E L S E V I E R Fuzzy Sets and Systems 93 (1998) 231-234

ImUI3V sets and systems

Fuzzy preference and Orlovsky choice procedure K u n a l S e n g u p t a a

a Department of Economics, Jawharlal Nehru University, New Delhi 110067, India I

Received June 1995; revised February 1996

Abstract

This paper provides an axiomatic characterization of the Oflovsky choice procedure for a decision maker whose preferences are fuzzy. In contrast to the result in Banerjee (1993), our characterization does not require that in a pairwise choice, the choice coincides with the Orlovsky set. Moreover, our characterization result holds for a larger domain of preferences than that in Banerjee (1993). © 1998 Published by Elsevier Science B.V.

Keywords: Fuzzy preferences; Orlovsky procedure

1. Introduction

Orlovsky [3] suggested a simple procedure for gen- erating choices for a decision maker whose prefer- ences are fuzzy. In a recent contribution to this journal, Banerjee [1] provides a set of axioms that uniquely characterizes this procedure. Broadly, the axioms in [ 1 ] can be broken up into two sets. The first set of ax- ioms (5.2 and 5.3) tells us how to translate the fuzzy preferences in a pairwise choice. In particular, these axioms ensure that in a pairwise choice, the choice set coincides with the Orlovsky set, i.e., x belongs to the choice set of {x, y} if and only ifR(x, y ) >~R(y,x). The second set of axioms (5.4 and 5.5) are the well known /~ and Chernoff conditions. The purpose of these ax- ioms is to extend the choices made in a pair to a larger set in a consistent fashion.

The purpose o f this paper is to provide an alterna- tive characterization o f the Orlovsky procedure. First we replace axioms 5.2 by a much weaker axiom that

Address for correspondence. E-mail: ktmal@jnuniv.ernet.in.

only requires that an alternative y be rejected in a pairwise choice with x when the preferences between x and y are exact and x dominates y, i.e., R(x, y ) = 1 and R ( y , x ) = 0. Secondly, we replace condition/~ by an axiom we call independence of rejected alternatives (IRA). This condition in the presence o f Chernoff con- dition is weaker than condition/~. As a consequence, our characterization result is shown to be valid on a larger domain of preferences than that is implied by the max-min transitivity or the transitivity assumption used in Banerjee.

2. Preliminaries

Let X be a nonempty set that stands for a fixed finite number of alternatives. Throughout we will as- sume that IX[/> 3. Also let Z be the set o f all possible nonempty subsets o f X.

Definition 1. A fuzzy binary preference relation (FBPR) is a function R : X 2 ~ [0, 1]. The relation-

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232 K. Sengupta/ Fuzzy Sets and Systems 93 (1998) 231-234

ship R is said to be exact 2 iff for all x,y, R(x,y) is either 0 or 1. R is said to be reflexive iff for all x EX, R(x,x) = 1. R is said to be connected iff for all x, y EX, R(x, y) + R(y,x) >t 1.

Denote by ~R, the set of all such reflexive and con- nected FBPR.

When preferences are fuzzy, there are many ways of defining the notion of transitivity. In this paper, I restrict my attention to the following notion of transi- tivity.

Definition 2. R E ~R is said to be weakly transitive if and only if for all distinct x, y,z EX, R(x, y) > R(y,x) and R(y,z) > R(z, y) implies R(x,z) > R(z,x).

Denote by Hw, the set of all R E 9~ which are weakly transitive.

Kolodziejczyk [2] has shown that this notion of transitivity is weaker than the notion of max-min tran- sitivity, the one most commonly used in the theory of fuzzy relationships. Also, the notion of T2 transitivity employed in Banerjee clearly implies this notion.

Definition 3. A preference based choice function (PCF) on domain f2 C 91 is a function C : X × f2 ~ X. A (PCF) is said to be well defined if for all R E f2 and AEX, ~ # C(A,R) CA.

It is known that if f2 C Hw, the Orlovsky PCF denoted as OV(A,R) is well defined and can be expressed as

OV(A,R)

-- {xEA\R(x,y)>.R(y,x) for all yEA}. (1)

Since in this paper, we work under Hw, we will use (1) to be the definition of Orlovsky PCF.

3. An alternative characterization O V(A, R)

Consider the following axioms. A PCF C is said to satisfy (a) Chernoffcondition (CC): For all x, y EX and

A E Z, A D{x, y}, x q~ C({x, y} ,R) implies x q~ C(A, R).

2 This is the usual Boolean case.

(b) Independence of rejected alternatives (IRA): Suppose x q~ C(A,R), then, C(A,R) = C(A/ {x},R ).

(c) Symmetry (S): For A E Z and R E ~R, if R(x, y) = R(y,x) for all x, yEX , then, C(A,R) = A.

(d) Axiom of rejection (AR): For any R, R(x,y) = 1 and R(y,x) = 0 implies that {x} = C({x, y},R).

(e) Independence (I): For any A E Z and R,R ~, if R(x,y) = R'(x,y) for all x,y E A, then, C(A,R) = C(A,R').

The first condition is the familiar Chernoff condition (also used in Banerjee) that requires that a pairwise rejected element be not chosen in a larger set.

The second condition tries to capture the idea that the optimal choice in a set should not depend on the alternatives that are already rejected. In place of this axiom, Banerjee uses condition fl (originally due to Sen [4]).

Condition ft. For all x, y E X and A E ~, A D {x, y}, {x, y} = C({x, y} ,R) implies x E C(A,R) if and only if y E C(A, R).

Although IRA and fl are logically independent conditions, it can be shown that in the presence of Chernoff condition, fl implies our axiom IRA. The converse, however, does not hold.

The third assumption is a very weak symme- try requirement that forces the decision maker to choose the entire set whenever his preference inten- sity across all the pairwise comparisons are the same. This axiom is a slight strengthening of the pairwise symmetry condition (5.3) in Banerjee. However, in presence of t , pairwise symmetry implies our notion of symmetry.

The fourth assumption just says that if preferences were exact and x is preferred to y, then y cannot be chosen in the pair wise choice between x and y. Note that this axiom is a substantial weakening of the axiom (5.2) in Banerjee that requires y to be rejected in a pairwise choice with x whenever R(x, y) > R(y,x).

Finally, the independence axiom ensures that for two different preference orderings R and R ~, the deci- sion maker's choice from A be the same if the pref- erence under R and R ~ is the same when restricted to the set A.

Theorem 1. Let (2 = Hw. A well defined PCF on the domain Hw satisfies axioms CC, IRA, S, AR and I if and only it coincides with the Orlovsky PCF.

K. SenguptalFuzzy Sets and Systems 93 (1998) 231-234 233

The sufficiency part of the theorem will require the following lemma.

Lemma 1. Suppose x ~ OV(A,R), then, there exists y E O V ( A , R ) such that R(y ,x) > R(x,y).

Proof. The lemma is clearly true for [A I = 2. So as- sume as an induction hypothesis that the lemma is true for all [A[ = 1,2, . . . , k and let [A[ ~- k + 1.

Consider now x EA/OV(A,R) = B. Now if x E OV(B,R), then R(x, y)>~R(y,x) for all

y E B. Thus, for x not to belong to OV(A,R), there must exist z C OV(A,R) such that R(z,x) > R(x,z) and the lemma is proved.

Now if x f~ OV(B,R). By induction hypothesis (since [BI ~ k), there exists z E OV(B,R) such that R(z,x) > R(x,z). Since z ~ OV(A,R), there exists y E OV(A,R) such that R(y ,z ) > R(z,y) . But since R E Hw, we have R(y,x) > R(x ,y) and thus the lemma is proved. []

Proof of Theorem 1 (Sufficiency). It is obvious that OV(A,R) satisfies the axioms (CC, S, AR, I). We now check that OV(A,R) also satisfies IRA. So let x ~ OV(A,R). We want to show that OV(A/{x} ,R) = OV(A, R). That OV(A,R) C OV(A/{x} ,R) follows from the definition of OV procedure. To see the reverse implication, consider z E OV(A/{x},R). I f z ~ OV(A,R), by Lemma 1, there exists some y E OV(A,R) such that R(y ,z ) > R(z,y). But y E OV(A/{x} ,R) and thus z~OV(A/ {x} ,R ) . []

To prove necessity, the following lemmas will be used.

Given a FBPR, R, we say that R is weakly transitive on A E Z if the restriction of R on A (denoted R(A)) is weakly transitive.

Lemma 2. Given any R E ~R, i f R(A) is weakly transitive, then there exists a FBPR, R' such that R'(A) = R(A) and R' EHw.

Proof. Let R' satisfies the following: R'(x ,y) = R(x ,y ) for a l lx , y CA, R ' (x ,y) -- 1 i f x ~ A and Rt(x, y) = 0 if x E A and y ~ A. It is easy to check that R' E Hw. []

Lemma 3. For any R E Hw and any A = {x, y} , we have C(A,R) = OV(A,R).

Proof. I f R ( x , y ) = R(y,x), then by axiom S, it fol- lows that C(A,R) = {x,y} = OV(A,R).

So assume without loss of generality that R(x, y) > R(y, x). We want to show that y (~ C(A, R).

Assume the converse, i.e., let yE C({x, y},R). Consider now z E X, z ¢ x, y and let R ~ be such

that R'(x, y) = R(x, y), R ' (y ,x) = R(y,x) , R'(x,z) = R'(z ,y) = 1 and R'(z,x) = R~(y,z) -- 0. It is easy to check that R'(A) is weakly transitive, where, A -= {x, y,z}. Thus, by Lemma 2, there exists R* such that R*(A) = R'(A) and R* EHw.

Now by axiom AR, z f~ C({x,z} ,R*) and y C({z,y},R*). Thus, by axiom CC and the fact that C is well defined, we have {x} = C( {x, y,z},R* ). Since IRA is satisfied, we have C({x ,y , z ) ,R*) = C({x,y} ,R*) = {x}. But R*(x,y) = R(x,y). Thus, by axiom (I) we have {x} -- C({x,y} ,R*) = C({x,y} ,R) , but this contradicts our hypothesis that y G C({x,y} ,R) . Thus, y f~ C({x,y} ,R) . Since, C({x ,y} ,R) is nonempty, the lemma is proved. []

Lemma 4. I f R E Hw, then for any A, there exists x E A such that R(x,y)<~R(y,x) for all yEA.

Proof. For xEA, define

Px = { y E A \ R ( y , x ) > R(x, y)}.

I fPx is empty for all x, then R(x,y) = R(y ,x) for all x, y E A and the lemma is proved. So assume that Px ¢ 0 for some x.

Let x* be such that IPxl is maximized at x*. I claim that x* satisfies the lemma.

If not, then, there exists some y such thatR(y,x* ) < R(x*, y). Now for every z E Px*, we have R(z,x*) > R(x*,z). Since, R E Hw, we then have R(z ,y) > R(y,z) . This implies that z E Py. Since x* E Py and x* q~ Px* (reflexivity), we violate our assumption that [Px[ is maximized at x*. []

Proof of Theorem 1 (Necessity). By Lemma 3, C(A,R) = OV(A,R) whenever IAI = 2. So assume as an induction hypothesis that C(A,R) = OV(A,R) for all A with IA[ -- 2, 3 . . . . . k and let [AI = k + 1.

By Lemma 4, there exists x* E A such that R(x*,y)<~R(y,x*) for all y E A. Now if this holds with an equality for all y, then by axiom S, C(A,R) = A = OV(A,R). Otherwise, there exists y E A such that R(y,x* ) > R(x*, y). Clearly, x* f~ OV(A,R).

234 I~ Senoupta l Fuzzy Sets and Systems 93 (1998) 231-234

Consider B = A/{x*}. By induction hypothesis, C(B,R) = OV(B,R). since IBI = k. Since x* OV(A,R) and OV satisfies IRA, we have OV(A,R) = OV(B,R).

Given that R(x*,y) < R(y,x*), by Lemma 3, x* q~ C({x*,y} ,R) = OV({x* ,y) ,R) . Thus by CC, x* f~ C(A,R). Since C satisfies IRA, we have C(B,R)=C(A,R) . Thus, C(A,R)=OV(A,R) . []

We now check that the axioms in Theorem 1 are independent of each other.

Theorem 2. The axioms CC, IRA, S, AR and I are independent.

Proof. 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: CC does not hold. Let C(A,R) = OV(A,R) if [A[ = 2, otherwise,

C(A,R) = A. Clearly, it satisfies S and I. Since for ]A]~>3, C(A,R) = A, IRA is vacuously satisfied. Finally, since AR applies to a pair only, it is also satisfied.

Case II: IRA does not hold. Pick some z E X and consider R* such that

R*(x, y) = R*(y,x) for all x, y ¢ z and R*(x,z) > R*(z,x) for x E X. Clearly, R* E Hw. Now, define C(A,R) = OV(A,R) whenever R ¢ R* or A ¢ X. For A = X and R = R*, define C(A,R) = {x} for some x E X. Clearly, this choice rule satisfies CC, S AR, I. Now C(X,R*) = {x} and C(X/{z},R*) = X/{z} since IX[ ~>3, IRA is violated.

Case III: S does not hold. Consider R* E Hw such that R*(x, y) -- R*(y,x)

for all x , y E X. Define now C(A,R) = OV(A,R) whenever R ~ R* or A ~ X. For A --- X and R -- R*, let C(A,R) = A/{x} for some x E X. It is easy to check that this satisfies CC, IRA, AR and I.

Case IV: AR does not hold. Let C(A, R) = A for all A and R. Clearly, it satisfies

the rest of the axioms.

Case V: I does not hold. Choose R* E Hw such that R*(x,y) > 0 for all

x , y E X. Define now C(A,R) = OV(A,R) for all R ~ R*. For R = R*, define C(A,R*) = A for all A. It clearly satisfies all axioms but I.

4. Concluding remarks

In this paper, we have characterized the Orlovsky PCF on the domain of preferences which are weakly transitive. It might be of interest to know whether one can enlarge this domain. In particular, it is well known that Orlovsky PCF as characterized by Eq. (1) is well defined if and only if R is acyclic. Recall that R is said to be acyclic if for all Xl,X2 . . . . . Xn E X , R(xi,xi+l) > R(Xi+l,Xi), i - - 1,2 , . . . ,n - 1, we have R(xl,xn)>>-R(xn,xl). Unfortunately, however, if R is acyclic, the Orlovsky PCF does not necessarily satisfy axiom IRA. To see that let X -- {x ,y ,z) , R(x ,y) = R(y ,z) = 0.9, R(y,x) = R(z ,y) --- 0.7 and R(x,z) = R(z,x) -- 0.8. This preference relationship is acyclic. However, OV(X,R) = (x} ~ OV({x ,z} ,R) = (x,z} violating IRA. Characterizing Orlovsky PCF on this larger domain remains at this point an open question.

Acknowledgements

I am grateful to Asis Banerjee and Prasanta Pattanaik for helpful discussions.

References

[1] A. Banerjee, Rational choice under fuzzy preferences: the Orlovsky choice fimetion, Fuzzy Sets and Systems 53 (1993) 295-299.

[2] W. Kolodziejczyk, Orlovsky's concept of decision making with fuzzy preference relations - further results, Fuzzy Sets and Systems 19 (1986) 11-20.

[3] S.A. Orlovsky, Decision making with a fuzzy preference relation, Fuzzy Sets and Systems 1 (1978) 155-167.

[4] A.K. Sen, Collective Choice and Social Welfare (Holden-Day, San Francisco, CA, 1970).