A new approach to sum-fuzzy rational choice functions

Fuzzy Sets and Systems 127 (2002) 241–246

www.elsevier.com/locate/fss

A new approach to sum-fuzzy rational choice functions Xiao Luo ∗ Department of Economics, National Taiwan University, Taipei 100, Taiwan, ROC Received 2 February 1999; received in revised form 21 November 2000; accepted 19 January 2001

Abstract The purpose of this paper is to present a new approach to sum-fuzzy rational choice functions. By making use of the model of perceptrons in neural theory, we establish a su1cient and necessary condition for sum-fuzzy rationality. Moreover, we provide a geometric characterization of sum-fuzzy rationality for single-valued choice functions. Based on the learning rules of perceptrons, we o3er an algorithm to 4nd a sum-fuzzy implementation of a choice function and, then, provide a c 2002 Elsevier Science B.V. All rights reserved. concrete example.
Keywords: Decision analysis; Fuzzy rationality; Perceptrons

1. Introduction There are many problems of choice in real life. The study of choice problems is of great signi4cance for all of the social sciences and, in particular, economics and decision theory (see, e.g., [1]). Richter [15] started to investigate the rationality of choice functions (see also [17]). Formally, let X be a universal set, and let P(X ) be the collection of nonempty subsets of X . A choice function C on X is de4ned as a mapping C : P(X ) → P(X ) such that C(A) ⊆ A for all A ∈ P(X ). 1 A choice function C on X is rational if there exists a binary relation R on X such that C(A) = fR (A) for all A ∈ P(X ), where fR (A) = {x ∈ A | (x; y) ∈ R for all y ∈ A}: ∗

Tel.: +886-2-2351-5468; fax: +886-2-2321-5704. E-mail address: [email protected] (X. Luo). 1 In particular, a choice function C on X is single-valued if C(A) is a singleton for each A ∈ P(X ).

However, human preferences are often vague in many practical situations and, thus, fuzzy theory is clearly useful (see, e.g., [6,8,14]). Kim [9] and Orlovsky [13], for instance, 4rst extended the concept of rationality to the concept of fuzzy rationality. 2 In the same vein, Luo et al. [11] introduced two kinds of fuzzy rational choice functions, namely maxminfuzzy rational choice functions and sum-fuzzy rational choice functions. As pointed out by Luo et al., it is an interesting question to characterize the sum-fuzzy rationality. The main purpose of this paper is to present a new approach to sum-fuzzy rational choice functions. More speci4cally, by making use of the perceptrons model in neural theory, we establish a su1cient and neces2 Cf. also Banerjee [2], Barrett et al. [3], Basu [4], Bouyssou [5], Dutta, Panda and Pattanaik [7], Sengupta [16], and Switalski [18] among others; see, in particular, Kulshreshtha and Shekar [10] for a recent discussion.

c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/02/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 0 1 ) 0 0 0 6 5 - 3

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X. Luo / Fuzzy Sets and Systems 127 (2002) 241–246

Table 1 A choice function

C

x1

x2

x3

x4

x1 x 2

x1 x3

x 1 x4

x2 x3

x2 x4

x 3 x4

x1 x2 x 3

x1 x2 x 4

x1 x3 x4

x 2 x3 x4

x1 x2 x3 x4

x1

x2

x3

x4

x2

x3

x4

x2

x4

x3

x1

x1

x3

x2

x1

sary condition for sum-fuzzy rationality. Furthermore, we provide a geometric characterization of sum-fuzzy rationality for single-valued choice functions. Based on the learning rules of perceptrons, we o3er an algorithm to 4nd a sum-fuzzy implementation of a choice function and, then, provide a concrete example. It is worthwhile to emphasize that the novel approach suggested by this paper itself is clearly of great interest in the applications of neural theory. The rest of this paper is organized as follows. Section 2 introduces the concept of sum-fuzzy rationality. Section 3 is a brief description of the model of perceptrons. Section 4 establishes a formal relationships between sum-fuzzy rational choice functions and perceptrons. Section 5 provides a geometric characterization of sum-fuzzy rationality for single-valued choice functions. Finally, Section 6 o3ers a learning algorithm to 4nd a sum-fuzzy implementation of a choice function and, then, provides a concrete example. 2. Sum-fuzzy rationality Let X = {x1 ; x2 ; : : : ; xn }. Let R be a binary relation on X , i.e., R ⊆ X × X . Denition 1. Let [0; 1] be the unit interval of the real line. If there exists a mapping rR : X × X → [0; 1] such that rR (x; y) ∈ (0; 1] if (x; y) ∈ R and rR (x; y) = 0 if (x; y) ∈= R, then (R; rR ) is called a fuzzy binary relation on X . Let (R; rR ) be a fuzzy binary relation on X . Let   rR (x1 ; x1 ) rR (x1 ; x2 ) · · · rR (x1 ; xn )    rR (x2 ; x1 ) rR (x2 ; x2 ) · · · rR (x2 ; xn )  : M (R; rR ) =     ··· ··· ··· ···  rR (xn ; x1 ) rR (xn ; x2 ) · · · rR (xn ; xn ) We call M (R; rR ) the matrix representation of the fuzzy binary relation (R; rR ). The following de4nition is taken from Luo et al. [11].

Denition 2. Let (R; rR ) be a fuzzy binary relation on X . De4ne, for any A ∈ P(X ),     fR (A) = x ∈ A  rR (x; z) ¿ rR (y; z)  z∈A



z∈A

for all y ∈ A : We call fR the sum-fuzzy choice function on (R; rR ). Denition 3. A choice function C on X is sum-fuzzy rational if there exists a fuzzy binary relation (R; rR ) such that C is the sum-fuzzy choice function on (R; rR ), i.e., C = fR . In particular, the matrix representation M (R; rR ) is called a sum-fuzzy implementation of C. Remark. It is easy to see that C is sum-fuzzy rational if, and only if, C is max-SF (cf. [3]). Thus, a sumfuzzy rational function can be viewed as one of nine preference-based choice functions proposed by Barrett et al. and it is of considerable interest to economists and social scientists. Example 1. Let X = {x1 ; x2 ; x3 ; x4 }. Consider a choice function C on X , which is given in Table 1. According to Kim’s notion of fuzzy rationality, C is a fuzzy irrational choice function (see [9, Theorem 1]); however, C is a sum-fuzzy rational choice function. Indeed, a sum-fuzzy implementation of C is as follows:   1:00 0:59 0:35 0:29  0:60 1:00 0:24 0:27     0:40 0:20 1:00 0:30  : 0:30 0:28 0:20 1:00 3. Perceptrons The model of perceptrons, 4rst proposed by F. Rosenblatt in 1957, is a neural network with a

X. Luo / Fuzzy Sets and Systems 127 (2002) 241–246

243

Table 2 Finding out all U [C(A)] U [C(A)]

Fig. 1. The model of perceptrons.

single array of computing units. The standard model of perceptrons composed of linear threshold elements is depicted in Fig. 1 (see, e.g., [12] for details). In Fig. 1, w = (w1 ; w2 ; : : : ; wn ) is a weight vector,  is a threshold, u = (u1 ; u2 ; : : : ; un ) is an input vector, and v is an output value.

x1 x2 x3 x4 x1 x2

∅ ∅ ∅ ∅ {(−1; −1; 0; 0; 1; 1; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0)}

x 1 x3

{(−1; 0; −1; 0; 0; 0; 0; 0; 1; 0; 1; 0; 0; 0; 0; 0)}

x 1 x4

{(−1; 0; 0; −1; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; 1)}

x 2 x3

{(0; 0; 0; 0; 0; 1; 1; 0; 0; −1; −1; 0; 0; 0; 0; 0)}

x2 x4

{(0; 0; 0; 0; 0; −1; 0; −1; 0; 0; 0; 0; 0; 1; 0; 1)}

x 3 x4

{(0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 1; 0; 0; −1; −1)} 

x 1 x2 x3  x 1 x2 x4 

4. Sum-fuzzy rational choice functions and perceptrons For any A ∈ P(X ), let ‘(A) = {l | xl ∈ A}, and let |A| denote the cardinality of A. For any u ∈ n×n , we also write u as

x 1 x3 x4  x 2 x3 x4

x1 x2 x3 x4

(u11 ; u12 ; : : : ; u1n ; u21 ; u22 ; : : : ; u2n ; · · · ; un1 ; un2 ; : : : ; unn ); and we denote by uT the transpose of u. For any A ∈ P(X ) and for any l; k ∈ ‘(A) and l = k, let    1 if i = l and j ∈ ‘(A); lk ij (A) = −1 if i = k and j ∈ ‘(A);   0 otherwise; where i; j = 1; 2; : : : ; n. Now, consider a choice function C on X . Let A ∈ P(X ). De4ne 3   U [C(A)] = {u ∈ n×n | uij = lk ij (A) l∈‘(C(A)) k∈‘(A)\{l}

for i; j = 1; 2; : : : ; n}; 3

If |A| = 1, then we de4ne U [C(A)] = ∅.

(1; 1; 1; 0; −1; −1; −1; 0; 0; 0; 0; 0; 0; 0; 0; 0) (1; 1; 1; 0; 0; 0; 0; 0; −1; −1; −1; 0; 0; 0; 0; 0) (1; 1; 0; 1; −1; −1; 0; −1; 0; 0; 0; 0; 0; 0; 0; 0) (1; 1; 0; 1; 0; 0; 0; 0; 0; 0; 0; 0; −1; −1; 0; −1) (−1; 0; −1; −1; 0; 0; 0; 0; 1; 0; 1; 1; 0; 0; 0; 0) (0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 1; 1; −1; 0; −1; −1) (0; 0; 0; 0; 0; 1; 1; 1; 0; −1; −1; −1; 0; 0; 0; 0) (0; 0; 0; 0; 0; 1; 1; 1; 0; 0; 0; 0; 0; −1; −1; −1)

   

   (1; 1; 1; 1; −1; −1; −1; −1; 0; 0; 0; 0; 0; 0; 0; 0)  (1; 1; 1; 1; 0; 0; 0; 0; −1; −1; −1; −1; 0; 0; 0; 0)   (1; 1; 1; 1; 0; 0; 0; 0; 0; 0; 0; 0; −1; −1; −1; −1)

and de4ne 4 U [C + (A)] =





{u ∈ n×n |uij

l∈‘(C(A)) k∈‘(A)\‘(C(A))

= lk ij (A) for i; j = 1; 2; : : : ; n}:  Finally, let U [C] = A∈P(X ) U [C(A)], and let U + [C]  = A∈P(X ) U + [C(A)]. Example 2. Consider the same choice function in Example 1. All U [C(A)] are listed in Table 2. Remark. If C is a single-valued choice function on X , then U [C(A)] = U + [C(A)] for all A ∈ P(X ). 4

If C(A) = A, then we de4ne U [C + (A)] = ∅.

244

X. Luo / Fuzzy Sets and Systems 127 (2002) 241–246

The following theorem establishes a formal relationship between a sum-fuzzy rational choice function and the model of perceptrons.

Proof. Note that U [C] = U + [C] if C is a singlevalued choice function on X . Corollary 2 follows immediately from Theorem 1.

Theorem 1. A choice function C on X is sum-fuzzy rational if and only if there exists w ∈ n×n such that wuT ¿0 for all u ∈ U [C] and; moreover; wuT ¿0 for all u ∈ U + [C].

5. Sum-fuzzy rationality: a geometric characterization

Proof. Necessity: If C is a sum-fuzzy rational choice function, then there exists a fuzzy binary relation (R; rR ) such that C is the sum-fuzzy choice function on (R; rR ). Let w ∈ n×n such that wij = rR (xi ; xj ) for i; j = 1; 2; : : : ; n. Then, it is easily veri4ed that wuT ¿0 for all u ∈ U [C] and, moreover, wuT ¿0 for all u ∈ U + [C]. Su4ciency: Let w ∈ n×n such that wuT ¿0 for all u ∈ U [C] and, moreover, wuT ¿0 for all u ∈ U + [C]. Let wM and w be the maximum and minimum of components of w. Clearly, the proof is complete if wM = w. Now suppose that wM = w. De4ne the following matrix:  ∗ ∗ ∗  w11 w12 · · · w1n  w∗ w∗ · · · w∗   21 22 2n   ··· ··· ··· ··· ;

Theorem 2. A single-valued choice function on X is sum-fuzzy rational if and only if co(U + [C]) does not contain the origin of n×n .

∗ wn1

∗ wn2

∗ · · · wnn

where wij∗ = [wij −w]=[w−w] M for i; j = 1; 2; : : : ; n. Note that wij∗ ∈ [0; 1] for i; j = 1; 2; : : : ; n. Therefore, there exists a fuzzy binary relation (R; rR ) such that C is the sum-fuzzy choice function on (R; rR ). Remark. By Theorem 1, we can transform the problem of sum-fuzzy rationality into the model of perceptrons. Corollary 1. A choice function C on X is sum-fuzzy rational if and only if there exists w ∈ n×n such that (i) wuT = 0 for all u ∈ U [C]\U + [C]; and (ii) wuT ¿0 for all u ∈ U + [C]. Proof. Note that −u ∈ [C]\U + [C] if u ∈ [C]\U + [C]. Corollary 1 follows immediately from Theorem 1. Corollary 2. A single-valued choice function on X is sum-fuzzy rational if and only if there exists w ∈ n×n such that wuT ¿0 for all u ∈ U + [C].

In many economic applications, e.g., demand functions in classical consumer theory (cf. [17, Chapter 2]), choice functions are usually assumed to be singlevalued. In this section we provide a geometric characterization of sum-fuzzy rationality for single-valued choice functions. Let co(U + [C]) denote the convex hull of U + [C].

Proof. Su4ciency: Let · denote the Euclidean norm in n×n . Since co(U + [C]) is compact, there exists w ∈ co(U + [C]) such that

w 6 u

for all u ∈ co(U + [C]):

We proceed to prove that wuT ¿wwT for all u ∈ U + [C]. Assume, in negation, that there exists uˆ ∈ U + [C] such that wuˆ T = wwT − , where ¿0. Let  ∈ [0; 1]. Then,

(1 − )w + u ˆ 2 = (1 − )2 wwT + 2(1 − )wuˆT + 2 uˆuˆT = wwT − 2(1 − ) + 2 [uˆuˆT − wwT ]: Now, choose ∗ ∈ [0; 1] such that [∗ =(1 − ∗ )][uˆ uˆ T − wwT ]¡2. Then, (1−∗ )w+∗ u ˆ 2 ¡wwT . However, ∗ ∗ + since (1 −  )w +  uˆ ∈ co(U [C]), we arrive at a contradiction. Thus, wuT ¿wwT for all u ∈ U + [C]. But since co(U + [C]) does not contain the origin of n×n , wuT ¿0 for all u ∈ U + [C]. By Corollary 2, C is a sumfuzzy rational choice function. Necessity: Suppose that C is a sum-fuzzy rational choice function. By Corollary 2, there exists w ∈ n×n such that wuT ¿0 for all u ∈ U + [C]. Assume, in negation, that co(U + [C]) contains the origin of n×n . Let U + [C] = {u1 ; u2 ; : : : ; um }. Then, there exists a

X. Luo / Fuzzy Sets and Systems 127 (2002) 241–246 Table 3 Single-valued choice functions

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

x1

x2

x3

x1 x2

x2 x 3

x1 x3

x1 x 2 x3

x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1

x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2

x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3

x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2

x2 x2 x2 x2 x2 x2 x3 x3 x3 x3 x3 x3 x2 x2 x2 x2 x2 x2 x3 x3 x3 x3 x3 x3

x1 x1 x1 x3 x3 x3 x1 x1 x1 x3 x3 x3 x1 x1 x1 x3 x3 x3 x1 x1 x1 x3 x3 x3

x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3

sequence, {i }m i=1 , in [0; 1] with m

m

i=1

i = 1 such that

i ui = 0:

245

Table 4 Sum-fuzzy implementations     1:00 1:00 0:50 0:67 0:50 0:00  0:00 0:50 1:00   0:17 0:33 1:00  0:50 0:00 0:50 0:17 0:17 0:33





   0:40 1:00 0:20 0:75 0:75 0:00  0:00 0:40 0:40   0:25 0:50 1:00  0:80 0:00 0:60 0:75 0:25 0:50





   1:00 0:67 0:67 1:00 0:57 0:29  0:00 0:30 0:00   0:71 0:57 0:80  0:00 0:67 0:33 0:00 1:00 0:71





   0:50 1:00 0:33 1:00 1:00 0:00  0:17 0:33 0:00   1:00 0:67 1:00  0:33 0:17 0:67 0:17 0:83 1:00





   0:33 0:17 1:00 0:50 0:00 1:00  0:50 0:67 0:00   1:00 1:00 0:50  0:17 0:17 0:33 0:00 0:50 0:50





   0:57 0:71 0:86 0:33 0:00 0:00  0:71 1:00 0:14   1:00 1:00 0:67  1:00 0:00 0:86 0:33 0:00 0:33





   0:50 0:25 1:00 0:75 0:00 0:75  0:75 0:75 0:00   1:00 0:50 0:25  0:25 0:75 0:50 0:00 1:00 0:50







   0:62 0:88 0:88 0:33 0:17 0:17  0:88 0:75 0:00   1:00 0:33 0:17  0:75 0:38 1:00 0:00 0:50 0:67

 1:00 0:14 0:71  0:00 0:71 1:00  0:71 0:86 0:57  0:40 0:40 0:00  0:00 0:60 0:80  1:00 0:20 0:40  0:80 0:20 0:40  0:00 0:40 0:40  0:40 1:00 0:40  0:50 1:00 0:00  0:00 0:50 0:50  1:00 0:50 1:00  1:00 0:00 1:00  0:33 1:00 0:83  0:83 1:00 0:67  0:40 0:00 0:40  0:20 0:80 0:40  1:00 0:40 0:40  0:75 0:00 0:75  0:75 0:50 0:25  0:75 1:00 0:50  0:50 0:00 0:50  1:00 0:50 0:00  0:50 1:00 1:00

i=1

Thus, m

i wuiT = 0:

i=1

It, therefore, must be the case that wuT = 0 for some u ∈ U + [C]: This is a contradiction.

6. An algorithm Based on the learning rules of perceptrons in neural theory, we now give an algorithm to 4nd a sum-fuzzy implementation of a choice function as follows: Step 1. Let i; j = 1; 2; : : : ; n. Input a beginning value of wij .  Step 2. Input u ∈ U [C] and calculate v = wij uij .

Step 3. Go to Step 5 if (i) v = 0 when u ∈ U [C]\ U + [C] and (ii) v¿0 when u ∈ U + [C]; go to Step 4 otherwise. Step 4. Modify wij into wij + "uij , where 0¡"¡1. Step 5. Go to Step 2 until v is stable for all u ∈ U [C], i.e., (i) v = 0 for all u ∈ U [C]\U + [C] and (ii) v¿0 for all u ∈ U + [C]. Step 6. Let wij∗ = [wij − w]=[wM − w], where w = min{wij } and wM = max{wij }. Step 7. Output the outcome, and stop. Example 3. Let X = {x1 ; x2 ; x3 }. Consider all singlevalued choice functions on X , which are given in Table 3. Note that U [C] = U + [C] since C is a single-valued choice function. By making use of the algorithm given in this section, we calculate all sum-fuzzy implementations of the single-valued choice functions in Table 4.

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X. Luo / Fuzzy Sets and Systems 127 (2002) 241–246

Acknowledgements The author would like to thank the editors for their encouragement and two anonymous referees for helpful suggestions that led to this revision. Financial support from the Social Sciences and Humanities Research Council of Canada (SSHRC), the National Sciences Council of Taiwan, and the Economic and Social Research Council of UK is gratefully acknowledged. The usual disclaimer applies. References [1] K.J. Arrow, Social Choice and Individual Value, Wiley, New York, 1963. [2] A. Banerjee, Rational choice under fuzzy preferences: the Orlovsky choice function, Fuzzy Sets and Systems 79 (1996) 407. [3] C.R. Barrett, P.K. Pattanaik, M. Salles, On choosing rationally when preferences are fuzzy, Fuzzy Sets and Systems 34 (1990) 197–212. [4] K. Basu, Fuzzy revealed preference theory, J. Economic Theory 32 (1984) 212–227. [5] D. Bouyssou, Acyclic fuzzy preferences and the Orlovsky choice function: A note, Fuzzy Sets and Systems 89 (1997) 107–111. [6] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.

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