e Sa(A,R), we then have R(z,y) = R(y,z). This gives us R{x,y) = R(y,x). Now by Lemma 1 or the fact the R satisfies condition T, we have R(x,z) = R(z,x). But this violates our supposition that z >-a x.

Case 2. R{x,y) < R(y,x). Since x s Sa({x,.y}, R), we have R(x,y) > a. Two sub cases need to be con- sidered.

2.1. R(y,z)>R{z,y). Now R{x,z) >min[R(x,y),R(y,z)]. Since R(y,z) > R{z,y), we have, R(x, z) > min[R(x,y),R(z,y)] > min^^,^), mm[R(z, jc), R(x,y)]]. Thus, either R(x, z) > R(z, x) or R(x, z) > R{x,y) > a, in either case, z does not a-dominate x.

2.2 R(y,z)<(z,y). In this, case we must have R(y,z)>oi since y e Sa(A,R). But then, R{x, z) > min[R(x, j), R(y, z)] > a and thus z does not a-dominate jc. ■

We now show that Sa procedure satisfies continuity. So consider a sequence Rn - y R° such that x e Sa({x, y}, Rn). If x is not in

s<x({*,y},R°), then, we have R°(x,y) < a and R°(x,y) < R{y,x). Since Rn converges to R0, there must exist TV such that for n > N, we have Rn(x, y) < a and Rn(x, y) < Rn(y, x). This implies that x is not in Sa({x,y}, Rn) for n> N, violating the hypothesis. ■




To prove the necessity part, we will require the following Lemmas.

Lemma 5. Suppose C(.,.) satisfies axiom N. Then, R(x,y) = R{y,x) implies thatC{{x,y},R) = {x,y}.

Proof, The proof of this immediate from axiom N by taking x, y, z, w to be x, y, y, x and R~R'. ■

Lemma 6. Suppose C(., .) salifies axioms N and M Then, R(x,y) > R(y,x) implies that x e C({jc, >>},/?).

Proof. Consider C,i? e SR and x,y elas specified in the statement of the Lemma. It can be easily checked that there exists E! e H such that R'{x,y) = R(x,y) = R'(y,x). Clearly, by Lemma 5, C({x,y},R') = {*,>'}. Hence, given Rf(y, x) > R(y, jc), noting M, we have x e C({x,y}, R). ■

Lemma 7. Suppose C satisfies (I,N, M, CC,ß). If forjóme Re H, R(x,y) > R(y, x) - m and y e C({x, v}, R), then, for any R' e H with R'{y, x) > m, we haveyeC({x,y},R').

Proof Consider a R* e H such that R*(x,y) = 1 and R*(y,x) = m. We first prove that y e C({x,y},R*). Assuming to the contrary, we have, y$ C({Xiy},ir).

Consider any z ̂ x,y and let A = {x, v,^}. Construct the following R°. R°(x,y) = l and R°(y,x)=m, R°(x,z) = R°(z,y) = R(x,y)9 R°(y,z) =

R°(z, x) = m. R°{w, k) = lifw*A and R°(k, w) =0ifkeA and w i A. IX is possible to check that R° so constructed satisfies T-transitivity. i.e, R° g H.

Now R°(x, y) = 1 = R*{x,y) and R°(y,x) = m^ R*(y,x), and y i C({x,y}, R*) by hypothesis. Thus, by condition I, y $ C({x,y}, R°). By axiom CC, we, then, have

>><¿C({A-,v,r},tf°) Factl.

Similarly, R°(x,z) = R(x,y) and R°(z,x) = R(y,x), and y e C({x,y},R) (hypothesis of the lemma). Thus, :eC({x,:},i?°) by axiom N. Since &(x,z) = R(x,y) >m = R°(zJx), by Lemma 6, xe C^x^}^0). Therefore, by axiom /?, we have

zeC({x,y,z},FP) if x e C({x, y,z},F») Fact 2.

Finally, R°(z,y) = R(x,y) and Ä°(y,z) = Ä(y,x), and yeC({x,y},R) (hypothesis of the lemma). Thus, ye C({j>,z},ii0) by axiom N. Since R°(z,y) >m = R°(y,z), by Lemma 6, z e c({y,z},R?). Therefore, by axiom /?, we have

yeC({x,y,z},R°) if z e C({x,y,z},BP) Fact 3. Since the choice function has to be nonempty, Fact 1, 2 and 3 leads to a

contradiction. This contradiction proves that y e C({x, v}, R*). We now complete the proof of the lemma. So consider any Rf such that

R'(y,x) >m, we want to show that ye C({x,y},R'). Now R'(x,y) < 1 =



268 K. Sengupta

R*(x,y) and R(y,x) > R°{y,x) = m and ye C{{x,y},R*). Thus, the result follows by applying axiom M. ■

We now prove the necessity part of the theorem.

Necessity

Pick any x*,y* eX and let

K = {ReH : {x'y*} = C({(x*,f)hR)} To check that 9Î is non empty, consider R* which has R*{x,y) = 1 for all

x, y e X. Clearly, R* e H. Moreover, by Lemma 5, {x,y} = C{{x,y},R*) for allx,je X. Let

A = {zR:<xR = R(y*,x*) and R e «} Denote by a* the infimum of ccr on A. We now show that C(A,R) = Sa.(A,R).

Stepl. C{A,R)czS(X*(A,R). Let x g C(A, R). Now, if x $ S** (A, R), there exists z e A such that z >a* x. This implies that R(x, z) < a*. Since x e C(A, R)9 by CC, x e C({x, z}, Ä). By Lemma 7, then x€ C({jc,z},^;), where /^(Zjjc) = 1 and R'x,z) = ä(jc,z). Thus, by axiom N, / e C({x'y*},R*) where ä*(jc*,/) = 1 and /**(/,**) = i?(x,z) < a*. This, however, contradicts the definition of a*.

Step 2. Sar(A,R) c C(i4,Ä). Let x e Sa* (i4, R). Consider any z e C(A, R). By CC, z g C({*, z}, /?).

Now, if Ä(x,z) > Ä(z,x), by Lemma 5 or 6, x e C{{x,z), R). Therefore, {x, z} = C({x, z},R). By condition ß then, x e C(A, R) whenever z g C(A, R).

Now if R(x, z) < R(z, x), we have R(x, z) > a* since x e Sa* (A, R). Since C(.) satisfies continuity, the infimum a* is attained on A, i.e, there

exists some R* such that a* = R*(y*,x*) and R* eÑ. By Axiom M, /* e C({jc*,/},H')> where, Rf(x%y*) = R(z,x) and

Ä'(/, a*) = R(x, z) > a*. Thus, by axiom N, we have x g C({x, z}, jR) There- fore, {x,z} = C({x,z},Ä). By applying condition ß (once more), we have xeC{A,R). M

Proof of Theorem 2

Sufficiency. Ma procedure clearly satisfies the axioms I, CC, N, M and CON. We show that it satisfies condition ß on the domain H.

Let {x,y} - Ma({x,y},R). Now suppose xe Ma(AyR) and y e A. Since x g Ma(A,R), we have R(x,z) > a for all z e A. Moreover, R(y>x) > a since y e Ma({x,y), R). Consider any z e A. Since ReH,we R(y, z) > min[R(y, x), R(x, z) > a. This implies that ysM^A.R). ■

Necessity. First note that since H c //, from theorem 1, it follows that any C satisfying all the axioms must be of the form 5«, 1 > a > 0. We now show




that if a > 1/2, then there exists Re H such that Sa(A, R) must violate the ß condition.

So let a>l/2. Consider now A = {x,yyz} such that R(xiy) = R(y,x) = .5, R{y,z) = R(z,y) = .5, R(x,z) = .8, R{z,x) = .5 R(wyk) = 1 if w $A and R(k, w) = 0 if k e A and w $ A . It is possible check that ReH.

Since 5a procedure satisfies N, by Lemma 5, we have {x,y} = Sadx.yj.R) and {y,z} = £«({>>, z},Ä). Therefore, by condition ß, A = Sa{A,R). By CC, z6Sa({jc,zj,Ä). But this is impossible as Ä(z,jt) = 1/2 < a. ■

Proof of Theorem 3

Sufficiency. The OV(A,R) procedure clearly satisfies the axioms /, CC,N,M and CON. We show that it satisfies ß on the domain Hw-

Let {x,y} = OV{{x,y}, R) and let x e OV(A, R) with y e A.We will show that y e OV(A,R). Otherwise, there exists we A such that R(wyy) > R(y, w).

Since {x,y} = OV({x,y},R), R(x,y) = R(y,x). Moreover, x e OV(A,R) implies that R(x, w) > R(w, x) for all w e A.

Now i{R(x, w) = R(w,x), then by Lemma 1 or condition T, we know that R(y, w) = R(w,y), contradicting the hypothesis that R{w,y) > R(y, w).

Now if R(x,w)> R(w,x), then, from Lemma 1.2 of Dasgupta and Deb (1996), we have, R(x,y) > R(y,x) contradicting our hypothesis that yeOV({x,y},R). ■

Necessity. First note that since H c Hw, from theorem 1, it follows that any C satisfying all the axioms must be of the form 5a, 1 > a > 0. We now show that if a 7* 1, then there exists ReHw such that S^A, R) must violate the ß condition.

So let for some a>0,a#l, Sa satisfies the ß condition. Consider A = {x,y,z) such that R{x,y) = l,R(y,x) = 0, R(y,z) = l,R(ziy)=0, ä(jc,z}= l,Ä(z,x) = a. R(w,k) = 1 if w$A and A(fe,w)=0 if keA and w 4 A.

Note that because of the restriction of applying weak max min transitivity, we do not require R(y,x) > min[R(y,z),R(z,x)]_ or R(zìy)^.rma[R(zìx)ì R(x,y)), therefore R as defined above belongs to Hw-

Now by the definition of 5«, we have {x,z} = Sa({x,z},Ä). Thus, by condition ß, zeSu(A,R) if xe Sx(A,R). But x a-dominates y and y a-dominates z, thus, {x} = Sa(A, R), a contradiction to condition ß. ■

Proof of Theorem 4

Here, we show that the axioms in Theorem 1 are independent. To show this, we will drop each of the axioms sequentially and show that in each case, there exists a PCF that satisfies the rest of the axioms.

Case I. I does not hold.

Choose any R* e H and define C(A,R) = OV{A,R) for all R^R*. For R = R' define C{A,R*) = A for all A. It clearly satisfies axioms CC, ß and



270 K. Sengupta

CON. Recall that condition N and M is not required to hold since axiom I is being violated.

Case II. CC does not hold.

Let C(A,R) = OV(A,R) if 'A' = 2, otherwise, C(A,R) = A. Clearly, it satisfies I. Since on pair, the choice rule is Orlovsky, it also satisfies N, M and CON since these are conditions on pair wise choice only. Finally, ß is satisfied since for 'A' > 3, C{A, R) = A.

Case HL ß does not hold.

Consider R* such that R*(x,y) = 1 for all x,yeX. Now, define C(A,R) = OV(A,R) whenever R ^ R* or A =¿ X. For A = X and R = R' define C(X, R) = {x*} for some x* e X. Clearly, this choice rule satisfies /. Again N, M and CON are satisfied as they apply to a pair. CC is satisfied since x* e C({x*,y}, R*) for all yeX. Clearly, condition ft is violated for R = R*.

Case IV. N does not hold.

Pick any a* e X. Define now C{AyR) = OV(A,R), whenever x* $A, otherwise, define C{A,R) = OV(A,R)kj {x*}. This trivially satisfies all axioms butN.

Case V. M does not hold.

Let

C(A,R) = {xeA'R(x,y)<R(y,x) for all y e X}

Essentially, this choice set chooses the least preferred elements in a set. Given that Re H, it is possible to check that this rule generates non empty choices for every possible A and R. It also satisfies all axioms but M.

Case VI. CON does not hold

Pick any a < 1/2 and define the following modified Ma rule:

C(AìR) = M%(AìR) = {xeA'R{x,y)>oL for all y s A} It is possible to check that M^ satisfies axioms I,CC, ß, M and N. However, it is possible to take asequence R¿ and pair x,y such that Ri(x,y) > a for all i but at the limit, R, R{xiy) = <x. Clearly, xeC^jJ.i,-) for all / but x t C({x,y},R) violating CON.

The proof that the axioms for the other two theorems are independent is similar and is left to the reader.

8. Conclusion

In this paper, we have provided an axiomatic characterization of the choice rules that a decision maker may follow when his preferences are fuzzy. Except for the last characterization (the Orlovsky rule), our characterization results,




however, are non unique. In particular, it is no longer true that under fuzzy preferences, two agents with identical preferences end up making the same choices. This ambiguity, however, is a useful one as it removes the one to one correspondence between preference and actual choices in situations when agents have exact preferences.

One can point out at least two weaknesses of the present paper. First, as is demonstrated in the theorems, our characterization results crucially depend on the domain of preferences, in particular, the transitivity requirements. It will be extremely important and interesting to analyze a scenario where this is not so. One obvious route could be to try to characterize the choice rules of an agent whose choice sets are restricted to only two element sets. Recently, some progress in this direction has been made in Pattanaik and Sengupta (1995).

The second problem concerns the use of the axiom ß. As was noted in Remarks 5 and 6, the primary reason that our characterization results depended on the transitivity requirement is because of the axiom ß. More importantly, with condition /?, the outcome of our choice rules have the property that once the choices are known, it can be 'rationalized' by an exact ordering. Since it is easy to visualize consistent choices that does not necessarily require transitivity of preferences (as in quasi transitive or acyclic preference), it might be of interest to relax this stringent requirement. Sengupta (1997), has made some progress in that direction. By substituting ß by a condition which is called 'independence of rejected alternatives', I have been able to characterize Orlovsky's procedure in a framework where Banerjee (1993) required the ß condition. However, in that structure, the domain of preferences is too large, i.e much larger than that of Hw- With too large a domain of preference, however, consistency requirements on choices have extremely strong bites as was shown in Sengupta (1997). It might be of interest to characterize choice rules under the domain of max-min transitive preference without using axiom /?. This remains at this point an interesting and open question.

References

[1] Banerjee A (1993) Rational choice under fuzzy preferences: The Orlovsky choice function. Fuzzy Sets Systems 53: 295-299

[2] Barrett CR, Pattanaik PK (1985) On Vague Preferences. In: Enderle G (ed) Ethik and Wirtschaftswissenschaft. Duncker & Humbolt, Berlin, pp 69-84

[3] Barrett CR, Pattanaik, PK, Salles M (1990) On Choosing Rationally when Pref- erences are Fuzzv. Fuzzv Sets Systems 34: 197-212

[4] Basu K (1984) Fuzzy Revealed Preference Theory. J Econ Theory 32: 212-227 [5] Dasgupta M, Deb R (1996) Transitivity and Fuzzy Preferences. Soc Choice

Welfare 13: 305-318 [6] Dutta B, Panda SC, Pattanaik PK (1986) Exact Choices and Fuzzy Preferences.

Math Soc Sci 11:53-68 [7] Kolodziejczyk W (1986) Orlovsky's concept of decision making with fuzzy pref-

erence relations - further results. Fuzzy Sets Systems 19: 11-20 [8] Orlovsky SA (1978) Decision making with a fuzzy preference relation. Fuzzy Sets

Systems 1: 155-167



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[9] Pattanaik PK, Sengupta K (1995) On the Structure of Simple Preference Based Choice Functions. Mimeo, University of California at Riverside

[10] Sen AK (1970) Collective Choice and Social Welfare. Holden-Day, San Fran- cisco, CA

[1 1] Sengupta K (1998) Fuzzy Preference and Orlovsky Choice Procedure. Fuzzy Sets Systems 93: 231-234



Choice rules with fuzzy preferences: Some characterizations

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