Information Processing Analysis of a Stochastic Model of Brand Choice

Luis Dominguez, Case Western Reserve University

Richard W. Olshavsky, Indiana University

A formal and detailed comparison between Morrison’s brand loyal model and its information processing counterpart reveals serious internal inconsistencies and other shortcomings of this stochastic model.

Morrison’s [14, 151 brand loyal model (hereafter BL) is among the simplest and most widely applicable stochastic brand choice models. Massy, Montgomery, and Morrison found that BL represented more closely the brand choice processes of a Chicago Tribune panel than a Bernoulli model and a last purchase model [ 13, pp. 135-1361. More recently, Blattberg and Sen [3, 41 relaxed the assumption that all consumers -must conform to the same particular type of model and expanded BL from a two-state to a five-brand specification that included two national brands and three private labels. They found that of the nine market segments that resulted for three major categories of frequently purchased grocery items, four could be described as BL, accounting for 52% of the purchase histories in the panel.

The purpose of this note is to reveal some serious internal in- consistencies and shortcomings of BL and by implication, of the contingent evaluation process that underlies not only BL but the stochastic modeling approach. This shall be done by formally comparing BL in its simplest, two-state formulation, with its information-proc- essing-model counterpart. This comparison and the resulting arguments parallel somewhat those of Gregg and Simon [8].

Morrison’s Brand Loyal Model

The brand switching matrix is

1 0

1 P 1 -P

0 kP 1 -kp .

Address correspondence to Richard W. Olshavsky, Marketing Department, Graduate School of Business, Indiana University, Bloomington, IN 4 7405.

JOURNAL OFBUSINESS RESEARCH 9,39-48 (1981) 0 Elsevier North Holland, Inc., 1981 39 52 Vanderbilt Ave., New York, NY 10017 0148-2963/81/01039-10$2.50

40 Luis Dominguez and Richard W. Olshavsky

Brand 1 is the subject’s preferred brand. Brand 0 represents all other brands; p is the probability that the subject will repeat purchase brand 1. BL is stationary in that neither k nor p varies over time. BL is heterogeneous because p is allowed to vary across subjects according to the parameters of the beta distribution.

Learning occurs in the model through the parameter k. Morrison [ 151 explains k thus:

The past purchase decision has less influence the higher the value of k. In other words, if the brand loyal model gives a good fit to the data, the higher the value of k, the more “Bemoullian” the universe of consumers.

If k = 0, brand 0 becomes the absorbing state immediately after its first purchase. So long as k > 0, the subject will not adopt brand 0 exclusively. As long as k < 1, learning will occur in the sense of changed purchase probabilities.

Therefore, BL involves two behavior mechanisms:

1. p represents the subject’s propensity to remain with brand 1. It is a response operator.

2. k represents a learning operator in the sense of changed response tendency following purchase of brand 0. If k = 1, past consumer purchase does not influence the current response propensity.

Choice Process Description of Morrison’s BL

In a very aggregative sense, BL can be regarded as a statement of an information processing model. Consider the following description that Morrison [ 141 has provided of this model:

The brand loyal model says that an individual with a high probability of remaining with brand 1 will also have a higher probability of leaving brand 0 to buy brand 1 than an individual with a low probability of remaining with brand 1. High loyalty (ifir exists in an individual) is directed toward a particular brand. Therefore, this model is classified as the brand loyal model.

These and similar statements [ 13, 141 and the basic form of the model (i.e., first-order Markov) suggest that the brand choice process de- scribed by BL is some sort of contingent evaluation process. A particular brand is preferred because it satisfies a subject’s evaluative criteria. Given that a preferred brand exists, the consumer will tend to reject other brands. The assumption of a first-order Markov process is used to depict consumers as evaluating brands with limited memory of past purchases.

Our objective here is not to formulate a model of how consumers choose among brands in an actual setting but rather to offer a very simple

A Stochastic Mode2 of Brand Choice 41

but plausible information processing model of the kind of behavior implied by Morrison’s description of the behavioral process that underlies BL. For instance, it suffices to assume that the consumer randomly selects brands for evaluation from a supermarket shelf that contains N available brands. Further, assume that the consumer tests the selected brand against a set of evaluative criteria. The brand will be purchased if it passes an initial screening test done on criteria observable in-store, such as price, color, size, and the like. Otherwise, a different alternative is selected (sampling without replacement of screened brands). After a brand is purchased it is evaluated against criteria observable in using the product. Brands that pass postpurchase evaluation criteria are considered again; those that fail are not considered on the next purchase occasion.

This model, call if P,, is restated operationally in Figure 1. El-E3 represent the external environment. S 1 defines the desired brand attributes and S2 is the subject’s representation of the external en- vironment. S3-S are the information processing steps that describe the individual’s brand choice process.

Comparing P, and BL

Table 1 shows all inputs required to generate predictions from P,. The brand attributes have been scored on a scale from one to five. The subject’s desired attributes make brands 1, 2, and 3 acceptable during in-store screening. Of these, only brand 1 is acceptable after purchase. Because of its favorable allotment of shelf space, brand 1 will be most frequently selected (as shown below, this condition is not essential for generating brand loyal behavior).

A detailed description of the data generated by P,, for a randomly selected subject is presented in Table 2. This model is programrned in Fortran. Brand 1 predominates. Every purchase of brand 2 or 3 leads to its postpurchase rejection and removal from consideration the next period. This heightens the chance of buying brand 1, thereby directing the subject’s behavior to the preferred brand as suggested by Morrison.

A detailed comparison of P, and BL reveals several serious internal inconsistencies in BL. We shall mention two. The first inconsistency regards the assumption that 0 < k < 1 and hence that @ < p. This has the curious effect that the probability of returning to brand 1 following purchase of brand 0 is less than the probability of remaining with brand 1. In other words, purchase of brand 0 directs the subject away from rather than toward the preferred brand, a peculiar form of brand loyalty to say the least. By contrast, P, implies that k 1 1. Then a brand rejected

42 Luis Dominguez and Richard W. Olshavsky

FIGURE I: Flow Diagram of the Process Model.

Table 1: Sample Input Data to PO

Item

Brand Attributes

Observable Observable

in Store after purchase 12 3 4 1 2 3 4

No. of brand shelf facings

Brand 1 44 4 4 4 4 4 4 6 Brand 2 44 4 4 3 3 3 3 2

Brand 3 44 4 4 2 2 2 2 2

Brand 4 33 3 3 4 4 4 4 3

Brand 5 22 2 2 2 2 2 2 2 Subjects’ desired

brand attributes 44 4 4 4 4 4 4

Tab

le

2:

Sam

ple

Out

put

Dat

a of

PO

Peri

od

1 2 3 4 5 6 I 8 9 10

Bra

nd r

ejec

ted

in

Bra

nds

sele

cted

pr

evio

us p

erio

d fr

om s

helf

- 1

_ 2

2 1

_ 5,

432

2 3

3 59

1 _

2 2

5,l

- 49

1 _

1

&

Bra

nds

scre

ened

A

ctua

l 03

1 in

sto

re

bran

d ch

oice

br

and

choi

ce

3 0.

2 -

1 1

- 2

0 -

1 1

5,4

2 0

- 3

0 5

1 1

_ 2

0 5

I 1

4 1

1 -

1 1

44 his Dominguez and Richard W. Olshavsky

after purchase would not be considered the next period. The probability of choosing brand j equals its allotment of shelf space (nj) relative to brands acceptable during in-store evaluation. Therefore, for the input data in Table 1, P, results in the brand switching matrix:

1 2 3

Q/N

n,lN-n, ,

0 I

which results in

1 2

1 .60 .20

2 .75 .oo

3 .75 .25

This can be collapsed to

1 0

3

.20

.25 ,

.oo I

1 .60 .40

0 [ 1 .75 .25 .

This implies that k = 1.25. Our formalization of the process model that preceded BL suggests that BL generates predictions inconsistent with the basic process model upon which it was based. Further, note that when k = 0, brand 0 becomes the absorbing state, contrary to the specification that brand 1 is the preferred brand. Therefore, the behavioral process that underlies BL involves some process other than that on which BL is claimed to be based.

The second inconsistency is that in general BL is nonstationary when the probabilities of returning to brand 1 are unequal. Under P, the probability of returning to brand 1 will fluctuate if nonpreferred brands are allocated unequal shelf space. Suppose that brands 2 and 3 were allocated three and one shelf spaces, respectively. Then the brand

A Stochastic Model of Brand Choice

switching matrix is

45

Hence the probability of returning to brand 1 is either .86 or .67 but never a constant like .75. When brands 2 and 3 am collapsed, stationarity occurs simply because of an inadmissible aggregration.

A more important comparison of BL and PO is that while the stochastic model can be derived from P,, the converse is not true. This is demonstrated simply enough. Brand loyal behavior can be generated by any number of variants of P,. A simple variant P , would assume that the subject samples from the list of acceptable brands, disregarding the allotment of shelf space among brands. .Brand 1 would continue to be the preferred brand, but it would tend to be purchased less frequently. Another variant P, representing a less strict brand evaluation process would assume that brand scores are required only to be within one point of the subject’s desired attributes. This would thereby make brands 2 and 4 fully acceptable. However, brand 1 would continue to claim the most purchases owing to its greater number of shelf facings. Notice, however, that if the features of P , and P, are combined, the subject will gravitate equally toward three fully acceptable brands ( 1,2, and 4) and the model will not exhibit brand loyal behavior in Morrison’s sense that the choice process is directed toward a particular brand.

To distinguish among PO, PI, and P2 would be impossible except by collecting detailed data on the marketing environment and on the decision processes of consumers. Thus the rejection of brand 4 would indicate that PO rather than P, is at work. The relative frequency with which brands are selected would indicate whether shelf space influences brand choice and hence whether P, or P, is at work.

The crucial point is that validation of a stochastic model affords no basis for generalization. In our example, a high degree of brand loyalty occurs whenever brand differences are so great in relation to the subject’s evaluation processes as to render only very few brands fully acceptable. For example, if the subject applied an exact conjunctive test to a large number of evaluative criteria, it would be comparatively more difficult for changes in the market environment to make more brands

46 Luis Dominguez and Rich&-d W. Olshavsky

acceptable than if lax criteria were applied to just a few attributes. Therefore tests of stochastic models are necessarily situation-bound; the behavior obtained results from the interaction of market and subject characteristics that will be unknown and unspecified unless decision process data are also analyzed.

Additional Comparisons

An additional comparison concerns the different ways in which stochastic features enter either model. In the process model, the behavioral mechanisms were fully specified. Randomness entered the model by assuming random search of the subjective shelf. All search and evaluation processes were fully defined. In the stochastic model, however, no effort is made to specify what is deterministic and what is stochastic about behavior. As a consequence, stochastic models furnish no basis for stating laws and regularities about consumer behavior. A stochastic model’s parameter estimates and goodness of fit are rooted in unspecified combinations of market variables and consumer charac- teristics. When these change, so does the model and its efficacy.

Finally, this analysis has demonstrated that the argument that stochastic models are more parsimonious is incorrect. For if the information processing model that underlies BL were spelled out at the minimal level of detail necessary to be operational, it would be no less lengthy than P,. Furthermore, we have demonstrated that information processing models such as P, are capable of generating parsimonious stochastic models whose behavioral underpinnings are unambiguous and that, of course, can lay claim to the same predictive accuracy that stochastic models do. Our formalization of P, indicates that 2, not 1, is the correct brand loyal model implied by the behavioral processes wielded in support of BL.

Conclusion

For over two decades since Bush and Mosteller’s [5] work and Kuehn’s [lo] seminal dissertation, a substantial portion of the literature on consumer behavior has centered on the stochastic modeling of brand choice processes. It has been argued that stochastic models are more predictive than other, more comprehensive modeling approaches [2]. Comprehensive simulations [ 1] and econometric models [7] are said to suffer not only from poor predictive ability but from a host of methodological difficulties [6, 9, 121. In a nutshell their problem is the high degree of a priorism required to formulate detailed hypotheses

A Stochastic Model of Brand Choice 47

about the perceptual and learning mechanisms that lead to brand choice behavior. Our knowledge of consumer behavior has not advanced to the point where such detailed processes can be prespecified with confidence WI.

The upshot is a paradoxical state in which stochastic models are advanced as feasible and predictive but are unsuited to generating the kind of detailed knowledge necessary to advance explanation. Fur- thermore, since the justifications for the formulation of stochastic models “are to be found in the model builder’s knowledge of the behavioral processes in question [ 141,” brand choice modeling seems to be caught irretrievably in a “chicken, egg” situation.

The formal comparison performed here between Morrison’s BL and its information processing counterpart raises two important points. First, it strongly suggests that the information processing approach, which is rooted in Newell and Simon’s studies of human cognition [16], may be the more viable approach to the study of brand choice. For not only does it afford a better explanation. of behavior but it is also capable of generating internally consistent models, as was illustrated in the preceding section when we translated P, into a stochastic model. Furthermore, the information processing approach is capable of specifying the conditions under which a stochastic model’s parameters will change and the conditions under which the model ceases to be an adequate portrait of the choice process in question.

The second point raised, albeit only indirectly, concerns the different manner in which information processing models and stochastic brand choice models are developed. Since the starting point of modeling information processing is a detailed analysis of only a few subjects, the question arises as to whether the resulting models can be used to generate predictions at the aggregate or group level. The answer is that the extent of generalizability depends on the degree of variability in choice strategies across consumers that exists in any particular brand choice environment. If choice strategies are highly idiosyncratic, then pre- dictions at the aggregate level will suffer. However, recent empirical studies of individual processes (e.g., [ 11, 171) strongly suggest that only a few basic types of choice strategies are used (e.g., lexicographic, conjunctive, disjunctive), and that the choice of strategy responds to the character of the task environment (e.g., the number of brands and the number and type of product characteristics present). Employing ap- propriate estimates of the relative proportions of consumers using each type of strategy, predictions at the aggregate level can be generated through computer simulation of each choice strategy. This is in many ways akin to the stochastic modeling practice of segmenting consumers

48 Luis Dominguez and Richard W. Olshavsky

according to the brand switching models that best describe their brand choice behaviors [3,4].

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