Stochastic Weights in Multiattribute Decision Making

Dov Pekelman, University of Pennsylvania
Subrata Sen, University of Rochester
ABSTRACT - Most multiattribute models assume that the consumer's brand perceptions, ideal point, and attribute weights do not change from one buying situation to another. However, since consumers express preferences for different brands in different purchasing situations, one or more of these parameters must be changing over time. To explain such multibrand preference, we present a model which allows one of the model's parameters, the consumer's attribute weights, to vary across purchasing situations.
[ to cite ]:
Dov Pekelman and Subrata Sen (1976) ,"Stochastic Weights in Multiattribute Decision Making", in NA - Advances in Consumer Research Volume 03, eds. Beverlee B. Anderson, Cincinnati, OH : Association for Consumer Research, Pages: 81-87.

Advances in Consumer Research Volume 3, 1976      Pages 81-87


Dov Pekelman, University of Pennsylvania

Subrata Sen, University of Rochester


Most multiattribute models assume that the consumer's brand perceptions, ideal point, and attribute weights do not change from one buying situation to another. However, since consumers express preferences for different brands in different purchasing situations, one or more of these parameters must be changing over time. To explain such multibrand preference, we present a model which allows one of the model's parameters, the consumer's attribute weights, to vary across purchasing situations.

An important problem in marketing is the design of new products or the modification of existing products. Products can be characterized by a set of descriptive attributes such as crispiness, sweetness, and calorie content for dry cereals. Product design, therefore, involves the determination of attribute levels which will result in a successful product. In order to do this, it is first necessary to understand how consumers combine and weight the various attributes in order to arrive at overall preference judgments for the existing brands of a product class.

Traditional economic demand theory is not equipped to aid the marketing researcher in this task because it does not utilize any information on product attributes. It starts with information on product preferences (expressed in terms of an indifference map) which already incorporates all information about product attributes. All that demand theory can say about product attributes is that the preference map would be different if the products were different. Since there is no theory about how product attributes affect preferences, traditional demand theory cannot predict how demand would be affected by changes in one or more attributes of an existing brand or how a new brand would affect the preference structure for existing brands (see Lancaster, 1971, pp. 1-12, for a more detailed discussion of these is sues ).

In recent years, however, various models of the process by which consumers combine product attributes to arrive at overall preference judgments have been presented in the marketing and decision making literature. The major models are the linear compensatory model (Bass, Pessemier, and Lehmann, 1972, Bass and Talarzyk, 1972, Lehmann, 1971, and Pekelman and Sen, 1974), the conjunctive and disjunctive models (Einhorn, 1970, and Wright, 1973), and various versions of the lexicographic model (MacCrimmon, 1968, Russ, 1971, and Tversky, 1972). Most of the models cited above deal with three parameters of the choice process: (1) the ideal point, defined as the consumer's vector of most-preferred levels for each attribute, (2) the attribute weights, defined as the consumer's vector of saliences indicating the relative importance of each attribute, and (3) the consumer's perceptions of the locations (or attribute levels) of each brand. Most models have implicitly assumed that these parameters are unique for each consumer and do not vary from situation to situation. But is this a reasonable assumption? We attempt to answer this question by exploring the value of a model which allows these parameters to vary across purchase situations.

We start by examining why one might wish to view these parameters as being stochastic. First, company advertising is often designed to alter the consumer's ideal point, attribute weights, or brand perceptions so as to increase the probability of purchasing the company's brand (see Boyd, Ray, and Strong, 1972). But even in the absence of such advertising, it might be reasonable to allow at least the consumer's ideal point and weight vector to vary across purchase situations. If the consumer's ideal point and weight vector always remained unchanged we would expect the consumer to provide the same preference ranking of the brands at all times. However, when confronted with different purchase situations, consumers typically provide preference rankings which change from situation to situation. For example, Belk (1974) reports that he obtained different likelihoods of purchase for snack items such as potato chips, popcorn, cookies, assorted nuts, and crackers when he varied the purchasing situation. The following examples are illustrative of the purchasing situations presented by Belk to his sample of consumers:

1. You are shopping for a snack that you or your family can eat while watching television in the evenings.

2. You are planning a party for a few close friends and are wondering what to have around to snack on.

3. You are going on a long automobile trip and are thinking that you should bring along some snacks to eat on the way.

Other examples of changing consumer preferences when confronted with different purchasing situations are provided in Bass, Pessemier, and Lehmann (1972), pp. 538-541, Day (1972), p. 284, Green and Rao (1972), pp. 152-180, Hansen (1969), p. 442, and Sandell (1968).

The increasing body of evidence regarding the existence of situational behavior has motivated the recent development of stochastic preference theories (Bass, 1974, and Herniter, 1973). However, these theories do not explain why a consumer's behavior changes across situations. We suggest, in this paper, a possible explanation. Specifically, we postulate that changes in situations alter a consumer's ideal point or weight vector, resulting in changes in his overall brand preference structure. The changed preference structure manifests itself in changed preference rankings or likelihoods of purchase as the situation is varied.

So far, the discussion has centered around consumers' stated preferences. What about their actual purchasing behavior? For inexpensive, frequently purchased products, diary panel records indicate that most consumers do not always purchase the same brand. Some of the brand switching is undoubtedly due to price variations, unavailability (out-of-stock) of the consumer's most-preferred brand, and word-of-mouth recommendations for the brand that the consumer switched to. However, even when these marketing factors do not exhibit much change (as frequently happens for limited time periods), we observe brand switching on the part of consumers. Again, it seems reasonable to postulate that this brand switching is occasioned by changes in either the ideal point or the weight vector. We should point out that an alternative explanation for such brand switching is provided by Lancaster (1971), pp. 25-49. However, see Pekelman and Sen (forthcoming) and Hendler (1975) for some problems with Lancaster's approach.

We believe that a model with stochastic parameters (i.e., a model whose parameters can vary from situation to situation) can be useful in explaining situational effects in buyer behavior. In other words, multi-brand preference (or purchase) can be explained fairly well by a model with stochastic parameters. Therefore, the principal objective of this paper is to present a model which allows the consumer's weight vector to vary from situation to situation. To simplify the model, we will continue to assume that his ideal point remains unchanged. We use the stochastic weight model to predict the frequency with which consumers purchase each of the brands in a particular product class. To test the predictive power of the model, we compare our predictions with the consumer's actual purchase frequencies as recorded in a diary purchase panel.

In this paper, we limit ourselves to validating the stochastic weight model. But if we can validate this model, we will obtain a clearer understanding of how consumers utilize product attributes to form preference judgments. This will enable us to make better predictions of consumers' purchases which will aid us in our ultimate objective of helping manufacturers to make better decisions regarding the design of new brands and the modification of existing brands.

The next section consists of a description of our model of how consumers combine product attributes to arrive at overall preference judgments. This preference model is then used to formulate the optimization problem which enables a firm to determine the best combination of attributes for its brands. The preference model is initially described with a deterministic ideal point and weight vector. We then allow the weight vector to be stochastic and provide a numerical example to illustrate the nature of our stochastic parameter model. Next, we provide a formal representation of the model. The paper concludes with a description of the data available to test the model.


Consider q brands each characterized by a set of n attributes. For each attribute, a consumer has a specified preferred position denoted as his ideal point. When the consumer evaluates a brand, he estimates the "distance" between the brand's position on each attribute and his ideal point. Once these distances are estimated for each attribute, the consumer computes an "overall distance" by means of a weighted combination of the individual distances. These ideas can be expressed more formally as follows: xij = location of consumer i's ideal point on attribute j, where i = 1, . . ., m, j = 1,. . ., n and xi = (xi1, xi2, . . ., xin). ykj = location of brand k on attribute j where k = 1, 2, . . ., q and yk = (yk1, yk2, . . ., ykn). dijk = distance of brand k from consumer i's ideal point on attribute j, i.e.,

dijk = |xij - ykj|   (1)

wij = weight assigned to attribute j by consumer i and wi = (wi1, wi2, . . ., win). Dik = overall distance of consumer i from object k, i.e.,


In Appendix A we distinguish between two main types of attributes and discuss the economic implications of each attribute type. Briefly, attribute type A is such that a consumer prefers as much (or as little) of it as possible. The following two automobile attributes are examples of type A attributes: miles obtained per gallon of gasoline and repair expenditures per month. If the upper and lower limits of zj (the level of attribute j) are zj and zj, the consumer's ideal point for type A attributes is located either at zj or at zj. zj is possibly ~ while zj is usually 0. The second type of attribute (type B) is such that the most preferred level is not at either extreme. Examples of type B attributes are: the amount of sweetness in a cereal or the amount of carbonation in a soft drink. Essentially, a consumer would generally prefer a cereal with a certain amount of sugar and his preference would decrease if the amount of sugar increased or decreased from that level.

We assume that cost differences involved in producing brands with different levels of type A attributes may be considerable, while for type B attributes it is negligible (see Appendix A for a discussion of this assumption). We concentrate on type B attributes in this paper primarily because we can separate the problem of finding the optimal combination of attributes from the problem of setting the optimal price (again, see Appendix A for a more detailed discussion of this point).

The preference model described above can be used to formulate the firm's optimization problem which enables it to determine the best combination of attributes for its brands. Since the primary concern of this paper is the validation of the stochastic parameter model, we relegate the formulation of the firm's optimization problem to Appendix B.


The solution of the optimization problem described in Appendix B depends upon the values of the preference model parameters. These parameters have typically been assumed to remain constant. However, as was argued earlier, it seems more reasonable to allow these parameters to vary across purchase situations. Consequently, we present below a model with stochastic parameters. To keep the model simple, we assume that the consumer's ideal point does not change. However, we allow his weight vector to shift from situation to situation. As the weights vary, the rank orderings, in general, will not remain the same since the Dik's (defined in equation (2)) will change for each situation. Changes in the rank orderings can result in changes in the most-preferred brand. The proportion of situations in which a particular brand attains the most-preferred position will constitute our prediction of how often the consumer will choose that brand. To validate the model, we need only to compare these predictions with the consumer's actual choice patterns.

In order to make such predictions, the first step is to determine all the situations that can possibly occur. This is an extremely difficult task since each situation may be characterized by a combination of a large number of events, and different situations may be characterized by different sets of events. This fact also suggests that it would, in general, be infeasible to relate each one of the elements characterizing a situation to choice behavior. For example, situations preceding a choice of a cereal brand may be characterized by the composition of the menu, the time of the day, temperature, appetite, etc. Each of these elements has some implicit relationship to choice behavior, but to investigate them all would be impractical. However, each situation corresponds to the consumer's choice of a vector of weights which, in turn, leads to the rank order specified by the consumer for the situation. Therefore, by delineating all possible rankings we essentially cover all possible situations. We are now ready to formalize the model which is based on the assumption that the probability of purchasing brand k given a set of weights is equal to 1 if Dk < Dk', for all k k', and equal to zero otherwise. We start by describing a simple numerical example which illustrates the basic ideas of our model. Numerical Example Let the distances of three brands (A, B, and C) from an individual's ideal point on three attributes (the dijk defined in (1)) be as follows:


Let w1, w2, and w3 represent the individual's weights on the three attributes and let the sum of the weights equal 1.0 for simplicity. The first step in the analysis is the specification (for every pair of brands) of the constraints which express the fact that one brand of the pair is preferred to the other. Starting with brands A and B, the constraint specifying that A is preferred to B is as follows:

(D"-DB) < 0 <=> (-2w1 + w2 + w3) < 0 <=> w1 > 1/3


(D"-DC) < 0 <=> (-w1+ w2) < 0 <=> w2 < w1

(DB-DC) < 0 <=> (w1 - w3) < 0 <=> w2 < (1 - 2w1).

These constraints define ranges of weights within which a certain rank order would hold as illustrated in Figure 3. For instance, the area PQRS in Figure 3 represents: (1) w1 > 1/3, implying A is preferred to B, (2) w2 > w1, implying A is preferred to C, and (3) w2 < (1 - 2w1), implying C is preferred to B. This gives rise to the rank order [ACB](which indicates that brand A is preferred to brand C and brand C is preferred to brand B). The basic logic of the analysis is unaffected if, instead of the City Block metric used in the above constraints, we use the Euclidean metric described in equation (2) (see Pekelman and Sen, 1974, for a discussion of the appropriate metric). The next step of the analysis involves the prediction of the frequency with which each brand is chosen by the consumer. This requires the determination of a distribution for the attribute weights. Initially; we will assume a uniform distribution of weights. Given this assumption, the choice frequency for each brand for a three-attribute problem is given by the ratio of the sum of the areas where the brand is most preferred to the total area. For example, the computation of Pr{A}, the probability of choosing A, involves first the summation of the areas where A is the most-preferred brand: areas PQRS and QTR in Figure 3. Pr{A} is then computed as the ratio of this area sum, PQTS, to the total area, UVS. Using this model, the predicted probabilities are:

Pr{A} = 0.39, Pr{B} = 0.44, and Pr{C} = 0.17.

The final step is to compare these predictions with the consumer's actual choice behavior as computed from the record of his purchases in a diary panel. If, for example, the consumer's purchase history indicated that he bought brand A 40 per cent of the time, brand B 45 per cent of the time, and brand C 15 per cent of the time, our predictions (Pr{A} = 0.39, Pr{B} = 0.44, and Pr{C} = 0.17) would compare very favorably with actual behavior.



We also present a second (and weaker) version of the model in which we attempt to incorporate information about unrealizable rankings since, for a given consumer, not all possible rankings are realizable. For example, if we consider three objects, the consumer may state that two of the six possible rankings have not occurred in the past and are highly unlikely to occur in the future. Such a statement would make sense if, for example, the consumer could not visualize any circumstances in which one of the three objects would be his most-preferred alternative. As shown below, unrealizable rankings modify our brand choice prediction in a simple manner. Clearly, this version of the model amounts to what we may call "partial prediction." Here we predict just part of the choice distribution, the realizable rankings, while the other part of the distribution, the unrealizable rankings, is given as input.

Returning to our numerical example, assume that the consumer states that the rank order [BAC] is unrealizable. This amounts to redefining the "total area" in Figure 3 as (UVS - QVT). The modified predictions are:

Pr{A} = 0.44, Pr{B} = 0.37, and Pr{C} = 0.19.

Formal Representation of the Stochastic Weight Model

The formal representation of the model utilizes the following notation:

su = a discrete situation in the situation space S, u = 1, 2, ..., h where h denotes the number of situations.

wu = a vector of attribute weights corresponding to su (one to one). wu is an n-dimensional vector in w-space where n equals the number of attributes.

Rr = a particular rank order of all objects, r = 1, 2, . . ., q! where q denotes the number of objects.

Wr = the set of vectors, wu, leading to the rank order, Rr. The set Wr contains ar members. In other words, ar weight vectors lead to the rank order Rr. Clearly, ar can equal zero.

ck = event of having object k (k = 1, 2, . . ., q) as the most preferred object. ck represents the decision to choose object k.

Rk= the set of all Rr's such that the most preferred brand in Rr is k. Let bk be the number of members in this set. bk will range between 0 and (q-1)! (In this model, ties are not allowed).

Wk = the set of  Rk defined in terms of attribute weights, i.e.,


Pr(su) = probability that situation s will occur. This is also equal to the probability that the weight vector, wu, will occur. Thus, Pr(su) = Pr (wu).

Pr(Rr) = probability that rank Rr would be realized.


Pr(ck) = probability that decision ck would be realized.


Our ultimate objective is to compare the computed values of Pr(Ck) with the consumer's actual frequency of choosing brand k.


The two data sets required for both versions of the model are:

1. Relative frequency of purchase of the various brands.

2. The distances between the ideal points and the locations of the q brands on each of the n attributes (the dijk of equation (1)).

The first data set can be obtained from existing consumer panel data. The second set of data should be collected from the same consumer group participating in the panel by direct questioning. For the weaker version of our model, there is an additional data require-merit, i.e., the specification of the set of unrealizable rankings. This can be obtained by direct questioning in the same questionnaire used to obtain the brand distances.

The M.R.C.A. Data

The Market Research Corporation of America (M.R.C.A.) has made available to us the coffee purchasing records of a panel of 900 households. The panel was operated by M.R.C.A. between October, 1972 and June, 1973 in Phoenix, Arizona. The panel members were chosen after ensuring that they had purchased both instant and regular coffee. From this panel data, the relative frequency of purchase of the leading coffee brands in Phoenix can be easily computed.

In addition to the panel data, we have available the responses of 542 of the 900 panel members to a series of questions dealing with their perceptions of the 11 leading coffee brands in Phoenix in terms of six product attributes. These 11 brands accounted for 70 per cent of all coffee purchases in Phoenix.

The purchase panel data and the brand perception responses provide us with the information needed to test the stochastic weight model presented in this paper.

We cannot test the weaker version of the model because it was not possible to gather data on a complex concept such as "unrealizable rankings" in a mail questionnaire. Such information can probably be gathered only through personal interviews. The questionnaire also provides information on brand preferences. Some limited lifestyle information is also available. In addition, M. R. C.A. has provided detailed demographic data on each panel family. The data were made available to us very recently. Thus, it has not been possible to present any empirical results in this paper.

Evaluation of the M.R.C.A. Data

Coffee data are particularly appropriate for testing a model of situational behavior since the choice of instant versus regular coffee is likely to be situation oriented. Further, coffee is a frequently-purchased product which will result in a relatively large number of purchases per family. This will provide us with the sample size necessary to compute market shares for the various brands with a reasonable degree of confidence.

However, it is important to point out some of the potential problems in using panel data to test the stochastic weight model. Our model cannot distinguish between brand switching due to marketing factors (such as price and promotion) and brand switching due to changing attribute weights. Hence, we expect that the model will perform better for those consumers who are not very sensitive to marketing variables and yet switch between brands frequently (see Blattberg and Sen, 1974, for descriptions of such consumers). Even for these consumers, the brand switching could result not only from changing attribute weights but also from changing ideal points, changing brand perceptions due to company advertising, and purchases made by different members of the family (Davis, 1970). For the purpose of testing the stochastic weight model, we will have to assume that these variables do not have a systematic effect on the brand choice frequencies.



The nature of the two types of attributes is illustrated in Figures la and lb where U(zj) equals the utility derived from consuming the amount zj of attribute j (given zi for all i j). Type A attributes (Figure 1a) are characterized by:


for the entire range of z.. Type B attributes (Figure 1b), on the other hand, are characterized by:


where zj* equals the ideal level of attribute j.

There exists another important difference between these two types of attributes. The cost differences involved in producing brands with different levels of type A attributes may be considerable, while for type B attributes it is negligible. To produce brands of cereal with different amounts of sugar within a limited range would usually involve negligible differences in cost. (Naturally, once we move out of this range, we may have technological problems which could increase the cost sharply.) Similarly, the manufacture of soft drinks with different levels of carbonation would not generally involve different costs. However, the addition of safety mechanisms in a car or the improvement of its miles per gallon performance would generally involve a considerable increase in cost.





These ideas are illustrated in Figures 2a and 2b which show demand and marginal cost (MC) curves for four brands: 1, 2, 3, and 4. The "level of attributes" increases from brand 1 to brand 4. In accordance with our discussion above, we make the following two assumptions: (1) for type A attributes (Figure 2a), each brand has a different MC curve (MC4 > MC3 > MC2 > MC1), and (2) for type B attributes (Figure 2b), each brand has an identical MC curve (MC4 = MC3 = MC2 = MC1 = MC). For both types of attributes each brand has a different demand curve. For type A attributes, demand increases with the "level of attributes" in the brand, i.e., D4 > D3 > D2 > D1. However, for type B attributes, D2 > D3 > D4 > D1. Note that brand 2 has the highest demand while brand 4, with the highest "level of attributes," is superseded in demand by both 2 and 3.

It is obvious, therefore, that if the firm produces a single brand, it would select the one with the highest demand in Figure 2b while it may select any of the levels (depending upon the corresponding MC) in Figure 2a. Essentially, the proposed research examines certain elements of a model for finding the highest demand function when type B attributes are involved.

Given the assumption that there exists an identical MC for all attribute combinations, decisions regarding price and the optimal combination of attributes are separable; i.e., the decisions can be made sequentially. In other words, we wish to identify first the attribute combination which yields the highest demand curve and then determine the price.







Let Pik be the probability that individual i would purchase brand k during a specified period of time.

Pik = fik(Di1, Di2, ..., Dik, ..., Diq;   p1, p2, ..., pk, ..., pq)    (5)

where p1, p2, ..., pq are the prices of the q brands and where

EQUATIONS   (6)  -  (8)

Equation (6) assures the separability of the two main problems: the optimal composition of attributes and the optimal pricing policy. And indeed, there is no reason, a priori, to assume that the effect of price (in terms of a change in the probability of purchase) would be different for brands with different overall distances. The function fik($) can therefore be separated as follows:

fik($) = gik(Di1, Di2, ..., Diq) + hik(p1, p2, ..., pq) .   (9)

The function gik($) is discussed in the literature (Schonemann and Wang, 1972 and Stefflre, 1972) where the commonly investigated forms are:

EQUATIONS   (10)  -  (13)

In general, the probability of purchase, Pik decreases as the distance, Dik, increases. Given that the consumer distributes his expenditures among brands in a product class according to the specified probabilities, the firm would wish to maximize the expected probability over all consumers. Therefore, if there exist ri consumers of type i and the firm wishes to produce a new product k, then the problem faced by the firm is: Find  ykj for all attributes j such that the expression Ei Pik$ ri would be maximized, or, equivalently:

max Ei rif ik(Di1, ..., Diq, Dik; p1, ..., pq, pk)   (14)


Dik = Ej(xij - yij)2wij for all k=1, 2, ..., q, k  .

Note that the parameters ri, ykj, xij, wij and, as a result, Dik, are known. Pekelman and Sen (1974) show how the weights, wij, for all i, j can be numerically derived. The parameters ykj for all k, j are generally obtained directly from the various individuals in the sample. The parameters xij for all i, j can be numerically computed using a technique described by Srinivasan and Shocker (1973). The decision variables for the firm are the attribute locations of the new product expressed by (yk1,yk2, . . ., ykn).

Optimal solutions of the problem posed by expression (14) are difficult to obtain numerically. Consequently, a practical approach is to search for an optimal or near-optimal solution by forming a grid on the attribute space (Shocker and Srinivasan, 1974). Of course, as the number of attributes increases the grid increases exponentially, thereby increasing the search costs considerably. But since the number of attributes is usually no larger than four (Pekelman and Sen, 1974) the search would be feasible. Also note that in general we would not search over the entire attribute space. Usually, technological constraints would restrict the search to a limited region. The problem can be easily extended to deal with the case when the firm wishes to introduce, on the described space, multiple brands. The gain achieved by introducing an additional brand can then be compared to the cost incurred by this action.


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