# Perspectives on Dynamic Modeling of Consumer Discrete Choice

ABSTRACT - A major objective of consumer research is to pre dict the choices of an individual consumer or a group of consumers when the conditions that influence choice change. This paper presents a unified approach to the modeling of dynamic discrete choice, which subsumes the two major approaches in the literature--econometric approach of qualitative choice and stochastic choice modeling.

##### Citation:

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Vithala R. Rao and Naufel Vilcassim (1985) ,"Perspectives on Dynamic Modeling of Consumer Discrete Choice", in SV - Historical Perspective in Consumer Research: National and International Perspectives, eds. Jagdish N. Sheth and Chin Tiong Tan, Singapore : Association for Consumer Research, Pages: 14-18.
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A major objective of consumer research is to pre dict the choices of an individual consumer or a group of consumers when the conditions that influence choice change. This paper presents a unified approach to the modeling of dynamic discrete choice, which subsumes the two major approaches in the literature--econometric approach of qualitative choice and stochastic choice modeling.

INTRODUCTION

Consumer research has essentially focused on discrete choice (or brand choice) whereas continuous choice behavior researchers using the economics approaches have focused on choice at the intensive margin. Further, consumer behavior models presented from the behavioral tradition seem to focus on cognitive processes rather than the outcomes of such processes; see Bettman (1979).

Essentially two views seem to emanate for building outcome (brand choice) models in the literature of marketing research: that the choice is essentially deterministic as discussed by Howard and Sheth (1969) and that the individual choice behavior for separate occasions is inherently stochastic as discussed by Bass (1974). The literature is abound with vast amounts of research from these two viewpoints.

The literature pertaining to the modeling of discrete choice behavior can essentially be classified under one of two approaches. These are the econometric analysis of qualitative choice [McFadden (1973)] and stochastic modeling of consumer choice [Bass (1974)]. The dynamics of discrete choice arise from: changes in parameters of the utility function; changes in the marketing mix variables; lagged effects; and shifts in preferences.

Against this background, the objective of this paper is to present a unified framework to encompass the sources of dynamics consistent with purposive behavior.

A UNIFIED FRAMEWORK FOR DYNAMIC ANALYSIS OF DISCRETE CONSUMER CHOICE

In its simplest form, a paradigm of consumer choice that has dominated current research can be described as a flow diagram shown in Figure 1. The portion of the diagram in dashed boxes deals with the process of formation of brand perceptions. Although these relationships are important to understanding consumer behavior, we will not delve into them in this paper.

Various feedback effects exist and only certain parts of the paradigm may apply to any given choice situation. We will utilize the core of this paradigm to present a unified framework to describe consumer choices of brands over time. Initially, we will develop a model to describe consumer choice at a point in time and later extend it to incorporate dynamics of choice for frequently purchased packaged consumer goods. The basic framework will, however, be applicable to choices in any category of goods and services.

Constructs

Several constructs are needed to describe dynamic choice of individual consumers. These are: product category, choice set (or evoked set), brand information, individual characteristics, situational descriptors, brand attributes, brand perceptions, brand preferences (utilities), and brand choice. Each of these is briefly discussed below.

Product Category. This construct may be defined as the total set of all items fulfilling prespecified consumer needs or wants and which has relevance to marketing planning. Given the situation of choice modeling, this set may be very broad or very narrow.

__Choice Set. __From the individual's perspective, all brands in a product category may not be deemed as alternatives for choice. Due to one reason or another, the relevant set of items would be a subset of the brands in the category. This subset will be called the choice set. The choice set may vary from one choice situation to another and is idiosyncratic to individual consumers; a term with a similar meaning is the evoked set.

Brand Information. This construct will be defined as the total information - objective as well as subjective - on the brands communicated to the consumers. Individual consumers differ with respect to the total brand information accessible to them; this variation is partly due to marketers' selective activities and partly due to the individual's selective perception. For a comprehensive discussion, see Howard and Sheth (1969).

Individual Characteristics. This construct is the set of all variables specific to an individual that would be relevant to the choice process. Examples are age, income, occupation, media exposure, and life style.

Situational __Characteristics. __This construct is an attempt to capture the aspects unique to a choice situation. Consumer research has identified the necessity to consider the characteristics of a choice situation in describing the choice process; see Belk 119751.

Various situational variables may be categorized into three broad groups on the basis of their anticipated effects on the choice process; these groups are:

(a) those variables that would restrict the choice set of an individual consumer;

(b) those variables which might enhance the utility of buying the brand; and

(c) other variables that are idiosyncratic to the buying situation such as intended use of the brand.

Brand __Perceptions. __While some of the brand information is objective, much of what consumers utilize in the choice process is subjective. For the purposes of simplicity of exposition, we will consider all of the information as perceptual; note that objective information will be perceived to be the same across consumers (or variance across consumers to be zero). These perceptions will be deemed to be stable over a period of time. In line with the practice in the literature, we will portray the brand perceptions as a multidimensional array of brand attributes (or dimensions). Further, these attributes will be assumed to be orthogonal.

Brand Preferences. Given the choice situation, the consumer will be deemed to have a preordering of preferences over the brands in the choice set. Our inability to construct a mathematical utility function does not imply the non-existence of a preference structure for the individual.

Brand Choice. This construct simply refers to the specific choice of the brand by the consumer under the specified choice situation. It is the revealed preference or overt action in the marketplace.

Assumptions

Our goal in this unified framework is to "reconcile" the two apparently distinct approaches of modeling discrete consumer choices over time. In order to accomplish this, we assume the following with regard to individual choice behavior.

(a) Any individual's brand preferences satisfy the three axioms of reflexivity, transitivity, and completeness; see Varian (1978);

(b) The behavioral assumption for the individual's choice is that he maximizes utility subject to certain constraints.

(c) In any choice situation, the individual chooses at most one item from the choice set or not choose at all.

Consumer Decision Problem: Static View

The basic behavioral assumption underlying the paradigm is that on any choice occasion k, the consumer chooses a particular brand so as to maximize utility, subject to certain constraints. The utility function of the consumer is not directly observable, but the outcomes of the choice process can be observed. Hence, there is a need to link the utility function (and through it the underlying preferences) to observable behavior.

The consumer's decision problem, in general, is one of constrained maximization. These constraints are imposed due to several factors. First, given that the consumer is faced with limited monetary resources, a budget constraint as well as a constraint on a maximum price (or reservation price or evoked price) beyond which a consumer may not be willing to buy the product.

The other constraints that are relevant, though not always observable and known to the modeler, include but are not limited to effects due to such factors as stockouts, urgency of the buying situation, and limits on expendable search effort. Various informational imperfections in the markets result in consumers searching for quality and/or prices (Stigler (1961)).

The consumer's decision problem can be formally stated using the following notation:

Let:

P = Product category.

W= Total set of brands (alternatives) in the category plus one.

J = Total number of alternatives in 0, subscripted by j = 1, 2, ..., J.

I = Number of individuals being modeled subscripted by i = 1, 2, ..., I.

K = Number of choice situations facing the consumers subscripted by k = 1, 2, ..., K

Wi = Evoked set of alternatives for the i-th consumer; Wik __c __W.

W ik = Choice set facing the i-th consumer for the k-th situation W ik __c __W.

J ik ~ Number of alternatives in the choice set of the i-th individual for the k-th choice situation.

L = Number of consumer characteristics, subscripted by I = 1, 2, ..., L.

T = Number of attributes describing brand perceptions subscripted by t = 1, 2, ..., T.

Q = Number of observable situational characteristics subscripted by q = 1, 2, ..., Q.

Zjt(i) = Score for the j-th brand on the t-th attribute specific to the i-th individual; j = 1, 2, ..., J: t = 1, 2, ..., T.

Z(i) = The JxT matrix (Z ~ specific to the i-th individual.

Zj (i)= The vector of the T scores corresponding to the j-th brand specific to the i-th individual.

Z = The JXT matrix (Zjt ) describing the common space for the group of I individuals.

Zj = The vector of the T scores corresponding to the j-th brand in the common space, Z.

Ail = Score for the i-th individual on the 1-th characteristic; i = 1, 2, ..., I; *I = 1, *2,

Ai = The lxL vector (Ai1, Ai2, ..., AiL).

Skq = Score for the k-th choice situation on the q-th situational characteristic; k = 1, 2, ... I K; q = 1, 2, ..., Q.

Sk= The lxQ vector (Sk1, Sk2, ..., SkQ).

Ri = Reservation price for the i-th individual consumer. [One may distinguish this concept from that of evoked price discussed by Rao and Gautschi (1982). Further, these reservation prices may generally be assumed to exist for each choice alternative. For the sake of simplicity and owing to the context of frequently purchased consumer goods, we will consider here only one reservation price applicable to the category as a whole.]

Pj = Price of the j-th brand; j = 1, 2, ..., J.

P = IxJ vector of brand prices, (p1, P2, ..., Pi ).

x ij = Quantity of brand j chosen by i-th individual.

m i = Expenditure on the product category for the i-th individual.

u(-) = Direct utility function.

V(-) = Indirect utility function.

We can write the decision problem for the i-th consumer under the k-th choice situation as:

(M): Maximize U ijk (3)

j e W ik

subject to:

pj < vj e W ik (4)

xij = 0, 1 V j e W ik (5)

xijxij= = 0 V j,j=(j=j=)e W ik; and

xij > 0 for some j e W ik (6)

D ijk = A dummy variable taking the value 1 if j-th representing brand is chosen by the i-th individual in k-th situation and the value 0 otherwise.

Further, additional constraints will be relevant specific to the choice situation. The solution to (M) subject to the constraints (4)-(6) will yield an indirect utility function:

V ik (p j*, Ri, Zij*) (7)

where j* is the brand chosen under the situation.

It is worth pointing out that price of a brand enters naturally the indirect utility function; further the value of the indirect utility would be dependent on the choice situation and, in general, would vary from one situation to another. Equation (7) in its general form demands a large.amount of data for the purposes of estimation for any one individual. It can become tractable by making certain approximations around a common specification for a group of relatively "homogeneous" individuals. There exist several approaches to identifying such groups of consumers; one such group would be those individuals who face the same choice set. In the sequel, we will derive tractable specifications for such homogeneous groups. We may thus write:

Vik(Pj*, Ri Zj*i) = Vi(Pj*, Ri, Sk, Zj*i) +Fj*k (8)

Where Fj*k is the error associated with this functio4a~ separation. It would capture the unobserved effects not accounted for in the Sk vector. If the model is to be at the level of an individual, the specification (8) will be adequate. However, to arrive at a model at a group level yet retaining individual effects, we may first utilize a common attribute space, Zj* in place of Zij* and include deviations from this common space in an individual/situation specific heterogeneous component. Thus, we may specify (8) as:

V(Pj*, Ri, Ai, Zj*, Sk) + nik +Fj*k (9)

where n ik is the unobserved heterogeneity specific to the i-th individual in the k-th choice situation.

To accommodate purely random effects not accounted for by our description of the choice process, we will need to include a stochastic error term,e j*k. Thus, we have

V ik (Pj*,Ri, Zj*) = V (Pj*, Ri, Ai, Zj*, Sk + nik +Fj*k + e j*k (10)

The error terms Fj*k and e j*k cannot be separately identified n the estimation process and hence will be treated as one term, x j*k. This treatment would have some implications for the distributional assumptions about this error term.

For example, the commonly used Weibull distribution in discrete choice models [McFadden (1973)] with constant variance and independent errors may not be appropriate in this framework.

Given the above simplifications, we may now write the indirect utility function, V for J-th brand for the i-th individual in the k-th situation as:

Vijk = V(Pj, Ri, Ai, Zj, Sk ) +hik +xjk (11)

The formulation (11) will enable us to focus upon changes in the __indirect utility __function with respect to changes in one or more of its arguments.

The error term, xjk specific to situation/brand can be treated a 'random. At any given situation, it is possible to assume that the error terms specific individuals, hi=s are independently distributed. For a given individual across various choice situations, the error terms xjk=s can be assumed to be independent; but the error terms i=s will be correlated over situations (or time). Depending upon the distributional assumptions of xjk=s and the structure of hik=s (e.g., fixed effect or random effects), various discrete choice models would result.

As discussed previously, for frequently purchased goods we can suppress the reservation price argument (Ri) from the V-function in equation (11). We may accordingly write the indirect utility function for such goods as:

V ijk = V(Pi, Ai, Zi, Sk ) +hik +xjk (12)

This formulation is quite general and can accommodate various utility formulations developed in the consumer behavior literature including the preference models in multidimensional scaling and conjoint analysis.

The complete model is:

V ijk = V ijk + hik +xj*k. (13)

In a static context, the choice model for a group of consumers can be stated as follows:

Dij*k= 1 if and only if Vij*k = max {V ijk; yeWik }. (14)

The decision rule (14) combined with distributional assumptions for hik and xj*k will lead to probabilistic choice brands in a given situation for an individual.

Dynamics of the Model

We have so far treated the consumer decision problem for one individual in a given situation. Given our enumeration of the factors describing a situation, it is quite obvious that dynamics of choice are captured by different situations. If one choice situation has nothing to do with a subsequent choice situation, the model described by equations (13) and (14) can be reapplied to the second situation by utilizing different sets of impact coefficients and other descriptor variables. This approach to modeling is akin to treating the choice process as a zero order process and it does not enable us to take advantage of the model structure.

__Sources of __Dynamics. Changes in the choice outcomes from one situation (time) to another (subsequent time) can arise due to various factors. Essentially, the effects of these factors will be to alter the value of the V(-) function from one situation to another. The sources of these changes may be listed as follows:

(i) Characteristics of a choice situation may differ; these are evinced by differing choice sets Wik=s or changes in the scores of the different situational descriptors, __S__k

(ii) The perceptions of brands on various attributes may change; i.e., the Z-matrix may vary from one choice situation to another. These changes may in part be attributable to marketing mix variables.

(iii) The value function may change due to the actual choice made in the previous occasions; how many previous occasions need to be included is essentially an empirical question. (This source is called the state dependence or feedback.)

(iv) The value function is continuously modified or updated by the individual as situations change. Some effects of this change may be mitigated by the persistence of habit on the part of' the individual. One way to model this is assuming that the consumer systematically updates the value functions from the prior occasions (or prior propensities of choosing various brands) in computing that for the current situation using some type of moving average. (This effect is different from that of the source (iii) identified earlier.)

(v) The impact coefficients themselves differ from one situation to another. Note that the model builder has no opportunity to come up with a common model if this source were to dominate the dynamic choices.

The equation (13) can be modified to capture the effects of the sources T *through (iv)*. The result will be the model [The reader may recognize the similarity of this development to that of the stochastic models for discrete panel data described in Heckman (1981); the basis of Heckman's development is more statistical, however.]:

where {v1, ..., vh1} and (q1, ..., qh2} are parameters and h I and h 2 are the periods of lag. The effects due to W and (ii) above are easily accounted for by the first term, Vijk in (15) which is computed for the current situation (or time period). The second term captures the effect of state dependence while the third term accounts for the updating effects of prior propensities of brand choice on the part of the individual. Given the importance of the construct, we will elaborate on state dependence.

__State Dependence. __One assumption in our unified framework of dynamic choice behavior is that memory in the system has a feedback in altering the value function and thereby influencing future brand choice behavior.

If past experience appears to influence future behavior, two distinct explanations can be given for such empirical evidence. The first is that as a consequence of experiencing an event (e.g., buying a particular brand), preferences governing choice are altered. In such a situation, past experience has a true behavioral effect in that, an otherwise identical individual not exposed to the event would behave differently in the future. Such dependence is structural in nature and can be referred to as true state dependence (Heckman, 1981).

A second explanation for such dependence is that individuals inherently differ in their propensities to experience an event and if these individual differences are correlated over time, then previous experience may appear to influence behavior purely because it is an instrument for temporally persistent unobservables that influence choice. If this unobserved heterogeneity is not properly controlled for, then a spurious relationship between the past and future behavior may be observed.

The probabilities of brand choice can be computed using equation (15) under various assumptions for the distribution of errors. Three selected cases, identified as A, B, and C will be described.

All of these cases A, B, and C assume that the individual-specific effects are fixed and the same for any situation and vary according to the distributional assumptions for the brand specific effects; in cases A and B the brand specific effects are assumed to be independent across both brands and situations and lead to the familiar conditional logit model [Chamberlain (1980)] for the logistic distribution and independent probit model [Goldberger (1964)] for the normal distribution. These two models are subject to the IIA assumption. The case C assumes that the brand specific effects are correlated across brands for any situation but are independent across situations; the resulting model here for the normal distribution is called the covariance probit [Hausman and Wise (1978)] and it is one way of dealing with the IIA assumption. Appropriate models are yet to be developed for the case of random individual-specific effects.

Conclusion

We have developed in this paper a comprehensive model for analyzing the discrete consumer choices over time. The two existing approaches for analyzing dynamic discrete choices -- stochastic modeling of buyer behavior and probabilistic models for discrete choice -- are special cases of our model. In a longer version of this paper [Rao and Vilcassim (1985)] we have shown these connections and have also discussed various research issues on this modeling approach.

REFERENCES

Bass, Frank M. (1974), "The Theory of Stochastic Preference and Brand Switching," Journal of __Marketing __Research, 11 (February), 1-20.

Bass, Frank M. et al. (1984), "An Investigation into the Order of the Brand Choice Process," __Marketing __Science, 3 (Fall), 267-87.

Belk, Russell W. (1975), "Situational Variables in Consumer Behavior," __Journal of Consumer Research, __2 (December), 157-64.

Bettman, James R. (1979), An __Information Processing Theory of Consumer Choice, __Reading, Massachusetts: Addison-Wesley.

Chamberlain, Gary (1980), "Analysis of Covariance with Qualitative __Data," Review of __Economic Studies, 47 (January), 225-38.

Goldberger, A.S. (1964), __Econometric Theory, __New York: Wiley.

Hausman, J.A. and D.A. Wise (1978), "A Conditional Probit Model for Qualitative Choice:

Discrete Decisions Recognizing Interdependence and Heterogeneous Preferences," __Econometrica, __46 ( ), 430-26.

Heckman, James J. (1981), "Stochastic Models for Discrete Panel Data," in Charles F. Manski and Daniel McFadden (eds.), Structural __Analysis of Discrete Data with Econometric Applications, __Cambridge, Massachusetts: The MIT Press, 114-78.

Howard, John A. and Jagdish N. Sheth (1969), __The Theory of Buyer Behavior, __New York: Wiley.

McFadden, Daniel (1973), "Conditional Logit Analysis of Qualitative Choice Behavior," in P. Zarembka (ed.), __Frontiers in Econometrics, __New York: Academic Press

Rao, Vithala R. and Naufel Vilcassim (1985), "Perspectives on Dynamic Modeling of Consumer Discrete Choice," Working Paper, Johnson Graduate School of Management, Cornell University (July).

Rao, Vithala R. and David A. Gautschi (1982), "The Role of Price in Individual Utility Judgments: Development and Empirical Validation of Alternative Models," Choice Models __for Buyer Behavior, __Greenwich, Connecticut: JAI Press.

Stigler, George J. (1961), "The Economics of Information," Journal of Political Economy, 49 (June), 213-25.

Varian, Hal R. (1978), Microeconomic __Analysis, __New York: Norton.

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##### Authors

Vithala R. Rao, Cornell University

Naufel Vilcassim, Cornell University

##### Volume

SV - Historical Perspective in Consumer Research: National and International Perspectives | 1985

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