Theoretial Eonomis Letters, 017, 7, 13- http://wwwsirporg/journal/tel ISSN Online: 16-086 ISSN Print: 16-078 A Funtional Representation of Fuzzy Preferenes Susheng Wang Department of Eonomis, Hong Kong University of Siene and Tehnology, Hong Kong, China How to ite this paper: Wang, SS (017) A Funtional Representation of Fuzzy Preferenes Theoretial Eonomis Letters, 7, 13- http://dxdoiorg/10436/tel0177100 Reeived: Otober 6, 016 Aepted: Deember 9, 016 Published: Deember 13, 016 Copyright 017 by author and Sientifi Researh Publishing In This work is liensed under the Creative Commons Attribution International Liense (CC BY 40) http://reativeommonsorg/lienses/by/40/ Open Aess Abstrat This paper defines a well-behaved fuzzy order and finds a simple funtional representation for the fuzzy preferenes It inludes the existing utility theory for exat preferenes (no fuzziness) as a speial ase It is a simple and intuitive extension of the utility theory under unertainty, whih an potentially be used to explain a few known paradoxes against the existing expeted utility theory Keywords Funtional Representation, Utility Representation, Fuzzy Preferenes 1 Introdution A funtional representation of individual preferenes is a ruial issue in eonomis, espeially when some puzzling experimental paradoxes against existing funtional representations were found (suh as the well-known Allais paradox and the St Petersburg paradox) The non-expeted utility approah is one way that many eonomists have tried in their effort to obtain a better funtional representation of individual preferenes 1 This paper looks into the issue from a different angle: fuzzy preferenes Traditional individual preferenes are exat preferenes, for whih the preferene of one onsumption bundle over another is simply a yes or no Fuzzy preferenes, however, allow for a degree of preferene of one onsumption bundle over another Exat preferenes impose ompleteness in the sense that any two onsumption bundles an be ompared Fuzzy preferenes, on the other hand, allow for a ontinuum of attitudes towards preferenes over any two hoies We may view fuzziness in preferenes over onsumption bundles as a version of weak ompleteness that is, fuzziness measures the degree of ompleteness or vagueness Completeness of preferenes, without any fuzziness, is used in the existing theory of utility representation and even in the theory of non-expeted utility representation However, many experimental investigations and real-life examples suggest that indi- 1 See, for example, Kreps and Porteus [1], Epstein and Zin [], and Weil [3] DOI: 10436/tel0177100 Deember 13, 016
S S Wang vidual preferenes do not neessarily satisfy the ompleteness axiom In fat, some eonomists have argued that inomplete preferenes may be a major ause of the problems found in the existing theories of utility representation of preferenes One way to get around this problem is to replae ompleteness with fuzziness As we will show, this generalization will still yield a simple funtional representation This new funtional representation is muh more general, but it still has many of the attrative properties of the existing theories of funtional representation for exat preferenes Fuzzy preferenes have been extensively studied in the literature Researhers in this area have found many interesting appliations of fuzzy preferenes to eonomi problem, espeially to problems of rational hoie and soial hoie, and, in partiular, to the Orlovsky hoie funtion and the Arrow impossibility theorem 3 Our interest is in a funtional representation of fuzzy preferenes, whih, to our knowledge, has never been done in the literature With a onvenient form of funtional representation, theories based on fuzzy preferenes beome readily available for many eonomi problems, espeially for those that employ funtion-based mathematial tools suh as dynami programming and funtional analysis This paper first finds a simple utility representation of fuzzy preferenes, whih inludes the utility representation of exat preferenes as a speial ase It then finds a funtional representation of fuzzy preferenes, whih also inludes the funtional representation of exat preferenes as a speial ase This paper proeeds as follows Setion provides three axioms that define a well-behaved fuzzy preferene order It inludes the existing exat preferene order as a speial ase Some omparisons with existing definitions of fuzzy orders are disussed Setion 3 finds utility and funtional representations of fuzzy preferenes, whih inlude existing funtional representations of exat preferenes as speial ases A partiularly intuitive and onvenient form of the funtional representation is disussed (Theorem 3) Setion 4 onludes the paper with a few remarks Fuzzy Preferenes Defining a proper lass of fuzzy preferenes is a major task in this paper We do not intend to define a general lass of fuzzy preferenes that inludes all possible fuzzy preferenes On the ontrary, we will impose enough restritions so as to obtain a neat funtional representation of fuzzy preferenes Our intention is to strike a balane between generality and simpliity We are interested in a lass of representable fuzzy preferenes n Let be a losed set and R : [ 1,1] be a mapping R is a fuzzy order on if it satisfies the following three axioms Axiom 1 (Reflexivity) R( xx, ) = 0, for all x Axiom (Symmetry) R( xy, ) R( yx, ) =, for all xy, Axiom 3 (Transitivity) For any xy,, if there exists z suh that R( xz, ) 0 and ( ) R z, y 0, then Rxy (, ) 0 A fuzzy order R is an exat order if its range ( ) R ontains only three values See, for example, Shmeidler [4], Flament [5], and Gay [6] 3 See, for example, Basu [7], Dutta [8], Barrett et al [9], Banerjee [10], Dasguptaand Deb [11], Rihardson [1], and Sengupta [13] and [14] 14
S S Wang 1, 0 and 1, ie, R ( ) = { 1, 0,1} That is, an exat order is a speial fuzzy order For an exat order, the three ases Rxy (, ) = 1, 0 and 1 are respetively interpreted as stritly less preferred, indifferent, and stritly preferred Exat orders desribe traditional preferenes, whih are reflexive, transitive, and omplete A fuzzy order is, in ertain sense, an extension of an exat order to an order with weak ompleteness In other words, the preferene of a onsumption bundle over another is no longer a simple yes/no, rather, it has a degree of preferene As we know, under ertain onditions, traditional preferenes have utility and funtional representations The task of this paper is to show that the existing utility and funtional representations of exat preferenes an be extended to fuzzy preferenes In other words, we will find onditions under whih fuzzy preferenes have utility and funtional representations, and we will show that the existing utility and funtional representations of exat preferenes are speial ases In order to ahieve a neat funtional representation of fuzzy preferenes, we must impose restritions over the standard notion of fuzzy preferenes found in the literature Our axioms impose suffiient restritions on fuzzy preferenes and at the same time allow enough generality to over a large lass of sensible preferenes The symmetry axiom above is the key restrition Speifially, the existing definition of a fuzzy order in the literature is a mapping R : [ 0,1], as opposed to our definition of a mapping R : [ 1,1] Although we an onvert the two definitions liter- ally by the formula R( xy, ) = 1 Rxy (, ), the interpretations of their values are different As opposed to interpreting R( xy, ) as the degree to whih x is better than y, we interpret Rxy (, ) as the degree to whih x is better than y and Rxy (, ) as the degree to whih x is worse than y Thus, aording to our interpretation, we should have R( xy, ) = R( yx, ) This is the symmetry axiom Tehnially, our symmetry axiom is a speial ase of the standard ompleteness axiom of R( xy, ) R( yx, ) 1 or R( xy, ) R( yx, ) in the literature With the symmetry axiom, we an intuitively define the onept of indifferene If 4 Rxy (, ) = 0, we say that x and y are indifferent and denote their relation as x y The reflexivity axiom requires a onsumption bundle to be indifferent to itself: x x Our transitivity axiom is new There are quite a few alternative formulations of transitivity in the existing literature, among whih the following three are popular: a) R( x, y) > R( x, z) R( z, y) 1, x, y, z 1 1 b) R( x, y) R( x, z) R( z, y), x, y, z ) R( x, y) min { R( x, z), R( z, y) }, x, y, z Eah of these formulations has its problem For (a), the problem an be explained by an example If x z and z y, we should expet x y But ondition (a) only suggests Rxy (, ) > 1, ie, x y For ondition (b), onsider R( xz, ) = 1 and R( z, y ) = 1 We should not expet these two onditions to tell us anything about the relation between x and y However, ondition (b) implies Rxy (, ) 0 4 Similarly, we an denote x y for ( ) Rxy, = 1, and x y Rxy, = 1 for ( ) 15
S S Wang For ondition (), as pointed out by Basu [7] (footnote 3), onsider R( x, z) = 0, R( x, z) = 1 and R( z, y ) = 0 Sine ondition () indiates Rxy (, ) 0, R x, y to be stritly larger than 0, but ondi- R x, y 0 we should expet the lower bound on ( ) tion () ontinues to indiate ( ) Our version of transitivity is stated in a way that appears to be a natural extension of the transitivity axiom for exat preferenes Beause of the symmetry axiom, we do not need an elaborate formula to define our version of transitivity, and our notion of transitivity appears to be less restritive than other existing notions of transitivity In par- R z, y have noth- tiular, for our notation of transitivity, the sizes of R( xz, ) and ( ) ing to do with the size of Rxy (, ) This avoids the above mentioned problems In addition, our transitivity axiom implies the usual transitivity axiom for exat orders as a speial ase Lemma 1 For any fuzzy order R (, ), if, for some z R( xz) R( z, y) 0 and one of them is strit, then Rxy (, ) > 0,, 0 and Proof: Suppose not Suppose R( xz, ) > 0 and R( z, y) 0 but Rxy (, ) = 0 we have R( z, y) 0 and R( yx, ) = 0 By transitivity, we then have ( ) R( xz, ) 0 This ontradits with the fat R( xz, ) > 0 Rxy (, ) > 0 Then, R zx, 0 or Therefore, we must have Lemma 1 suggests that the usual transitivity ondition for exat orders is a speial ase of our transitivity ondition for fuzzy orders Lemma If x z and z y, then x y Proof: By transitivity, R( xz, ) = 0 and R( z, y ) = 0 implies Rxy (, ) 0 By symmetry, we also have R( yz, ) = 0 and R( zx, ) = 0, whih then implies R( yx, ) 0, ie, Rxy (, ) 0 Thus, Rxy (, ) = 0 Lemma is a standard property of preferenes This property is essential for a funtional representation 3 Funtional Representation We say that a funtion u : R : 1,1 if, for all xy,, u( x) > u( y) R( x, y) > 0 and u( x) = u( y) x y It turns out that the utility representation theorem for fuzzy preferenes is similar to the one for exat preferenes It has similar axioms and similar proofs In fat, utility and funtional representations of exat preferenes are speial ases of the ones for fuzzy preferenes R : 1, 0,1, under ontinuity, we know that there exists a For an exat order { } ontinuous funtion u : represents fuzzy preferenes [ ] and a monotone funtion ( ) ϕ ( ) ( ) ϕ : suh that Rxy, = ux, u y, xy, (1) In fat, this funtion ϕ is the sign funtion ( ab, ) sign ( a b) funtion sign : { 1, 0,1} is defined by 1, if t < 0; sign ( t) = 0, if t = 0; 1, if t > 0 ϕ = where the sign 16
S S Wang For a fuzzy order R : [ 1,1], under what onditions, an we find a ontinuous funtion u : and a monotone funtion ϕ : [ 1,1] suh that (1) holds? We all u a utility representation and ( u, ϕ ) a funtional representation If (1) is true for fuzzy orders, it means that the utility representation and funtional representations for exat preferenes are speial ases 31 Abbreviations and Aronyms To establish (1), just as for exat preferenes, we need the following two axioms Axiom 4 (Continuity) x R( xx, 0 ) 0 and t R( xx, 0 ) 0 are losed in, x0 Axiom 5 (Monotoniity) x y Rxy (, ) 0 We also need the following two lemmas for the proof of the utility representation theorem Lemma 3 Any A is onneted A is an interval k Lemma 4 A funtion f : is ontinuous f 1 ( ab, ) is open in k, ab, Theorem 1 (Utility Representation)There is a ontinuous utility representation u : for any ontinuous fuzzy preferenes R : [ 1,1] Proof: This proof is similar to the proof of the same result for exat orders For simpliity of the proof, we assume stritly monotoni preferenes, that is, in addition to n Axiom 5, we further require x> y Rxy (, ) > 0 Also, let and e ( 1,1,,1) For any x, we need to find a unique number u( x ) suh that x u x e Given x, define ( ) A t R ( te, x) 0, B t R ( te, x) 0 It is obvious that A B = By ontinuity, AA and B are losed sets in By monotoniity, A and B If A B =, then A \ A= B and B \ B = A, implying that A and B are open sets in Sine is onneted by Lemma 3, A or B must be empty This is a ontradition Therefore, A B Suppose t1, t A B Then Rtex ( 1, ) = Rtex (, ) = 0, implying te 1 te By strit monotoniity, t1 = t Therefore, A B only ontains a single point Let u( x ) denote this point Then, x u( x) e (Figure 1) We now show that this funtion u( x ) represents the preferenes By the definition of u( x ) and strong monotoniity, ( ) ( ) ( ) ( ) ( ) ( ) ( ) u x > u y u x e> u y e R u x e, u y e > 0 R x, y > 0, and u( x) = u( y) x u( x) e= u( y) e y x y u thus represents the preferenes We now only need to show the ontinuity For any ab,, we have 5 5 For any sets, 1 1 A B, and funtion f, we have f ( A ) = f ( A) and ( ) 1 1 ( ) 1 f A B = f A f ( B ) 17
S S Wang I( x) uxe ( ) 45 o x Indifferene Curve Ix ( ) yy x Figure 1 Utility representation 1 1 1 u ( a, b) = u (, a] [ b, ) = u (, a] [ b, ) 1 1 1 1 = u (, a] u [ b, ) = u (, a] u [ b, ) = x Xu( x) a x Xu( x) b Sine u ( x) a R u ( x) e ae R ( x ae), 0, 0, we then have (, ) (, ) 0 = (, ) 0 1 u a b x R x ae x R x be 1 By ontinuity, u ( ab, ) is thus open Hene, by Lemma 4, u( x ) is ontinuous As we know, the ontinuity ondition is neessary for the existene of a utility representation for exat preferenes It must therefore be neessary for fuzzy preferenes The transitivity ondition is suffiient for Lemma 1 Sine Theorem 1 implies Lemma 1, the transitivity ondition must also be neessary for the existene of a utility representation for fuzzy preferenes Remark 1 The utility representation of fuzzy preferenes appears to be the same as the traditional utility representation of exat preferenes However, the funtional representation of fuzzy preferenes in the following two setions will show the ruial differenes In fat, the interesting part of a representation theory for fuzzy preferenes is its funtional representation, from whih the key differenes between fuzzy and exat preferenes are revealed learly 3 Funtional Representation We now proeed to find a funtional representation for fuzzy preferenes Lemma 5 For exat preferenes R: { 1, 0,1 }, x x and y y imply R( x, y) = R( x, y ) Proof: If Rxy (, ) = 0, ie x y, then, by Lemma, x y By Lemma again, using x y and y y, we have x y R x, y = R x, y Thus, ( ) ( ) 18
S S Wang Rxy =, ie x y, then, by Lemma 1, x y By Lemma 1 again, using x y and y y, we have x y, ie, R( x, y ) = 1 Thus, R( x, y) = R( x, y ) Similarly, for Rxy (, ) = 1, we also have R( x, y) = R( x, y ) The lemma is thus proved For fuzzy preferenes, we will impose Lemma 5 as an axiom, whih is needed for a funtional representation Axiom 6 (Independene) x x and y y imply R( x, y) = R( x, y ) The independene axiom states that the degree of preferene stays the same after an indifferene transformation In the expeted utility literature, this axiom is known as the independene axiom, whih is ruial for the existene of an expeted utility representation of preferenes under unertainty For our fuzzy preferenes, this axiom is ruial for the existene of a funtional representation of preferenes under fuzziness The following lemma is from Protter and Morrey [15] Lemma 6 (Tietze Extension Theorem) Let A be a losed set in a metri spae S and f : A a ontinuous and bounded funtion Define M = sup x A f ( x) Then there is a ontinuous funtion g: S suh that If (, ) 1 ( ) ( ) ( ) g x = f x, x A; and g x M, x S Theorem (Funtional Representation)Given a ontinuous fuzzy order R : [ 1,1] and its utility representation u : let u ( ) Then, the fuzzy order is independent if and only if there exists a funtion ϕ : [ 1,1] suh that ϕ ux ( ), u( y) = Rxy (, ) for all xy,, and a) ϕ( t1, t) = ϕ( t, t1), t1, t, b) ϕ ( tt, ) = 0, t, ) ϕ ( t1, t) 0 and ϕ( t, t3) 0 ϕ( t1, t3) 0 Conversely, given a ontinuous funtion u : and a funtion ϕ : [ 1,1] satisfying the above three onditions, define a funtion R : [ 1,1] by Rxy (, ) ϕ ux ( ), u( y) Then, this R is a fuzzy order Furthermore, if is ompat and R is ontinuous, ϕ is also ontinuous and an be defined on Proof: Independene is obviously neessary We only prove its suffiieny By Theorem 1, there exists a ontinuous utility representation u : By independene, we an simply define ϕ : [ 1,1] suh that, for ( u1, u), ϕ u, u = R x, y, where u = u x, u = u y ( ) ( ) ( ) ( ) 1 1 u, u, we have By independene, given any ( 1 ) R( x y) = R( x y ) ( x y ) s t u = u( x ) u = u( y ),,, for any other,, 1 The funtion ϕ is thus well defined By symmetry R( xy, ) = R( yx, ), we immediately have ϕ( t1, t) = ϕ( t, t1), for all t1, t u( ) By reflexivity, we have ϕ ( tt, ) = 0, t u( ) The transitivity implies ϕ t, t 0 and ϕ t, t 0 ϕ t, t 0 ( ) ( ) ( ) 1 3 1 3 If is ompat and R is ontinuous, then is losed and ϕ is also ontinuous By the Tietze Extension Theorem, we an extend ϕ to be a ontinuous funtion on 19
S S Wang The onverse is straightforward The three onditions (a), (b) and () in Theorem are neessary for a funtion ϕ : [ 1,1] that defines a fuzzy order ϕ ab, sign a b A good example of Theorem is the exat preferenes with ( ) ( ) For an exat preferene order R : { 1, 0,1}, the funtional representation is ( ) ( ) ( ) Rxy, = sign ux u y, xy, This is a speial ase of Theorem, in whih ( t1, t) sign ( t1 t) obviously have ϕ( t, t ) = ϕ( t, t ) and ϕ ( tt, ) = 0, t1, t, t 1 1 33 A Speial Form of the Funtional Representation ϕ In this ase, we We now proeed to find a more onvenient form of the funtional representation R Theorem 3 Given a utility representation u : of a fuzzy order : [ 1,1], if and only if u( x) u( y) u( x ) u( y ) R( x y) R( x y ) there exists ϕ : [ 1,1] suh that Rxy (, ) = ϕ ux ( ) u( y), for all xy, = implies, =,, () (3) Proof: The proof is straightforward what we need to do is to define the funtion ϕ ϕ : 1,1 by diretly We define the funtion [ ] ( ) ( ) ( ) ϕ ux u y = Rxy,, for xy, By ondition (), this funtion is well defined Formula (3) is partiularly onvenient to use It is also very lose to the utility representation of exat preferenes The interesting part of the representation (3) is that we an treat the differene of utility values u( x) u( y) as the preferene intensity differene between two onsumption bundles x and y, and the role of ϕ here is to onvert a value t to a value ϕ ( t) [ 1,1] so that we an all it a degree of preferenes With Theorem 3, we an hoose speial forms of ϕ to define interesting fuzzy or- u : 0,1, define, = (4) ders R For example, for [ ] Rxy ( ) ux ( ) u( y) Here, impliitly ( t, t ) t t ϕ 1 1 Sine this ϕ satisfies the three onditions (a), (b) and () in Theorem, the fuzzy preferenes in (4) is hene well defined We an also define Rxy (, ) = sign ux ( ) u( y) (5) Here, impliitly ϕ ( t1, t) sign ( t1 t) This ϕ also satisfies the three onditions (a), (b) and () in Theorem The fuzzy preferenes in Banerjee is hene also well defined In fat, this preferene order desribes the traditional exat preferenes Remark We an now understand Remark 1 intuitively The differene between fuzzy and exat preferenes is not in the diretion of preferenes but in the strength of preferenes, and suh strength may be distributed disproportionally aross the utility spae This explains the similarity of utility representations for fuzzy and exat preferenes The utility funtion u indiates the diretion of preferenes, and the fun- 0
S S Wang tional ϕ represents the distribution of strength of preferenes In speial ases, the strength an be distributed proportionally as in (4), and it an also be distributed in a yes/no fashion as in Banerjee 4 Conluding Remarks What this paper has aomplished are: first, to define a well-behaved fuzzy order; and seond, to find a simple funtional representation for the fuzzy preferenes It is a general utility and funtional representation theory, whih inludes the existing utility and funtional representation theory for exat preferenes as a speial ase In doing so, we have tried to obtain a theory that is simple and intuitive Suh a theory may be readily inorporated into a utility theory under unertainty A more general/flexible utility theory has the potential to explain a few well-known paradoxes, suh as the Allais paradox and St Petersburg paradox The advantage of this approah is that it is still within the framework of expeted utility, as opposed to non-expeted utility Referenes [1] Kreps, D and Porteus, E (1978) Temporal Resolution of Unertainty and Dynami Choie Theory Eonometria, 97, 91-100 https://doiorg/10307/1913656 [] Epstein, LG and Zin, SE (1989) Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretial Framework Eonometria, 57, 937-969 https://doiorg/10307/1913778 [3] Weil, P (1990) Nonexpeted Utility in Maroeonomis Quarterly Journal of Eonomis, 105, 9-4 https://doiorg/10307/937817 [4] Shmeidler, D (1969) Competitive Equilibria in Markets with a Continuum of Traders and Inomplete Preferenes Eonometria, 37, 578-585 https://doiorg/10307/1910435 [5] Flament, C (1983) On Inomplete Preferene StruturesMathematial Soial Sienes, 5, 61-7 https://doiorg/101016/0165-4896(83)90085-9 [6] Gay, A (199) Complete vs Inomplete Preferenes and Eonomi Behavior In Pasinetti, L, Ed, Italian Eonomi Papers, Volume 1, 13-188, Oxford University Press, Oxford, UK [7] Basu, K (1984) Fuzzy Revealed Preferene Theory Journal of Eonomi Theory, 3, 1-7 https://doiorg/101016/00-0531(84)90051-6 [8] Dutta, B (1987) Fuzzy Preferenes and Soial Choie Mathematial Soial Sienes, 13, 15-9 https://doiorg/101016/0165-4896(87)90030-8 [9] Barrett, CR, Pattanaik, PK and Salles, M (1990) On Choosing Rationally When Preferenes Are Fuzzy Fuzzy Sets and Systems, 34, 197-1 https://doiorg/101016/0165-0114(90)90159-4 [10] Banerjee, A (1993) Rational Choie under Fuzzy Preferenes: The Orlovsky Choie Funtion Fuzzy Sets and Systems, 53, 95-99 https://doiorg/101016/0165-0114(93)90375-r [11] Dasgupta, M and Deb, R (1996) Transitivity and Fuzzy Preferenes Soial Choie and Welfare, 13, 305-318 https://doiorg/101007/bf0017934 [1] Rihardson, G (1998) The Struture of Fuzzy Preferenes: Soial Choie Impliations Soial Choie and Welfare, 15, 359-369 https://doiorg/101007/s003550050111 [13] Sengupta, K (1998) Fuzzy Preferene and Orlovsky Choie Proedure Fuzzy Sets and Systems, 93, 31-34 https://doiorg/101016/s0165-0114(96)00079-6 [14] Sengupta, K (1999) Choie Rules with Fuzzy Preferenes: Some Charaterizations Soial 1
S S Wang Choie and Welfare, 16, 59-7 https://doiorg/101007/s003550050143 [15] Protter, MH and Morrey, CB (1991) A First Course in Real Analysis Springer-Verlag, New York https://doiorg/101007/978-1-4419-8744-0 Submit or reommend next manusript to SCIRP and we will provide best servie for you: Aepting pre-submission inquiries through Email, Faebook, LinkedIn, Twitter, et A wide seletion of journals (inlusive of 9 subjets, more than 00 journals) Providing 4-hour high-quality servie User-friendly online submission system Fair and swift peer-review system Effiient typesetting and proofreading proedure Display of the result of downloads and visits, as well as the number of ited artiles Maximum dissemination of your researh work Submit your manusript at: http://papersubmissionsirporg/ Or ontat tel@sirporg