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Consumer Preference Judgments: An Exposition with Empirical Applications Author(s): Jehoshua Eliashberg Source: Management Science, Vol. 26, No. 1 (Jan., 1980), pp. 60-77 Published by: INFORMS Stable URL: http://www.jstor.org/stable/2630745 . Accessed: 15/04/2013 10:30 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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MANAGEMENT SCIENCE Vol. 26. No. 1, January 1980 Printed in U.S.A.

CONSUMER PREFERENCE JUDGMENTS: AN EXPOSITION WITH EMPIRICAL APPLICATIONS* JEHOSHUA

ELIASHBERGt

The modeling of preferences for multiattribute alternatives has received an increased attention in marketing (consumer behavior) and management science (decision analysis). The research in the two disciplines is closely related and can be applied to predicting consumer preferences for multiattribute options. The purpose of this paper is to illustrate and discuss several preference models and measurement techniques that have been used mainly by decision analysts and which are applicable in the consumer preference judgment context. A pilot application of the measurement techniques which provides some insight on their relative predictive accuracy and on the usefulness of empirically verifying the conditions necessary for the existence of the preference models is reported, too. (MARKETING-BUYER UTILITY/PREFBEHAVIOR; MULTI-ATTRIBUTE ERENCE THEORY; DECISION ANALYSIS)

1. Introduction Many judgments made by consumers involve outcomes characterized by several attributes (dimensions). For example, in evaluating automobiles, the consumer may consider attributes such as price and quality. Frequently the attributes are conflicting in the sense that an achievement of more from one attribute (higher quality) involves giving up some amount of another attribute (higher price or lower saving). In some situations there is no uncertainty in the evaluation; the consumer knows the multiattribute consequence of each alternative. In other cases, the consequences may not be known in advance. This paper is concerned with modeling issues regarding preferences among such multiattribute alternatives. In recent years, there has been a tremendous increase in research on multiattribute decision-making in marketing (consumer behavior) and management science (decision analysis). Consumer behavior theory postulates preference models and experimentally tests them. Marketing scholars have been mainly concerned with consumer decisionmaking under conditions of certainty where the decision maker is assumed to have complete and accurate knowledge of the various decision outcomes that would follow selection of each course of action [2], [4], [8], [17], [23], [24], [47], [48], [59], [60]. A recent study [501, however, suggests that incorporating the perceived risk due to uncertainty concerning the decision outcomes may be important in the formation of preference models. Decision analysts have also studied extensively alternative forms of multiattribute preference models and alternative methods for obtaining the models' parameters under conditions of uncertainty as well as certainty. Their approach is primarily prescriptive and draws on deductive theory which axiomatically implies the functional form of the multiattribute preference structure, based on behavioral assumptions which can be readily tested. Reviews of field studies are provided in [32]-[34]. Most empirical applications have been based on directly assessing the utility function of one or a small number of decision makers. Since the assessment of the preference structure relies very heavily on consistent responses, interactive computer programs have been developed to aid the decision maker and the analyst in detecting inconsistencies [13], [37]-[39]. *Accepted by Donald R. Lehmann; received January 10, 1979. This paper has been with the author 3 months for I revision. tNorthwestern University. 60

025-1909/80/2601/0060$0l.25 Copyright ?k 1980, The Institute of Management Sciences


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Recent research has focused on integrating these two approaches. For example, Hauser [28] has shown that the von Neumann-Morgenstern utility theory axioms [57] are consistent with descriptive models of consumer decision-making. In these descriptive models, preference and ultimately choice result from comparison of product attributes and thus the relationship can be modeled with a von NeumannMorgenstern utility like function. This means that the results of prescriptive utility theory which build up unique functional forms through direct and explicit assessment of their components, can be used to study consumer behavior. These forms are important because they allow direct measurement of constructs such as risk attitude and attributes importances. Hauser and Urban [27], [29] have presented an empirical example of one possible utility assessment procedure in the context of students' preferences for health care systems, and discussed the managerial implications. From a measurement standpoint, the preference models' parameters can be obtained through a compositional [4], [47], [50], [59], [60] or a decompositional approach [8], [21], [22], [24]. The first approach assesses utility functions over each attribute separately, and then combines them consistently into a multiattribute utility function. In the second approach, the objective is to decompose a set of overall responses to stimuli so that the utility of each stimulus attribute can be inferred from the respondent's overall evaluation of stimuli. The purpose of this paper is twofold. The first objective is to illustrate and discuss various formal compositional assessment procedures used by decision analysts in constructing multiattribute preference models. The second objective is to present results of a pilot empirical application of the techniques, in the context of consumer preference judgments. The stimuli chosen for the pilot study were housing locations characterized by monthly rent and walking distance. The choice of two-attribute stimuli was based on two major considerations. First, in a study like the one reported here, a tradeoff exists between the number of attributes of interest and the amount of information required to obtain from respondents (this observation has also been reported in [29]). Second, empirical studies on consumer preference judgments suggest that consumers are engaged in explicit tradeoffs between the two most important attributes, once the less important attributes pass their acceptable levels [48], [50], [58]. This is attributed to the limited nature of human information-processing capability [46], [53]. Hence, it was felt that the stimuli reported here represented realistic situations in which consumers are required to make explicit tradeoffs between attributes. In the following sections of the paper, attention will be restricted to two-attribute alternatives. ?2 discusses and illustrates the underlying theory and the measurement procedures required to construct the preference models. ?3 reports on the pilot study conducted to investigate the feasibility of the assessment techniques. ?4 presents findings that shed some light on the predictive accuracy of the preference models and on the usefulness of empirically verifying the conditions necessary for the existence of these models. Summary and suggestions for future research are given in ?5. 2. Theoretical Background The discussion in this section draws heavily on work by Keeney and Raiffa [37, Chapters 3 and 5]. For other related work and survey of theory and applications, readers are referred to [14] and [15]. We shall first introduce some definitions and concepts from which the major theorems are derived. For ease of exposition, our concern in this section is with two-attribute alternatives whose levels are denoted by Xl and X2. Conditions and theorems for the existence of preference structures for alternatives characterized by more than two attributes are generalizations of these results. We select or redefine XT, for any attribute i, such that it is a monotonic


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evaluator. That is, the more an alternative possesses Xi, the more it is preferred by the decision maker, or in our case, the consumer. The theory can also handle nonmonotonic models such as the popular ideal point models by redefining the attribute and measuring it in terms of the absolute distance from the ideal point. We shall first discuss decisions under conditions of certainty and then decisionmaking under uncertainty. In both cases the prescription provided by decision analysts is to select the alternative which offers the greatest utility, or some index of preferability, and thus the assumption that individuals are maximizers of this index from a descriptive standpoint is crucial and is made throughout the paper [1], [6], [19], [27], [28], [29]. 2.1

Value Functions

In many decision problems under certainty, it is the value function [37, Chapter 3] that is used to prescribe or predict the preference judgment. A value function v is a scalar-valued function defined on the consequence space with the property that v(X, X2) > v(Xjl, X2) if and only if (XI, X2) : (Xj, X2), where the symbol > reads "preferred or indifferent to." Here the problem is one of value tradeoffs. The decision maker is faced with a problem of trading off the achievement of one objective against another. The formal decision rule is to combine the various attributes into a scalar index of preferability (compensatory rule) and to choose the alternative that guarantees the highest value of this index. Various multiattribute preference models have been proposed in the literature for descriptive purposes: linear models [2], [26], [31], lexicographic models [44], [56], conjunctive and disjunctive [11], [20] and the exponential discrepancy model [12]. Some empirical studies found the predictive performance of the linear models to be remarkably successful in many situations [20], [54]. From normative perspectives, most of the literature has reported the applications of additive or multiplicative preference models [9], [36]-[38], [40], [49], [55]. An advantage of the decision analysis approach is that the necessary conditions for the existence of some functional forms are testable. For example, the following condition is necessary for additivity. 2.1.1. Necessary Condition for an Additive Value Function. The value function discussed in this paper (also known as an ordinal utility function) has been used for several decades in economics, finance and related disciplines to model consumer preferences. The assessment of the value function is greatly simplified if the corresponding tradeoffs condition is found to hold. This condition is based on the marginal rate of substitution and implies the existence of an additive valuefunction. DEFINITION1. If the decision maker is given an alternative specified by Xl and X2 and if he is willing to trade off an increase A in X2 with a decrease XAin XI, then his marginal rate of substitutionof XI for X2 at (X', X2) is equal to A. DEFINITION2. Consider the four points (alternatives) in the consequence space: (X1, X2), (X1, X2"), (X1'', X2) and (X"', X2"). Suppose that the following rates of substitution hold: 1. At (XI1,X2) an increase of b in X2 is worth a payment of a in X1; 2. At (X1, X2") an increase of c in X2 iS worth a payment of a in X1; 3. At (XI' , X2) an increase of b in X2 is worth a payment of d in X,. If at (X"' , X2")an increase of c in X2 is worth a payment of d in X1 regardless what X1, Xi", X2, X2", a, b, c and d are, then the correspondingtradeoffs condition is said to be satisfied. Figure 1 depicts the corresponding tradeoffs condition graphically for some arbitrarily selected (X1,X2), (X1,X2X'),(X"',X2) and (X"', X"'). At each of these four points, the arrow indicates both the magnitude and the direction of the tradeoff. For example, consider the point (alternative) (X1, X) which corresponds to statement 1 in


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the definition. Since an increase in X2 can be obtained by some decrease in XI, the arrow is directed toward the northwest. In this example the decision maker is willing to give up a units of X1 for an increase of b units in X2, where b < a. X2

eN

c

a

d

a

d

x1 FIGURE1. A Graphical Illustration of the Corresponding Tradeoffs Condition.

THEOREM1. A two-attribute value function is additive and has the form: V(X1, X2) = kIv1(XI)

+ k2V2(X2)

(1)

where vi(X1), i = 1, 2, are the conditional value functions and ki, i = 1, 2, are scaling constants, if and only if the correspondingtradeoffs condition (stated in Definition 2) is satisfied. REMARK. The product kivi(Xi) is referred to as part-worth utility in conjoint measurement [24, p. 43]. PROOF: Clearly, given the additive value function (1), the corresponding tradeoffs condition is met. The converse was proved by Luce and Tukey [43]. 2.1.2. Assessment of a Two-AttributeAdditive Value Function. Some implications of Theorem 1 can be readily seen. If the corresponding tradeoffs condition is verified, the assessment problem reduces to the elicitation of single-attribute value functions and the determination of the scaling constants. However, it should be noted that the verification of the condition may become a tedious task since one has to vary Xl, X2, X' and X2' to test whether preferences do not depend on any specific levels of the attributes. This, of course, determines the length of the required questionnaire. The verification of the necessary condition for additive preference structure when the number of attributes is greater than two requires an even larger number of questions since independence conditions need to be checked for each pair of attributes while keeping fixed the levels of the other attributes [37, p. 108]. Hence, the required length of the questionnaire is an important consideration in the determination of the number of attributes to be studied. When the attributes characterizing the alternatives are continuous (e.g., an automobile price) and their associated value functions are monotonic, it is convenient to bound the value functions between zero to one. Hence, if Xi. < Xi < Xi* for i = 1, 2, where Xi* is the upper possible level of attribute i (most preferred) and Xi* is the lowest possible level of this attribute (least preferred), we can arbitrarily scale the value functions such that:

v1(X*) =0O and v1(X*) =l1

fori=l1,2


(2)

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and v(X* ,X2) = 1.

v(X1*,X20) = 0,

(3)

Substituting Xl, X2 in (1) indicates that the scaling constants are constrained now by: k1 +

k2=

1.

(4)

This gives us one equation that the scaling constants must satisfy. One way to determine the values of the scaling constants, once the single-attribute value functions have been assessed, is to look for alternatives for which the assessor is indifferent. If some alternative (Xl, X2) is as equally preferred as some other alternative, (Xl', X2'), then v(X', X2) = v(X", X2') and k1vl(X)

+ k2V2(X2) =

klv1(X')

+ k2v2(X2 )-

(5)

This is a second equation which can be used to solve for k,,k2 and determine their values when the single-attribute value functions are known. 2.1.3. Assessment of a Single-Attribute Value Function "Midvalue-Splitting" Technique. The assessment of the single-attribute value functions for an additive multiattribute value function can be made through several procedures [37, pp. 91-100]. One of the procedures, which is called the "Midvalue-Splitting" Technique, has been applied successfully to help physicians establish some health-status index and rank order their patients [49]. This procedure is described in detail in [37, p. 94] and can be summarized as follows (see also [49, p. 1013]): DEFINITION 3. The pair of two different levels of attribute i [Xi', Xi"] is said to be differentially value-equivalent to some other pair [Xi"',Xi""] (where Xi' < Xi" and Xi..' < Xi""..),if whenever one is just willing to go from Xi" to Xi' for a given increase of XJ, one would be just willing to go from Xi.[.. to Xi['. for the same increase in Xi (j

i).

For any interval [Xi', Xi"] of Xi, its mid-value point Yi (where Xi' < Yi < Xi") is defined so that the pairs [Xi', YJ] and [Yi, Xi"] are differentially value equivalent. Assuming that the corresponding tradeoffs condition is valid, we seek a singleattribute value function vi(Xi) via the following procedure (the subscript i is dropped for notational simplicity): (1) Find the mid-value point of [X, X*], call it X05 and let v(X05) = 0.5. (2) Find the mid-value point, X075, of [X05, X*] and let v(X075) = 0.75. (3) Find the mid-value point, X025, of [X*, X05] and let v(X025) = 0.25. (4) As a consistency check, ascertain that X0.5 is the mid-value point of [X0.25, XO.75]. (5) Fit in the v(X) curve passing through points (Xk, k) for k = 0.25, 0.5, 0.75. (X*, 0) and (X*, 1). 2.1.4. Assessment of a Single-Attribute Value Function and Scaling Constants"Client-Explicated" Technique. The "Midvalue-Splitting" procedure requires subjects to respond in terms of levels of an attribute (abscissa) and hence to consider explicitly tradeoffs regarding the attribute. An alternative assessment procedure, called "Client-Explicated" [30], [32], requires subjects to provide information in terms of values for given attribute levels (ordinate). A version of this measurement approach has been used to test the "expectancy" theory. According to this theory, individuals' predispositions to given behaviors are governed by two components: the set of satisfactions resulting from such behavior and the probabilities of obtaining these satisfactions through such behavior [17], [511.The products of the two components are concatenated in an additive manner and it is assumed that any nonadditive coefficient is equal to zero. In applying this assessment procedure for building multiattribute preference structures, however, one does not necessarily have to assume that the coefficient associated with the interaction term is zero. By postulating that the


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interaction coefficient is different from zero [24, p. 44], it is possible for instance, to construct nonadditive value functions of the forms described in ?2.2.1 (6). This coefficient must then be inferred from a set of appropriately constructed indifference questions. In this measurement procedure, which has been widely used for predictive purposes in marketing [3], [24], [42], the subject is first asked to indicate his value for various levels of the attribute. This is often done on a 0% to 100% scale where the most preferred level is assigned a level of 100%and the least preferred level is assigned a level of 0%. For each intermediate level, the subject then indicates his relative value. In this way, the value functions over the separate attributes, vi(Xi), can be obtained and graphed. (See question A8 in the Appendix for an example.) To obtain the scaling constants, ki's, the subject is asked to assign the most important attribute a value of 1.0 and then to indicate the relative importance of the remaining attributes, using as a reference the most important attribute. For the purpose of comparisons across subjects and measurement methods, the scaling constants may be normalized by dividing each of them by their sum. In doing so, assurance is obtained that an item whose attributes are all at the most satisfactory level will achieve an overall value of exactly 100%. Finally, the vi(Xi) and ki values are combined to get the specific values of the postulated preference model. Note that the measurememt procedure does not rely upon empirically verifying the condition necessary for the existence of the preference model. That is, the ki's and vi(Xi)'s are assessed independently and then put together according to the a priori form assumed for the preference model. If the "MidvalueSplitting" technique can be thought of as a sequential measurement procedure in which the scaling constants are determined from the already assessed conditional value functions, the "Client-Explicated" procedure appears to be more direct in its nature and more of an independent method with respect to the conditional value functions and the scaling constants. Consequently, since measurement errors may be compounded in using the "Midvalue-Splitting" technique, it is likely that this assessment technique is more sensitive to measurement errors and inconsistent responses than the "Client-Explicated" technique. These two assessment procedures are not the only possible ones; however, they have been widely used and were found to be practical in many situations. For other proposed assessment methods see [18]. 2.2

Cardinal Utility Functions

When the decision consequences are uncertain and the decision-maker needs to express his preferences for probability distributions over these consequences, the decision situation is said to be uncertain or risky. The preference modeling is quite different for decision-making under uncertainty. In this case the tradeoff issue remains, but difficulties are compounded because it is not clear what the consequences of each of the alternatives will be. For example, in making purchasing decisions for major products such as automobiles or housing, consumers do not perceive an attribute such as durability with certainty, and hence the decision is characterized as a risky one. The preference model to be used in this case should explicitly incorporate the decision-maker's attitude toward risk in uncertain (risky) situations. Although various risk theories have been proposed in the literature [52], perhaps the most established and dominant one is the Subjective Expected Utility Theory. According to this theory, the preference structure that needs to be assessed is the von Neumann-Morgenstern cardinal utility function [57] which explicitly incorporates the decision-maker's attitude toward risky alternatives. The criterion to use in choosing among alternatives is the expected utility. This construct is computed by multiplying


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the utility of each consequence by the subjective (or objective) probability that the consequence will occur, and then adding these products across all possible consequences. An alternative theory, called "Prospect Theory" [35], has been proposed recently for descriptive purposes. According to this theory, the preference structure which is different from the cardinal utility function, should be combined with some transformation of the probabilities (decision weights) to yield a measure of goodness for each risky alternative. It should be noted, however, that at this point no formal assessment techniques have been proposed to be used in conjunction with this theory. In working with value functions we investigated additivity. Similarly, in cardinal utility theory there are testable preferential conditions that imply particular functional forms. For example, the following condition is necessary for a functional form called multilinear. 2.2.1. Necessary Conditionfor a Multilinear Cardinal Utility Function. As for value functions, if certain independence conditions hold between X1 and X2, the assessment of the two-attribute utility function, u(X1, X2), can be simplified by expressing it as a function of u1(XI) and u2(X2).One such condition is mutualutilityindependence. (Note that X1 and X2 denote now two random variables.) DEFINITION 4. X1 is utility independent of X2 when conditional preferences for lotteries [37, p. 711 on X1 given X2 do not depend on the particular level of X2. When X1 and X2 are utility independent of each other, the condition of mutual utility independenceis satisfied. THEOREM 2. If XI and X2 are mutually utility independent, then the two-attribute utility function is multilinear and has the following form:

u(X1,X2)

= k1u1(XI) + k2u2(X2) + k12u1(X1)u2(X2)

(6)

PROOF. See [37, p. 234]. The intuitive meaning of the mutual utility independence condition is that there is no interaction of risk preference between the two attributes. The condition is likely to be violated when the level of one attribute affects the preferences for lotteries over the other attribute. For example, in some situations, the decision-maker may be willing to assume more risk with respect to lotteries over X2, where X1 is held fixed at its highest level, compared with the case where X1 is held fixed at its lowest level. Hence, the above condition is violated. Unlike additive value functions discussed in ?2.1.1 (see (1)), the multiattribute preference model discussed in this section is of different form and possesses an interaction term, k12. Of course, as k12 approaches zero, the multilinear form can be approximated by an additive form. Keeney and Raiffa [37, p. 240] have discussed the interpretation of the interaction scaling constant k12. They noted that when k12 is positive, attributes X1 and X2 can be thought of as complements, whereas when k12is negative, attributes Xl and X2 can be viewed as substitutes. 2.2.2. Assessment of a Two-Attribute Multilinear Cardinal Utility Function. When we scale the utility functions such that:

ui(Xi.)=0

and ui(Xi*)=1

fori=1,2

(7)

and u(Xl *, X2*) = 0,

u(Xi*, X2*) = 1,

(8)

the scaling constants are constrained by: k1 + k2+

k12=1.


(9)

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Hence, once the single-attribute utility functions have already been assessed, we need to generate two more independent equations and solve for the three unknown scaling constants. One method for generating the equations involves utilizing "Probabilistic Scaling" [5], [29] in which the subject is asked to provide the probability p that makes him indifferent between the risky alternative that gives (X*, X*) with probability p and (X*, X2*) with probability 1 -p, and the sure alternative that gives (X*, X2*). Equating the expected utilities of the two alternatives yields the following value for the scaling constant k1: k1 =p.

(10)

Alternatively, pairs of indifference judgments between sure (riskless) consequences can be used to generate the necessary equations in a fashion similar to the one described for the value function (e.g. (5)). This scaling procedure, called "Certainty Scaling" [39], has been proposed for use in cases when the decision-maker does not feel comfortable in responding to the complex lottery question where all attributes vary between their extreme values. 2.2.3. Assessment of a Single-Attribute Cardinal Utility Function. The assessment of the single-attribute utility functions can be made through standard gambling methods [18] and the "Probabilistic Midpoint"method has been applied quite often. According to this assessment method, the subject is asked to estimate X05 such that he is indifferent to a 50-50 gamble resulting in either X* or X*. Then u(X05) = 0.5, considering the scaling convention. This procedure continues by finding levels of attribute indifferent to a 50-50 gamble between X* and X05 and between X05 and X* and so forth, until a satisfactory number of points has been obtained. A consistency check can be made by ascertaining the midpoint values, utilizing other assessed values. For example, one can ascertain that the subject is indifferent between X05 and a 50-50 gamble resulting in X025 or X075 after these values have been assessed. To sum up the theoretical framework, it should be clear that since certainty condition is only a special degenerate case of uncertainty, then theoretically a cardinal utility function is also a value function, but a value function is not necessarily a utility function. Thus, theoretical justification exists for using the cardinal utility functions to predict consumers' preferences in situations of certainty [27], [29]. However, this will not necessarily yield the best results in terms of predictive performance. The empirical study reported in the next section addresses this issue. 3.

An EmpiricalApplication

Eighty-five undergraduate business students in a large state university participated in a pilot study which was intended to investigate the predictive accuracy of the measurement techniques and the usefulness of empirically verifying the conditions necessary for the existence of the preference models. As noted by Hauser and Urban [29], measurement techniques such as those utilized in this study lack an error theory and thus prevent one from making statistical statements about parameter estimates. Consequently, in modeling consumers preferences, consistency must be assumed or checked by repeated assessment. Therefore, the subjects were given self-checking consistency questions and were urged to adjust their judgments if they felt that some response was not consistent with previous ones. A pre-test of the questionnaire indicated that the subjects did not face major difficulties in providing the necessary information. To have sufficient time, the subjects were given a questionnaire which they were to return in forty-eight hours. Informal communication with the subjects later revealed that they related well to the tasks and found them interesting.


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The stimuli employed in this study were housing locations, expressed as concept statements. It was felt that students in a college town are familiar with judgments regarding these stimuli, especially at the time when school begins. The subjects were instructed to consider only two attributes: rent and distance from campus. These two attributes were bounded such that the relevant range considered for rent was $100$160 per month and the subjects were told that housing is available only 0.5-3.0 miles from campus. The questionnaire administered' started with a set of questions which were intended to verify the corresponding tradeoffs condition. (See Appendix, Al -A4 for an example.) The subjects were then asked regarding some qualitative characteristics of the conditional value functions: monotonicity and marginal utility. The next set of questions was used to construct conditional value functions, for rent and distance separately, through the "Midvalue-Splitting" method (e.g., A5). Two types of consistency check questions then followed in order to test the validity of the obtained conditional value functions. The first consistency check question (its format is similar to A5) attempted to determine X05 again based on the information already obtained for X025 and X075 (recall that X05 is first determined through X* and X*). A consistent response was defined to be one which showed less than a 10%difference between the two assessed values. The second consistency check question tested the values of the utilities obtained (as opposed to the level of the attribute) for some predetermined levels of the attribute. For example, by using a question such as A6, we can determine whether v($130) is greater, less, or equal to [v($100) + v($160)1/2 and compare it with the assessed value of v($130). Next, the subjects were given an indifference type of question (see A7). This question was used to obtain a second equation in accordance with the "Certainty Scaling" procedure. The first equation used is (4) (see ?2.1.2) which is derived from the scaling convention. The next set of questions used in the value function assessment part of the questionnaire determined the conditional value functions and the relative importances of the attributes through the "Client-Explicated" approach (A8), (A9). The second part of the questionnaire dealt with the assessment of the cardinal utility functions. Here the subjects were instructed to make their judgments in uncertain situations. For example, uncertainty about rent is present when students have to make decisions concerning next year's housing at the time when the market price is uncertain. A situation for making judgments concerning uncertain distance may arise when the student does not know where in the campus the bulk of his classes will be given, and hence he is not sure at the decision-making time what his normal walking distance to campus will be. The mutual utility independence condition had to be first verified in this part of the questionnaire (see (A10-A12)). Once this was done, a set of questions was used to assess the conditional utility functions over rent and distance through the "Probabilistic Midpoint" method (see A13), when the other attribute was held fixed (at 0.5 mile and $100, respectively). Consistency checking questions similar to the ones used for the value functions were then presented. This part of the questionnaire was completed by indifference types of questions for the "Certainty" and "Probabilistic" (see A14) Scaling procedures, which are needed to generate additional equations for assessing kI, k2 and k12. The last part of the questionnaire consisted of five questions concerning new contending pairs of riskless alternatives specified by rent and distance. In each 'To conserve space, only a sample of the actual questions' formats is given in the Appendix. The complete questionnaire is available from the author upon request.


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question the subjects were asked to check one of three possibilities: (a) alternative a is most preferred, (b) alternative b is most preferred or (c) indifference between the two alternatives (e.g., (A15)). 4.

Analysis

In order to estimate the conditional value and utility functions for any possible level of the attribute, five points (X0o125 X0250, X0.500 X0.750,X0.875)were selected from the domain of each conditional function. By assessing the value of the conditional function at each of these five points through the questionnaire, we obtained a piecewise linear estimate of the desired function. This approximation is also reported in [9], [10]. From the eighty-five questionnaires administered, sixty-six were usable in that they were completely filled out. Table 1 shows the distribution of these sixty-six subjects with respect to the two major necessary conditions: corresponding tradeoffs and mutual utility independence. A chi-square analysis performed on the data indicates that there is no statistical association between the two conditions (p > 0.1). Further analysis was conducted on subjects who exhibited consistent responses (defined earlier). Tables 2, 3, 4, 5 illustrate the actual vs. predicted preferences obtained through value functions for these subjects. In situations where ties in preferences are allowed, it is important to recognize the concept of just noticeable difference.This concept which is central to many psychological experiments, assumes that two alternatives having utilities (values) which are not sufficiently distinct will cause the subject to exhibit indifference type of preference. This threshold level was measured and found to vary over various stimulus domains, subjects, and testing situations [25]. Krishnan [41] found that preference models utilizing the concept of just noticeable difference predict choices better than models that assume that an individual must prefer one or the other alternative and cannot remain indifferent. Therefore, we have to choose a threshold level sufficiently large to be above the just noticeable difference. In our case this was done judgmentally with a resulting threshold equal to 0.03. Thus, a preference for alternative a over b is predicted if and only if the difference between the two multiattribute values (utilities) evaluated at these alternatives exceeded this level.2 Based on Tables 2 through 5 it appears that the "Client-Explicated" method is more accurate than the "Midvalue-Splitting" method, independent of whether or not the corresponding tradeoffs condition is satisfied. TABLE 1 Distributionof the Sample over the Necessary Conditionsfor Additive Value Functions and Multilinear Utility Functions. Mutual Utility Independence Condition

Corresponding Tradeoffs Condition

Verified

Not Verified

Total

Verified

24

29

53

Not Verified

7

6

13

31

35

66

Total

2There is no empirical evidence in similar situations for the magnitude of the threshold. However, for comparison across preference models and assessment methods, the results are not likely to be influenced by the choice of this particular level which has been reported in [25].


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TABLE 2 Actual vs. Predicted Preferencesfor Value Functions Obtained throughthe "Midvalue-Splitting"Method and for which the CorrespondingTradeoffs Condition Was Satisfied. Predicted Preference

Actual Preference

a3 b3

a3

b3

c3

Total

27 9

11 47

11 7

49 63

c3

9

14

5

28

Total

45

72

23

140

Percentage of correct prediction: 56.43% Maximum chance criterion: 45.00% Proportional chance criterion: 36.50% n = 28

TABLE 3 Actual vs. Predicted Preferencesfor Value Functions Obtained throughthe "Client-Explicated" Method andfor which the CorrespondingTradeoffs Condition Was Satisfied. Predicted Preference

Actual Preference

a b c Total

a

b

c

Total

55 9 8 72

13 60 13 86

12 13 2 27

80 82 23 185


TABLE 4 Actual vs. Predicted Preferencesfor Value Functions Obtainedthroughthe "Midvalue-Splitting"Method and for which the CorrespondingTradeoffs Condition Was Not Satisfied. Predicted Preference

Actual Preference

a b

a

b

c

Total

8 4

7 14

4 2

19 20

21

6

40

c

1

Total

13

1

Percentage of correct prediction: 55.00% Maximum chance criterion: 50.00% Proportional chance criterion: 47.62% n= 8 3a, b and c are the three preferential possibilities concerning the new contending pairs of riskless alternatives discussed at the end of ?3.


CONSUMER

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71

JUDGMENTS

TABLE 5

Actual vs. Predicted Preferencesfor Value Functions Obtained throughthe "Client-Explicated" Method and for which the CorrespondingTradeoffs Condition Was Not Satisfied. Predicted Preference

Actual Preference

a b c Total

a

b

c

Total

20 1 3 24

3 22 2 27

5 3 1 9

28 26 6 60


In addition to comparing the preference models across assessment procedures, it is also interesting to test the predictive ability of the elicited models in comparison to chance models. Morrison [45] has shown that two chance criteria may be considered: the proportional and the maximum chance criteria. Under the proportional chance criterion, we predict the judgments in proportion to the actual judgments. With the maximum chance criterion, we predict the judgments according to the maximum frequency of the actual judgments. The results indicate that all of the assessed preference models predicted preferences better than both maximum and proportional chance models. (Another possible criterion is an equal chance naive model that would predict correctly 33% by assuming equal probability over the three possible responses for each alternative.) Tables 6, 7, 8, and 9 illustrate the results of predictive performance of cardinal utility functions obtained through "Certainty" and "Probablistic" Scaling with and without the verification of the mutual utility independence condition. The results appear to indicate no effect for the verification of the necessary condition for utility functions obtained through "Certainty Scaling" and slight improvement (48.7% vs. 50.0%) in the predictive performance of utility functions obtained through "Probablistic Scaling." Hence, the data generally suggest that the various preference models are predictively robust with respect to deviations from their TABLE 6

Actual vs. Predicted Preferencesfor Utility Functions Obtained through "Certainty Scaling" and for which the Mutual Utility Independence Condition Was Satisfied. Predicted Preference

Actual Preference

a b c Total

a

b

c

Total

7 3 2 12

7 18 3 28

7 4 4 15

21 25 9 55

Percentage of correct prediction: 52.73% Maximum chance criterion: 45.45% Proportional chance criterion: 37.92%

n= 11


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ELIASHBERG

TABLE 7

Actual vs. Predicted Preferencesfor Utility Functions Obtained through "Probablistic Scaling" andfor which the Mutual Utility Independence Condition Was Satisfied. Predicted Preference

Actual Preference

a b c Total

a

b

c

Total

25 8 10 43

19 42 11 72

12 10 3 25

56 60 24 140

Percentage of correct prediction: 50.00% Maximum chance criterion: 42.86% Proportional chance criterion: 37.31% n = 28 TABLE 8

Actual vs. Predicted Preferencesfor Utility Functions Obtained through "Certainty Scaling" and for which the Mutual Utility Independence Condition Was Not Satisfied. Predicted Preference

Actual Preference

a b c Total

a

b

c

Total

11 1 2 14

6 17 2 25

6 7 3 16

23 25 7 55

Percentage of correct prediction: 56.36% Maximum chance criterion: 45.45% Proportional chance criterion: 39.77%

n= 11 TABLE 9

Actual vs. Predicted Preferencesfor Utility Functions Obtained through "Probablistic Scaling" and for which the Mutual Utility Independence Condition Was Not Satisfied. Predicted Preference

Actual Preference

a b c Total

a

b

c

Total

19 10 4 33

21 34 4 59

9 11 3 23

49 55 11 115


necessary conditions. This result is parallel to the robustness of linear models as reported by Dawes and Corrigan [7]. It is also interesting to note the similarity of the result in Table 7 with Hauser and Urban's study. They used a similar assessment technique and reported 50% correct prediction of the first choice among the four concepts given to their subjects [29]. The comparison of the two scaling procedures is slightly in favor of the "Certainty Scaling" procedure (52.73% vs. 50.0%) and as for


73

CONSUMER PREFERENCE JUDGMENTS TABLE 10 Summary of Percentage CorrectlyPredicted by Preference Model and by Satisfaction of Necessary Condition. \Pref erence \ odel Necessary Condition

Value Function

\

Verified Not Verified

Cardinal Utility Function

Midvalue-

ClientExplicated

Certainty Scaling

Probablistic

Splitting

63.24% 71.67%

52.73% 56.36%

50.00% 48.70%

56.43% 55.00%

Scaling

the value functions, all of the assessed cardinal utility functions predicted the preferences more accurately than the two chance models. When an overall comparison between the predictive performance of value and cardinal utility functions is made, the data generally suggest that in situations of certainty, value functions predict preferences more accurately than cardinal utility functions (see Table 10). This finding, which is consistent with Fischer's experiment [16], seems to indicate that the cost of the additional complexity in assessing cardinal utilities may not be warranted for predictive purposes in situations of certainty. 5.

Summary

Many preference judgments made by consumers involve outcomes having several important attributes. A further classification of these judgments deals with the notion of certainty and uncertainty. That is, with situations where the decision consequences are known or uncertain to the consumer. In order to predict the consumer's preferences for multiattribute alternatives, we need to construct a preference model and assume that the consumer's judgment is dictated by some index of preferability maximization. In situations of certainty, the value function is appropriate to dictate the judgments, whereas in uncertain situations, the cardinal utility function is needed to capture the complexity of the decision. However, since certainty is a special case of uncertainty, theoretically a utility function is also a value function but a value function is not necessarily a utility function. This paper presented and discussed several formal assessment procedures that can be used for constructing consumers' multiattribute preference models. The procedures often require the respondents to provide different types of information. An empirical pilot study was conducted to investigate the predictive performance and the implications of empirically verifying the conditions necessary for the existence of the preference models. The results indicate no statistical association between the conditions necessary for the existence of additive value functions and multilinear utility functions. Furthermore, the predictive performance of the models studied appears to be robust with respect to these conditions. Examination of the data reveals that in situations of certainty, value functions predict preferences more accurately than utility functions and that all of the assessed preference models predict preferences more accurately than chance models. Although some of the results of the pilot study are consistent with some other findings obtained in different decision situations, the study involves several factors which limit its generalizability. For example, the sample was composed of undergraduate students who were given sufficient time to check for self-consistency. Further testing is required to determine whether an average respondent could consistently provide the necessary information in less time. It is also possible that some order bias existed in the presentation of the measurement techniques to the respondents. Without an error theory, it is difficult to answer in a statistical sense, the question as to


74

JEHOSHUA

ELIASHBERG

whether the elicited preference models are significantly different from each other in terms of their parameters. Without a probability distribution theory regarding the coefficients of the preference models, one cannot test them for significance. A small sample size makes any other statistical inference even more difficult. Several directions are possible for future research. First, stimuli characterized by larger number of attributes that are also perceptual rather than physical may be studied in a similar fashion. Second, other preference models may also be investigated and shed more light on the tradeoff between the predictive ability of these models and the task required from the subjects in providing the necessary information. Finally, a very important avenue for future research appears to be in identifying various consumer groups for which different assessment procedures such as the ones discussed here will yield the best predictive performance.4 Appendix A Sample of Actual Questions'Formats Al. Suppose you have arrived early in the semester to Columbia and are seeking an apartment. The unit you locate is 1.0 miles from campus and the rent is $110. Before signing the lease you heard about an equivalent apartment 0.5 mile from campus (or 0.5 mile closer to campus). If the owner is willing to negotiate the rent, how much more would you be willing to pay for this unit? Answer: $ more A2. Suppose you have rented an apartment 1.0 mile from campus for $150. How much more would you be willing to pay for an equivalent unit 0.5 mile from campus? Answer: $ more A3. If the unit you located is 2.5 miles from campus and the rent is $1 10, how many miles would you like to move closer to campus for the same increase in rent you specified in your answer to Question A 1? Answer: miles closer to campus A4. Now think hard about this one. Suppose that you arrived late in the semester to Columbia and located a unit 2.5 miles distance from campus for $150 rent. For the same increase in rent that you specified in your answer to Question A2, would you be willing to move closer to campus by the same distance you specified in your answer to Question A3? Yes No (please briefly explain why) A5. Specify a rent, x, such that you would give up the same distance to save from a rent of $160 to x as from x to a rent of $100. X =?

_

A6. Suppose that you have rented a unit at a given distance from campus. Would you give up more distance to change the rent from $160 to $130 or from $130 to $100? ______more from $160 to $130 more from $130 to $100 indifferent A7. Specify a rent x such that you are indifferent between a and b: a. $x rent and 3.0 miles from campus 4 Some preliminary results of this article were presented at the 1978 National AIDS Meeting, St. Louis, Missouri, October 30-November 1, 1978. I gratefully acknowledge the help of my research assistant, Patrick Koelling, in analyzing the data. I wish to thank my colleague John R. Hauser for his helpful comments and suggestions on an earlier version of this paper. I am also grateful to the two reviewers and a Departmental Editor for suggesting many improvements in the exposition of the paper. Comments by George John, Lou Stern, Brian Sternthal and Andy Zoltners also contributed to the clarity of this article.


CONSUMER

PREFERENCE

JUDGMENTS

75

b. $160 rent and 0.5 mile from campus A8. If a rent of $100 is assigned a level of 100%which indicates your satisfaction with the rent, and a rent of $160 is assigned a level of 0%,please indicate your relative satisfaction on a 0% to 100%scale of the following rents: Rent Level of Satisfaction %100 $100 $110 % $120 % $130 % $140 % $150 % $160 % 0 A9. Assign the number 1 to the factor (rent or distance) which you consider more important in making your housing judgments. Now, on a 0 to 1 scale, how would you rate the relative importance of the second factor? Factor Relative Importance rent distance

AlO. Assume that the distance is known to be 0.5 mile. What is the sure rent r that will make you indifferent between receiving it, and a lottery that will give you a unit with rent $160 at 0.5 chance, and at the same time, 0.5 chance of receiving a unit with rent $100? r =$ All. Would the amount r that you specified in question A 10 be changed if the distance was known to be 2.5 miles and you had to respond to question AlO again? Yes (please briefly explain how) No A12. Would the amount r that you specified in question AlO be changed if the distance was known to be different from 0.5 to 2.5 miles but within the range of 0.5-3.0 miles from campus and you had to respond to question AIO again? Yes (please briefly explain how) No A13. Please specify the sure amount of rent x that will make you indifferentbetween receiving it and a lottery that will give you a unit for $160 rent with 0.5 chance and $100 rent with 0.5 chance: X =?$

A14. For what probability p are you indifferent between alternatives a and b: a. receiving the lottery giving a p chance of receiving a unit with $160 rent and 3.0 miles from campus, and at the same time, a 1 - p chance of receiving a unit with $100 rent and 0.5 mile from campus. b. receiving for sure a unit with $160 rent and 0.5 mile from campus. P =

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