A Lancasterian Logic of Taste and Preference Antoine Billoty LEM-Université Panthéon-Assas, Paris 2 PSE-Ecole d’Economie de Paris November 12, 2010

Special Issue of the IJET in honor of James Friedman correspondence : Université Panthéon-Assas, Paris 2, 92 rue d’Assas, 75006, Paris, France, tel: 33(1)55425033, e-mail address : [email protected] y For

1

Abstract The aim of this paper is to reconcile the natural meaning of a taste de…ned as a subset of attributes and the theoretical modelling of a preference de…ned as a binary relation over alternatives. For that purpose, in a Lancasterian framework we …rst introduce attributes, addresses, alternatives and subjective associations between attributes. Second, we de…ne tastes as subsets of attributes resulting from a speci…c mapping T over the powerset of attributes and we establish the two conditions— monotonicity and normalization of the mapping T — under which tastes can be said well-formed, i.e. formally relevant as a representation of the subjective associations between attributes (Theorem 1). Third, we exhibit the formal properties— re‡exivity and transitivity— of the subjective associations that structure the attributes set if tastes are well-formed (Theorem 2). Finally, we prove that whenever tastes are consistently represented by preferences, tastes are well-formed i¤ preferences are totally weak-ordered (Theorem 3). Jel Classification: D11 Keywords: Tastes, Preference, Rationality, Decision

2

1

Introduction

The concept of preference holds what is called a pivotal position in the theory of value, then in general microeconomics. It is not considered controversial by economists when purely individual (the Arrovian paradigm is indeed of di¤erent nature). However, philosophical logicians tend to regard it as much more problematic insofar as there is no practical agreement among logicians about the basic principles of a logic of preference (von Wright, 1963, and more recently, Morris, 1996). This situation recalls what happens to the probability theory: everyone extensively uses and successfully applies probability in both physical and economic …elds while philosophers notoriously disagree about its true meaning (see e.g. Gilboa et al., 2008). Generally, a statement of preference is seen as a subjective value judgement which expresses an agent’s preference for something (called an alternative) over something else (another alternative). Now, one can ask for the basic motivations of such a preference, that is the reasons why an agent, call him Bob, prefers alternative x to alternative y. The most frequent answer to this question is that Bob likes x better than y. This is obviously not a reason, i.e. an appropriate answer, but a new verbalization of the preference itself: ‘liking better’ is just another term for ‘preferring’(von Wright, 1990, p. 150). Sometimes, there is a more suitable answer at hand: Bob can give practical reasons for his preference. For instance, he prefers cars to bicycles as a vehicle to go from his home to his o¢ ce because cars are, say, roomier and faster. Such an explanation— à la Lancaster— is decomposable in the two following statements: (1) ‘Cars are roomier and faster than bicycles’— which is an objective factual statement, i.e. which can be considered extrinsically true or false by an external modeller, and (2) ‘Bob prefers roomier and faster vehicle to less roomy and slower one’— which is a subjective valuation, i.e. which cannot be considered intrinsically true or false by an external modeller. Hence, by assuming a suitable reason to be expressed thanks to factual statements translated through a subjective valuation— which then avoid ambiguities between the reasons for a preference and the preference itself— , it seems that preferring a particular alternative to another one can always be justi…ed (roughly speaking) in terms of intensity (full presence and absence being two polar cases) of attributes (as roomy or fast in the above example).

2

Motivation

Generally, in standard microeconomics, tastes and preferences are considered similar. Both terms seem to describe the same concept. However, in the current language, talking about tastes often means to make reference to some social standards such as beauty, cultivation, re…nement, etc. This way, tastes are rather de…ned with an implicit reference to ‘good’tastes— i.e. a particular association of several attributes: ‘speed’ with ‘comfort’, ‘price’ with ‘reliability’ and so forth. Hence, this suggests a sort of a cultivated habit, of selecting one

3

set of attributes over another set... In the same time, distinctions of good tastes vs. bad tastes do not seem to be relevant while talking about preferences (which is perhaps the reason why most of the economists choose to use the term ‘preferences’rather than ‘tastes’): preferences are just binary comparisons between alternatives— independently of the comparison context or the individual perception of the social standards. One possible consequence of such a di¤erence between tastes and preferences is to consider that tastes are a kind of …rst-order or primary concept— essentially independent of the quantities of the di¤erent attributes that are embodied in a given alternative and being a product of the individual assessment of what is good and what is not. Preferences then refer to a representation of these primary tastes— essentially designated for decisions. A baseline de…nition of this two-step principle might be that tastes organize the set of attributes (for instance as a partition or a nesting according to the way the individual subjectively associates the attributes...) while preferences order precisely the alternatives relatively to the prior organization of the attributes in order to allow the individual to choose between alternatives. In the current literature (see the seminal paper of Strotz, 1955, and more recently, Gul and Pesendorfer, 2005) when someone tries to distinguish tastes from preferences, he generally merges each taste with a dated utility function and then studies the relevance of the global preferences by means of a dynamicconsistency analysis of choices (e.g. Kreps, 1979, for an original interpretation of Strotz, 1955). In our paper, there is no more dynamic set-up. Thus, the goal is quite di¤erent of Strotz’s. The aim is rather to reconcile the natural meaning of a taste de…ned as a subset of attributes (say, Bob’s taste for sedan or saloon cars, that is cars which are roomy, fast, safe...) and the theoretical modelling of a preference de…ned as a binary relation over alternatives.

2.1

Subjective Associations of Attributes

Concerning the possibility to use a language based on attributes, we already know that the modeling of any alternative as an associated set of attributes has been systematically presented and exploited by Lancaster (1966, 1971).1 According to this approach, an alternative is a composite of attributes that it embodies. For instance, the attributes of a car include gas consumption, comfort, power, reliability, design and so on... ‘The motivation of such an approach— generically called the ‘address’one— stems from the observation that commodities are not desired in themselves but rather it is the attributes they possess that form the basis for consumer preferences.’ (Anderson, de Palma and Thisse, 1992, p. 101). In our ‘Bob’ example, the natural consequence of this particular analysis is to de…ne Bob’s preferences on the powerset of car attributes rather than on the set of cars directly. In other words, Bob considers all the available bundles of car attributes before deciding the car he prefers. Hence, he formally processes the attributes as abstract commodities (precisely 1 Other famous authors, like Quandt (1956) or Baumol (1967), developed the same kind of intuition which can be traced back to the pioneering work of Hotelling (1929).

4

as in Baumol, 1967).2 Moreover, in de…ning a car as a composite of attributes, one allows, one one hand, the attributes which are embodied within a car to be eventually connected in Bob’s mind such that these connections introduce some fuzziness in the way Bob perceives the di¤erent cars (see Section 3.1); on other hand, these connections do not necessarily possess the same properties as the current relations used in network economics (where connections are central). For instance, it can be reasonable to consider that ‘speed’, ‘comfort’and ‘price’are probably linked in some way for cars as well as for bicycles... Even in this case— that is with the simplest association principle— it is well-known in psychology that the connections between, say, ‘speed’and ‘price’, may change according to the attribute Bob …rst considers— i.e. the connection is not necessarily symmetric. This corresponds to what is called by Chomsky (1968) the instability of the mind-association-process and it is clearly a close idea to that of a framing e¤ect as emphasized by Kahneman and Tversky (1981) in the case of a simple symmetric reversal of the frame. More precisely, we suppose in this paper that Bob thinks in terms of attribute associations. Then, connections between attributes are oriented (as in graph theory) and Bob’s path of associations between car attributes obviously depends on the …rst attribute (the root) Bob faces: for instance, if he faces an expensive car (in the sense where the seller considers that the most determinant information about this car is its price and then shows its price before its speed performance or its comfort capacity...), he wants it to be fast and roomy. Then, he associates the attribute ‘price’with the attributes ‘speed’and ‘comfort’. But, on the other hand, if he faces a fast car (the seller shows the speed performance of the car before its price or its comfort capacity...), he may desire it to be safe and then associates the attribute ‘speed’with the attribute ‘safety’... Intuitively, assocations between attributes are not always symmetric (which is seldom if ever the case in network models). Now, from a particular root, it can be designed an association class which re‡ects the area of association around the given root, that is a structured subset of attributes which corresponds to a particular address à la Hotelling— i.e. a particular point of the attribute space. Hence, we can distinguish the subjective addresses stemmed from Bob’s associations and the objective ones— i.e. the elements of the powerset of attributes. Each objective address is a natural candidate to be a subjective one as well. And we claim that the internal process of recognition of the association classes among the objective addresses can be assimilated to the dynamic genesis of Bob’s tastes. Suppose an address to be given— that is an abstract alternative. If one of Bob’s association class is included in it, it means, for this association class, that the given address is the best possible one since all desirable attributes are inside (we will say: the class …ts well the address). Nevertheless, if there are some nondesirable attributes within the address, in restricting this last to the only attributes whose association class is included, Bob de…nes one of his tastes— 2 In this approach each attribute gives birth to an implicit market. Nevertheless, such a theoretical design of an implicit market in terms of attribute supply, attribute demand, then equilibrium and price for the attribute, is de…nitely far from our purpose.

5

i.e. he cleans the address until the elicitation of the relevant attributes. This de…nition of a taste is purely qualitative, its characteristic function is a usual Kronecker function de…ned from the Cartesian product of the attribute set with its powerset toward the Boolean pair f0; 1g. This approach recalls epistemic logics. Actually, the standard ‘semantics of possible worlds’(see Kripke, 1963, and Aumann, 1976, for application to knowledge modeling) are here replaced by a ‘semantics of attributes’ and the relation of accessibility between worlds, by a relation of association between attributes. Accordingly, events correspond to addresses and the Aumannian knowledge structure to the tastes structure (see Billot, 2002, and Billot and Walliser, 1999, for further details about epistemic properties of Kripke structures).3

2.2

Tastes and Preferences

Derivation of (attribute-based) preferences (for comparing the alternatives according to one particular attribute) leading to a consistent or at least a tractable behavior requires to formally explicit the relationship between tastes and preferences— Clostermann and Seitz (2002), for example, consider the relationship between preferences and tastes, but in a paradoxical manner, since they assume preferences to be exogeneous while tastes would be endogeneous. It is standard in microeconomics to justify Bob’s global preferences (when he compares alternatives according to an implicit aggregation of all their possible attributes) as a formal translation of his tastes (see e.g. Debreu, 1959, chap. 4) even if there is no mathematical rule allowing such an interpretation, except the common sense (as illustrated by Gul and Pesendorfer, 2005)... Nevertheless, using tastes as primitives needs …rst to de…ne the concept and, then, to relate it to preferences. For that purpose, we assume tastes and preferences to be axiomatically consistent if and only if, when preferences are never in‡uenced by any attribute for which a given address is not the best, then there exists a taste based on this given address. From this de…nition, it is natural to investigate the way taste properties in‡uence preference properties and more generally to exhibit the correspondence between the association relation between attributes and the preference relation between alternatives. What …nally happens is quite intuitive: when Bob’s preferences are totally (weak) ordered, that is de…ned à la Arrow-Debreu, his tastes must be well-formed— i.e. there exists a subjective association relation which structures speci…cally the set of attributes in such a way that this structure forms a partition. From a methodological point of view, the theoretical problem we face is exactly a problem of representation. In standard utility theory, most of the axioms are assumed on the primitive preferences (continuity, weak-ordering...) in order to show that it is possible to set up a numerical representation that is called a utility function. Here, we assumed several axioms on tastes in order to show that there exists some binary relations which represent them through preferences: in some sense, we propose the infra side of the preference theory while the utility one would be the supra 3 Methodologically,

this paper deeply uses this analogy.

6

side. The following Section formally introduces alternatives, attributes and association between them. Then, we establish the conditions under which tastes are said to be well-formed (Theorems 1 and 2). Section 3 exhibits the link between preferences and tastes and proves that total weak-ordered preferences are equivalent to well-formed tastes (Theorem 3).

3 3.1

The Logic of Taste Alternatives, Attributes, Addresses

Features of an alternative which are thought to appeal consumers can be conveyed through a lot of attributes. For instance, safety, color, speed, price, comfort and reliability might be considered as the main attributes of a car. Denote A this set of attributes. Note that, in the current language which is naturally used by consumers to describe a car or any other alternative, these attributes are rather caught throught adjectives according to an intuitive syntactical process: ‘price’leads to expensive and cheap, ‘comfort’to roomy and small, ‘speed’to fast and low, etc... Therefore, in all Lancasterian models (McAlister 1979, Barsmish 1984...), a real number translates each adjective and, consequently, indicates how and how much of each attribute is embodied in one unit of the alternative. Thus, an alternative x can always be identi…ed as a vector of Rm — this last being called the attribute space— if the number of attributes in A is m where xj 2 R, for all attribute j. All attributes are depending on the general possibilities, that is technical, cultural and so on, of the economy. More generally, we suppose a given …nite set X of alternatives, X Rm . By the way, call address any bundle of attributes— i.e. any event of 2A .

3.2

Association, Class

Suppose that X is a set of cars among which an agent called Bob has to choose. Assume that Bob is …rst interested by safety— i.e. he gives priority to safety— and assume, in addition, that he considers, on one side, a light-colored car as more safe than a dark-colored one (it is statistically true!) and, on the other side, a safe car as necessarily roomy. Hence, when he compares several cars, he immediately focuses on light roomy cars rather than on dark small ones because of the two simultaneous associations between ‘safety’and ‘colour’, ‘safety’and ‘comfort’. Next, he shrinks the set of the selected cars according to the viewpoint of their speed capacities... etc. until he …nds the optimal car whose address is close to his best car address— i.e. the car he likes the best. More or less, this process looks like a lexicographical one (except the fact that any attribute— e.g. ‘safety’— can be associated to several other attibutes— e.g. ‘color’ and ‘comfort’). Moreover, it is reasonable to consider that Bob associates, step by step, the potential attributes which help describing a car in accordance with his individual knowledge, his psychological background, his cultural identity...

7

However, if one wants to understand Bob’s organization of the attributes which leads, nesting after nesting, to his choice— i.e. the reason why ‘safety’ is followed by ‘color’and ‘comfort’, the later being, in turn, followed by ‘speed’...— , one needs to explain why ‘safety’and ‘color’, ‘safety’and ‘comfort’, then again ‘comfort’ and ‘speed’, are psychologically linked in Bob’s mind in such a way that these associations are …nally consistent with his preferences over cars. As an answer, we suggest (1) that these successive associations of attributes re‡ect Bob’s tastes and (2) that Bob’s tastes can be decentralized through his preferences if and only if they are su¢ ciently contrasted— i.e. when the set 2A of addresses is organized by association as a partition. In other words, tastes and preferences are not conceptually identical (that was already the case in Stigler-Becker, 1977’s famous paper, De Gustibus non est Disputandum) but, while they do not naturally coincide, they can become consistent under some particular conditions. Take now as a primitive representation of the Bob’s behavior a binary relation between the attributes denoted y, call it association, and write a y b when Bob associates the attribute a with the attribute b. This means that, a being given (we call it the root), Bob wants any car to possess ideally the attribute b. Denote [a] the class of all attributes b 2 A for which the attribute a is the root— i.e. which are associated by means of the association y. We call it an association class: [a] = fb 2 A : a y bg . (1) The association class [a] is the set of all attributes Bob wants to associate with the root a. Since it is reasonable to consider that each attribute is associated with itself, for instance, we know that, for Bob: [safety] =Bob fsafety, color, comfortg and [comfort] =Bob fcomfort, speedg.

3.3

Tastes

When a class [a] is fully contained in a given address , all associated attributes— i.e. 8b 2 [a]— naturally belong to . We suggest that, for the root a, there is no other address , ! , that can be considered by Bob strongly better than . Actually, insofar as Bob does not mind the other qualities of n [a], he then focuses on [a] and he appreciates the address thanks to some implicit distance between the class [a] he faces and the address (independently on the other qualities of n [a]). Example 1 : Suppose that the following attributes compose the set A: s(peed), l(abel), r(eliability), f (ame), p(rice), c(omfort), e(co-aware). Morover suppose that from the root s, Bob associates s, l and r: [s] = fs; l; rg. Furthermore, suppose [r] = fr; f g, [l] = fs; lg , [f ] = ff ; eg, [p] = fs; p; cg, [c] = fp; cg and [e] = fs; r; f ; eg. The association classes [r], [f ] and [e] are included in the address 1 = fs; r; f ; eg. Then, the address 2 = fs; l; r; f ; eg ( 1 ) is not better for the root r (nor for the roots f or e) than 1 since one has just added nondesirable attributes (l) 8

but 2 is better for s and l than 1 . On contrary, 3 = fr; f ; eg ( 1 ) is a less good address for e than 1 and a less good address for e; s and l than 3 since some of the associated attributes— i.e. s for 1 and s; l for 3 — vanished; …nally, [r], [f ] and [e] …t well the address 1 , [s],[l],[r], [f ] and [e] …t well the address 2 and [r] and [f ] …t well the address 3 . We propose to translate formally this idea of …t between an association class and a given address in de…ning a taste T as the address which is only based on all roots for which is the best possible address: [a]

,a2T .

More precisely, we have: De…nition 1 : Tastes are de…ned by a mapping T : 2A ! 2A which represents the association y such that, for all 2 2A , T = a 2 2A : [a]

.

(2)

In the example above, T 1 = fr; f ; eg while T 2 = fs; l; r; f ; eg and T 3 = fr; f g. Intuitively, Bob cleans the address (e.g. 1 = fs; r; f ; eg) in order to de…ne the taste (T 1 = fr; f ; eg) by eliminating the attributes whose association classes are not relevant or just touched by the address but not fully contained in it (here [s] \ 1 = fs; rg = 6 [s]). Then, he eliminates all roots (here s) whose association class does not …t the address he faces.

3.4

Well-formed Tastes

Basically, tastes are de…ned by a collection of di¤erent association classes. Then, these tastes are said to be well-formed if they proceed from the association which speci…cally organizes the whole set of attributes as a partition.4 De…nition 2 : Tastes T are well-formed if there exists an association y such that the mapping T represents y. By extension, the association y is also said to be well-formed if there exists a mapping T such that y is represented by T — (2) holds. Theorem 1 : Tastes T are well-formed if and only if they satisfy for all ( ; ) 2 2A 2A : (C1) )T T , (C2) TA = A. 4 The term ‘well-formed tastes’is clearly reminiscent of the so-called ‘well-formed formulas’ within consistent systems in modal logics. (See Hughes and Cresswell 1989, chap. 2.)

9

Proof. (i) Suppose that T represents y. Then, by de…nition, T

\

= fa 2 A : [a]

\ g

= fa 2 A : [a]

g \ fa 2 A : [a]

= T \T

g

which obviously yields T \ = T \ T which is equivalent to T T for — i.e. C1. Moreover, by construction, TA = fa 2 A : [a] Ag = A, then C2 holds as well. (ii) Now, suppose that T satis…es C1 and C2. Consider as the complement of in A and de…ne the association y by: [a] = fb 2 A : a 2 =T

bg

(3)

where b An fbg. Let us show that the mapping T represents the association as de…ned in (3). Suppose …rst an attribute a 2 T . Then, b involves T T b by C1 and a 2 T implying a 2 T b , we have: b 2 = [a]. Thus, if b 2 [a], it means 6 b which implies b 2 and then, [a] . Now, suppose [a] . When = A, a 2 T by C2. Consider now that 6= A— i.e. A. Then, = \b2= b. Therefore, if a 2 = T b , then b 2 . This means also that, if b 2 = , then a 2 T b . Since a 2 T b , for any b 2 = , it implies naturally that a 2 T . Conditions C1-C2 de…ne a monotone mapping where the normalization conditions correspond to C2. The …rst condition C1 expresses the fact that, for a given address, the larger it is, the larger is the subset of roots for which the address can be the best— i.e. the larger is the taste (cf. the example above). The second condition C2 corresponds to the case where the given address is the whole set of attributes. Indeed, in this situation, Bob directly considers it as a taste since all attributes being inside, all association classes are also inside by de…nition! Furthermore, this means that any attribute of A can be a taste root. It is also remarkable that (3) means that an attribute b is considered as belonging to a taste whose root is the attribute a if and only if all attributes which are not b do not lead to another possible taste. Theorem 2 : Tastes T are well-formed if the association y is re‡exive and transitive. Proof. Suppose y to be represented by T . (i) By De…nition 1, C1 always holds. (ii) If y is re‡exive, then 8a 2 A, a 2 [a]. Hence, we have T 2 2A . More particularly, TA A. If y is transitive, then 8 (a; b) 2 A2 , b 2 [a] implies [b] T

fa 2 A : [a] 10

T g,

, for all (4)

[a]. Hence, we have:

where

T = AnT , for all T?

2 2A . More particularly, fa 2 A : [a]

T? g .

(5)

Now, since by (4), TA A, then T? = ?— i.e. by negation: A = T? . Hence, by (5), A TA while by (4), TA A, that is TA = A— i.e. C2 holds. Well-formed tastes as described above are based on the organization of the attributes by means of Bob’s association. This means that well-formed tastes are directly expressed by the properties of the relation y, these ones corresponding to the mental associations Bob makes. Each property involves a particular manner to consider the possible attributes and …nally, each group of properties correspond to a particular way to translate the tastes through associations between attributes. Hence, if Bob is a well-formed agent— i.e. whose tastes are well-formed— , he behaves in terms of attributes as follows: - By re‡exivity of the association y, a 2 [a], for all a 2 A. This means that any attribute belongs to its association class: Bob is aware about the attribute he faces, he distinguishes it. - By transitivity of the association y, b 2 [a] implies [b] [a], for all a; b 2 A. This means that Bob can manage the association between three attributes with a very minimal ‘rationality’expressed by transitivity.

4 4.1

The Logic of Preference Attribute-Based Preferences f%d gd2D

Consider X as a …nite set of available alternatives that are comparable. Suppose that Bob has to choose a vehicle to go from his home to his o¢ ce. Bob’s choice set is made of a private car (x), a private bicycle (y) and a public bus (z). Because of their basic heterogeneity, x; y; z are here assumed to be fully described by the help of four attributes— i.e. a particular subset A0 of A: A0 = fprice, comfort, speed, safetyg corresponding to several factual statements measured by the associated number of R that indicates how and how much of each attribute is embodied in the considered alternative. Assume for convenience, that these factual statements are just three: (+1) good, (0) average and ( 1) bad. For instance, when a = a1 corresponds to the attribute ‘price’, then the a1 -element of alternatives x; y; z, respectively denoted xa1 ; ya1 ; za1 , can be described as follows: xa1 = 1 (expensive), ya1 = +1 (cheap) and za1 = +1 (cheap). More generally, we have:

A0 nX a1 =price a2 =comfort a3 =speed a4 =safety

x = car 1 (expensive) +1 (roomy) +1 (fast) 0 (average-in-safety)

y = bicycle +1 (cheap) 0 (uncomfortable) 1 (slow) 1 (unsafe) 11

z = bus +1 (cheap) 0 (average-in-comfort) +1 (fast) +1 (safe)

Now, Bob is supposed to rank the alternatives of X according to 3 attributebased (AB) preference relations, each one being denoted %a , 8a 2 A. These AB-preference relations are subjective by de…nition. Formally, they are binary relations over X X. If, 8a 2 A, x %a y, this means that according to the attribute a, alternative x is considered by Bob at least as good as alternative y. Moreover, two alternatives x; y are considered indi¤erent according to a particular attribute a, that is x a y, when simultaneously, x %a y and y %a x. By de…nition, any relation %a is assumed to be total when x %a y or y %a x, 8x; y 2 X, 8a 2 A, and transitive when x %a y and y %a z implies x %a z, 8x; y; z 2 X, 8a 2 A. Notice that totality and transitivity imply re‡exivity— i.e. x a x, 8x 2 X, 8a 2 A. For instance, a bicycle (y) is cheaper (a1 =price) than a car (x) while a car is more comfortable (a2 =comfort) than a bicycle— i.e. y %a1 x while x %a2 y. At …rst glance, it seems naturel to have: x %a y , x a

ya , 8a 2 A,

(6)

where xa ya means that xa dominates ya according to the natural language that is conveyed through the three values f+1; 0; 1g. For instance, cheap expensive (cheap, i.e. good-in-price, dominates— is better than— expensive, i.e. bad-in-price), roomy uncomfortable, etc. Nevertheless, the psychological process of Bob’s evaluation can be more fuzzy. Consider the following case: in the above example, xa3 = za3 . And it is actually reasonable to consider that, in most cities, cars and buses are more or less equivalently fast. Hence, we expect x a3 z. But, suppose now that speed and safety are directly linked in Bob’s mind, which translates that if a vehicle is fast, Bob wants it to be safe as well: speedysafety. Since xa4 = 0 (average-in-safety) while za4 = +1 (safe)— i.e. za4 > xa4 — , Bob’s preference can …nally conclude in such way that z a3 x. The process works here in a kind of lexicographical manner. Bob thinks: ‘A car is as fast as a bus but I don’t want to be injured!’ He associates immediately speed and safety and this association introduces some trouble in his sujective evaluation of both alternatives to the point that an expected indi¤erence (eq. 6) becomes a strict preference. Besides, consider now an address A. Then, x denotes the tuple fxa ga2 . If an alternative x can be de…ned as (y ; z ), it means that xa = ya for all attribute a 2 and xb = zb for all the other attributes b 2 An .

4.2

Tastes Consistency and AB-Preferences

De…ne a taste T in terms of preferences as follows: if Bob’s preferences never depend on any attribute for which the address is not the best, then T is one of Bob’s tastes; on the other hand, if Bob’s preferences ever depend on the attributes for which the address is not the best, then there is no taste stems from .

12

De…nition 3 : Tastes T are consistent with AB-preferences f%a ga2A if T = fa 2 A : (x ; y

)

a

(x ; z

) for all x; y; z 2 Xg .

This means that T is a taste if the complement subset in A— i.e. the union of all other addresses— is not determinant for AB-preferences. This requirement is intuitively weak because being generally heterogeneous, there are not so much situations for which can be relevant for Bob. Indeed, it is easy to check that tastes de…ned by a given mapping T can be consistent with many families of relations f%a ga2D . Basically, De…nition 3 is of same nature than that of order-preservation in utility theory.5 It links the primitives— i.e. tastes— with the representation— i.e. preferences. The following result shows that when tastes are well-formed (cf. Theorem 1), they are consistent with the AB-preferences as soon as the latter are totally weak-ordered. Theorem 3 : If AB-preferences f%a ga2A are totally weak-ordered, then tastes T which are consistent with f%a ga2A are well-formed (resp. the association y is re‡exive and transitive). Proof. (i) We show C1 holds: (a) By construction, )T T is equivalent to T \ T = T \ . Hence, suppose an attribute a 2 A and two addresses and such that a 2 T \ T . According to the de…nition, for all x; y; z 2 X: a 2 T ) (x ; y ) %a (x ; z ) ) x \ ; y ( \ ) %a (x \ ; y

\

;z

)

and symmetrically: a 2 T ) (x

;y

\

;z

\

) %a x

\

;z

( \ )

.

Now, since %a is assumed to be transitive: x

\

;y

( \ )

%a x

;z

\

which implies: a 2 T \ . (b) Suppose an attribute a 2 A such that a 2 T x Thus, (x ; y

\

;y

( \ )

) %a (x ; z

%a x

\

\

;z

) and (x ; y

( \ )

. Then, for all x; y; z 2 X: ( \ )

.

) %a (x ; z

)

5 For any two alternatives x and y and u (:) is the utility function, there is order-preservation of the preference in utilities whenever

x % y , u (x)

13

u (y) .

which implies: a 2 T \ T . (ii) We show C2 holds: by totality, x %a x for all alternative x 2 X and all attribute a 2 A. Hence, we always have: (xA ; y? ) %a (xA ; z? ) which immediately means by de…nition that TA = A. This theorem is a representation one. It can be interpreted as follows: whenever Bob’s tastes are (consistently) represented by his preferences, the former are well-formed i¤ the latter are totally weak-ordered. This way, the standard micro-economic requirement of total weak-ordered preferences— i.e. totality and transitivity of the individual relations— can be directly interpreted in terms of well-formed tastes, which means that the association between attributes is re‡exive and transitive. Corollary 1 . For tastes T which are consistent with AB-preferences f%i gi2A , there exists an association y such that T represents y.

5

Final Discussion

The standard rationality paradigm— from Arrow to Savage— that is used in economics in general— and in decision theory in particular— explains Bob’s choices by both his beliefs (cognitive side) and his preferences (instrumental side, see Billot, 1998). This way, Bob’s preference for a white car against a black one might be explained, for instance, by (1) Bob’s beliefs about safety-on-road (he is supposed to believe that the probability to be injured in a light-colored car is lower than the probability to be injured in a dark one) and by (2) a speci…c taste for safety (safety is a genuine root for Bob’s tastes when he faces cars). However, while Bob’s beliefs may change thanks to a new information (if someone acquainted gives Bob the true probabilities to be injured according to the color of the car and if these probabilities di¤er from Bob’s priors), his preferences— when reinterpreted in a Hotelling/Lancaster/Baumol setting involving bundles of attributes rather than direct primitive alternatives— are supposed to be totally insensitive to any change of the subjective associations Bob makes between all the attributes (even if someone acquainted reveals to Bob that, contrary to what he thought, safety is objectively correlated with price...). Consequently, Bob’s well-formed tastes are assumed to be de…nitely …xed, unable to be updated even when he learns something about an actual attributes association. It is then straightforward that one of the most patent shortcomings of the standard paradigm is that reasons for preference— i.e. in our language, associations between attributes and well-formed tastes that are consistent with preferences— play no explicit role for the preference elicitation.6 6 The intuition here— the role plays by tastes within the preference elicitation process— is close to the question studied by Kadane and Winkler (1988) for probabilities and utilities.

14

In the model we propose, where tastes and preferences are basically independent, such a shortcoming vanishes since changes in tastes are possible provided these changes are consistent with preferences which translate updated tastes.

6

References

Anderson, S., de Palma, A. and J.-F. Thisse (1992), Discrete Choice Theory of Product Di¤ erentiation, MIT Press: Cambridge, Massachusetts. Aumann, R.J. (1976), “Agreeing to Disagree,” The Annals of Statistics 4, 1236-1239. Barmish, B.R. (1984), “A new approach to the incorporation of attributes into consumer theory,” Journal of Economic Theory 32, 93-110. Baumol, W.J. (1967), “Calculation of Optimal Product and Retailer Characteristics: The Abstract Product Approach,”Journal of Political Economy 75, 674-685. Billot, A. (1998), “Autobiased Choice Theory,” The Annals of Operational Research 80, 85-103. Billot, A. (2002), “The Dark Side of Preference Theory,” Theory and Decision 53, 243-270. Billot, A. and B. Walliser (1999), “Epistemic Properties of Knowledge Hierarchies,” Journal of Mathematical Economics 32, 185-206. Chomsky, N. (1968), Language and the Mind, Harcourt Brace and World: New York. Clostermann, J. and F. Seitz (2002), “Exogeneous Preferences and Endogeneous Tastes,” Journal of Economics and Statistics 222, 513-530. Debreu, G. (1959), Theory of Value, Wiley: New York. Gilboa, I., Postlewaite A. and D. Schmeidler (2008), “Probabilities in Economic Modeling,” Journal of Economic Perspectives 22, 173-188. Gul, F. and W. Pesendorfer (2005), “The Revealed Preference Theory of Changing Tastes,” Review of Economic Studies 72, 429-448. Hotelling, H. (1929), “Stability in Competition,’Economic Journal 39, 4157. Hughes, G.E. and M.J. Cresswell (1989), An Introduction to Modal Logic, Routlegde: New York. Kadane, J.B. and R.L. Winkler (1988), “Separating Probability Elicitation from Utilities,” Journal of the American Statistical Association 83, 357-374. Kahneman, D. and A. Tversky (1981), “The Framing of Decisions and the Psychology of Choice,” Science 211, 453-458. Kreps, D.M. (1979), “A Representation Theorem for ‘Preference for Flexibility’,” Econometrica 47, 565-576 Kripke, S.A. (1963), “Semantical Analysis of Modal Logic,” Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 9, 62-67. Lancaster, K.J. (1966), “A New Approach to Consumer Theory,”Journal of Political Economy 74, 132-157.

15

Lancaster, K.J. (1971), Consumer Demand: A New Approach, Columbia University Press: New York. McAlister, L. (1979), “Choosing Multiple Items from a Product Class,”Journal of Consumer Research 6, 213-224. Morris, S. (1996), “The Logic of Belief and Belief Change: A Decision Theoretic Approach,” Journal of Economic Theory 69, 1-23. Quandt, R.E. (1956), “A Probabilistic Theory of Consumer Behavior,”Quarterly Journal of Economics 70, 507-536. Strotz, R.H. (1955), “Myopia and Inconsistency in Dynamic Utility Maximization,” Review of Economic Studies 23, 165-180. Stigler, G.J. and G.S. Becker (1977), “De Gustibus non est Disputandum,” American Economic Review 67, 76-90. von Wright, G.H. (1963), The Logic of Preference, Edinburgh University Press: Edinburgh. von Wright, G.H. (1990), “Preferences,”Eatwell, J., Milgate, M. and P. Newman, eds, Utility and Probability, The New Palgrave Series, 149-156, London: Macmillan Press.

16