Mathematical models of brand choice behavior

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH

ELSEVIER

European Journal of Operational Research 82 (1995) 1-17

Invited Review

Mathematical models of brand choice behavior Ajay K. Manrai College of Business and Economics, Universityof Delaware, Newark, DE 19716-2710, USA Received August 1994

Abstract

A critical review of recent developments in the single-stage and two-stage models of brand choice behavior is provided. The rich and complex history of development of choice models is organized via the two fundamental principles that drive most of the literature on single-stage choice models. The first principle of utility maximization has its roots in economic theory and it led to models such as the independent multinomial logit model (McFadden, 1976). The other principle of attribute-based sequential elimination has originated in psychology. Tversky's (1972) Elimination-By-Aspects model is a prime example of the models based on the second principle. The recent two-stage models of brand choice behavior (Gensch, 1987; Manrai and Andrews, 1994) use an attribute-based sequential elimination principle in the first stage to obtain a smaller consideration set from a full feasible set of brands and a utility maximizing logit model in the second stage to select a single brand from the reduced consideration set. We also provide conclusions and directions for future research.

Keywords: Choice theory; Choice models; Utility maximization; Attribute-based elimination; Random utility models; Marketing models; Perceptual mapping 1. Introduction

T h e goal of this p a p e r is to provide a review of recent developments in the field of mathematical modeling of brand choice behavior. These models represent the underlying process by which an individual consumer integrates information t o select a brand from a set of competing brands. The choice models have b e e n developed with varying assumptions and purposes and they differ in the underlying logic structure that drives them. There are three broad categories of these models, namely, (i) multi-attribute choice models (Manrai and Sinha, 1989; Roberts and Urban, 1988), (ii) preference a n d choice m a p p i n g models (DeSarbo et al., 1994), and (iii) conjoint analysis ( G r e e n and Srinivasan, 1990). The multi-attribute choice

models have assumed an increasingly important role in marketing applications. M a r k e t researchers have used these models for determination of market structure, d e m a n d forecasting, product positioning and buyer segmentation, and prediction of consumer choice (DeSarbo et al., 1993; Eliashberg and Manrai, 1992; G r e e n and Krieger, 1989). A possible reason for the wide use of the multi-attribute choice models is their operationalization based on the type of data that is readily available in marketing, particularly when dealing with large numbers of brands to be evaluated on m a n y attributes. In similar situations, the conjoint analysis runs into practical problems because of its requirement of experimental design to gather consumer preferences and its difficulty in obtaining sufficient data per consumer for esti-

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A.ICL Manrai/European Journal of Operational Research 82 (1995) 1-17

mation of t h e utilities or part worths. The multidimensional mapping approach works on the basis of paired comparison data and the data requirements from consumers increase rapidly as the number of brands in the choice set increases. Early literature in the field of marketing (prior to the mid-seventies) on multi-attribute choice models is largely driven by empirical results based on the additive compensatory models proposed by Bass and Talarzyk (1972). Fishbein and Ajzen (1975) extended this work by proposing a l i n e a r additive model of behavioral intentions involving a second component driven by the n o r m a t i v e influences, Ni, due to significant referent others, in addition to the consumer's own attitude towards a brand, A b. Bagozzi (1982) suggested a correlational driven structural equation methodology to estimate such models. LaTour and Manrai (1989) applied a M A N C O V A approach to data obtained from two field experiments to show that the two components of the Fishbein and Ajzen (1975) behavioral intention model, i.e., A b and Ni, may interact to influence consumer choice behavior. In this paper we are seeking to provide a critical review of the multi-attribute choice models which are introduced since the mid-seventies. There are two fundamental ways of classifying the multi-attribute choice models. The dichotomous classification is driven by two different principles, namely, (i) the principle of utility maximization founded in economic theory, and (ii) the psychological principle of feature- or attributebased sequential elimination or "attribute-based processing". The difference lies in the assumptions about the way consumers process information. T h e principle of utility maximization postul a t e s that a consumer uses all relevant available information and selects the brand t h a t maximizes h i s / h e r utility. Here, the basic choice process assumes that all of the attributes are considered in a simultaneous compensatory structure, thus assigning a utility value to each brand. After that, th e brands are compared and the brand with the highest total utility is selected. This is also called the "brand-based processing" approach. The independent multinomial logit ( M N L ) m o d e l

(McFadden, 1976) is an example of the models driven by the principle of utility maximization. Luce's (1959) model is also a brand-based processing model which is driven by the response potency measure (RPM) of a brand. The R P M is a utility-like measure and under certain assumptions, discussed in Section 2.1, the Luce model reduces to a random utility model. If RPM(i) is the response potency measure of a brand i among a set S = {1,2 . . . . . N} brands, then the probability o f choice of brand i as per the Luce model is given by:

P(i/s)=

RPM(i) ~RPM(j)'

forj~s.

(1)

J The principle of "attribute-based processing", on the other hand, suggests that a consumer makes a selection through a simplified heuristic and may not use all the relevant information available at the time of choice. Here, the choice is made by comparing brands on attribute-by-attribute basis. The models driven by this principle generally assume that there is a random or hierarchical sequence in which the attributes are considered, Elffnination-By-Aspects (Tversky, 1972) is a prime example of the models driven by the principle iof "attribute-based processing". The brand choice models that are prevalent in marketing usually assume that consumers consider a set of brands in a deterministic flamework, i.e., the consumers have the needed knowledge: about the relevant brand characteristics to select a brand. However, this assumption may not hold true when consumers face new innovative brands in the market place, as is the case in the automobile market, computer market, telecommunication market, and many other markets. Thus, w h e n the consumers are uncertain about the values of characteristics of the new innovative brands, it may not be appropriate to treat the set of brands in a deterministic framework. Under such circumstances, it may be more appropriate to treat the set of brands in a probabilistic flamework, in which the consumer may assign probabilities to various possible values of characteristics of brands. The work of yon Neumann and Mor-

A.K. Manrai /European Journal of Operational Research 82 (1995) 1:17

genstern ( 1 9 4 7 ) a n d its extensions by Hauser (1978) and many others provide a rigorous theoretical framework to handle such situations in which a consumer views a set of brands probabilistically. In this approach, one maximizes the expected utility. A formal mathematical structure of the von N e u m a n n - M o r g e n s t e r n model is given as follows: Maximize f u ( W S ) f ( w ) dw, subject to p'S <<.y, where u is the utility; W = ( W n , . . . W~zu), and any wij is a random variable indicating the values of the i t h attribute for the j t h brand; for each wii, the existence of a continuous probability distribution function F,.j, with corresponding probability density function fii is assumed.; S' = (Sa,... ,SN), a vector of N brands measured in completely divisible quantities, each brand b e i n g character-

3

ized by K attributes which are also measured in completely divisible quantities; f is the joint probability density function corresponding to F, the joint probability distribution function of the random variables in W = ( w n . . . . ,WKN); q¢= ~(~[rij , for i = 1. . . . ,K, and j = 1. . . . . N, and ~i] ~ [0,~) is a compact set such that F/i(gtij) = 1; here stands for the Cartesian product; p ' = (Pl, .... PN), the price vector; and y is the budget or income. The solution to the above optimization problem results in a choice brindle of quantities of various brands: (S~ . . . . ,S~). The above approach has received some attention in the marketing literature, b o t h via empirical applications as well as theoretical research on the calibration of utility functions in probabilistic choice situations, e,g., Eliashberg (1980), Eliashberg and Hauser (1985), Hauser (1978), and Roberts and Urban (1988). There is some empirical evidence which suggests that the axiomatic

Table 1 Summary of mathematical models of choice behavior Single-stage models based on the principle of

Two-stage models

Utility maximization

Sequential attribute-based elimination

Combine the Two Principles

Brand-based processing models (see Section 2)

Attribute-based processing models (see Section 3)

Stage 1: Attribute-based processing Stage 2: Brand-based processing (se e Section 4)

1. Elimination-By-Aspects 1. Generalized Logit Models (GLM) (EBA) model (Tversky, 1972) (Dalai and Klein, 1988) 2. Generalized Multinomial Probit 2. EBA-like Models Elimination-By-Dimensions (EBD) (MNP) model (Currim 1982) 3. Generalized Extreme Value (Genscb and Ghose, 1992) (GEV) model and Nested MNL Model - Elimination-By-Cutoffs (EBC) (Manrai and Sinha, 1989) (McFadden, 1981) 4. Extensions of Multinomial Logit EBA with price (Rotondo, 1986) (MNL) model (McFadden, 1976) - PRETREE - Gaudry and Dagenais (1979) (DOGIT) Meyer and Engle (1982) and others (Tversky and Sattah, 1979) 5. Multiplicative Competitive 3. Others Interaction (MCI) models A review of several non-spatial tree models (Cooper and Nakanishi, 1988) (DeSarbo et al., 1993) - Maximum-Likelihood-Hierarch (MLH) Model (Gensch and Svestka, 1984) -

-

-

-

1. Consideration-By-Aspects (CBA) hybrid model (Andrews and Manrai, 1994) 2. Dynamic heuristic model (Siddarth et al., 1993) 3. Cost-benefit models (Andrews and Srinivasan, 1995; Roberts mad Lattin 1991; and others) 4. Promotion screening model (Fader and McAlister, 1990) 5. MLH and LOGIT combined (Gensch, t987)

Note: Only the non-IIA single-stage choice models are summarized in the table. The MNL Model (McFadden, 1976), Independent Probit Model (IPM) (Currim, 1982), and Luce's Model (Luce, 1959) are IIA-single-stage models Which are also described in the paper.

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conditions necessary for the existence of the expected utility theory are not fulfilled in certain situations. Kahneman and Tversky (1979) have proposed prospect theory to explain uncertainty effects besides other effects observed in empirical studies of choice behavior. The fundamental difference between expected utility theory and prospect theory is that in the former values are assigned to final profiles which are weighted by probabilities, while the later gives asymmetric values to gains and losses and substitutes the "probabilities" by "decision weights". Tversky and Kahneman ( 1 9 9 1 ) h a v e deve!oped a multi-attribute generalization of the prospect theory's value function in a reference-dependent model. Based on the idea of value function, Hardie et al. (1993) propose a model in which the brand choice is influenced by the position of brands relative to multi-attribute reference points, and consumers weigh losses from a reference point more than equivalent sized gains. The probabilistic models of brand choice behavior that are reviewed in this paper work with a deterministic set of brands, i.e., the consumers are assumed to have no uncertainty regarding various relevant characteristics of brands to select a brand. The paper is organized around the two principles discussed earlier, i.e., brand-based processing and attribute-based processing (see Table 1 for a quick and simple summary). In the next section, we discuss the developments relating to mathematical models of brand choice behavior which are founded on the economic principle of utility maximization or "brand-based processing" models. In Section 3, we focus on attribute-based sequential elimination or "attribute-based processing" models. In Section 4, we provide a discussion of some of the recent developments dealing with two-stage models of brand choice behavior and finally in Section 5, we provide conclusions and directions for future research. The relatively new two-stage models use both of the above stated princ!ples. In the first stage, these models typically use an "attribute-based processing, strategy to reduce a large set of brands to a smaller consideration set and in the second stage a utility maximization model, such as multinomial logit, is employed to compute the probability of

selection of a brand from the consideration set (Gensch, 1987; Manrai and Andrews, 1994).

2. Choice models driven by the economic principle of utility maximization The models of brand choice behavior in this class are also called "brand-based processing" models. Most economists apply utility theory to model brand choice behavior. The common assumption of these models is that a brand is a bundle of multiple attributes relevant to the choice process. In most cases the utility function is a linear compensatory model of attributes of the brand. Furthermore, as stated before, consumers are assumed to consider the set of brands in a deterministic framework, i.e., the consumers have no uncertainty regarding various relevant characteristics of brands to select a brand. The information processing paradigm that is prevalent in marketing suggests that a consumer uses a process of abstraction to reduce a large number of physical features/attributes, such as, price, shape, color and looks, gas mileage, performance, maintenance, electronic sensors, air bags, anti-lock brakes, country of origin, advertising etc. of an automobile, to a few perceptual attribute dimensions, e.g., economy, style, reliability, and safety. The consumer then integrates the reduced set of perceptual attribute dimensions to form a preference which influences the selection or choice of a single preferred brand. Thus, in this class of models, competing brands are represented in terms of one or more perceptual attribute dimensions believed relevant to the choice process. Scaled preference or choice data are used to derive attribute weights at the individual or aggregate level. A combination of these derived attribute weights and attribute values for various competing brands results in utility values for those brands. These utilities are transformed to choice probabilities via Luce's (1959) choice axiom and probabilistic choice models, e.g., the independent multinomial logit (MNL) model (McFadden, 1976) or independent probit models (Currim, 1982). Next, we discuss the development of the Luce type, independent multinomial logit

A.K. Manrai /European Journal of Operational Research 82 (1995) 1-17

model in the random utility framework. The Luce model and the M N L model have met increasing amounts of criticism because of the independence from irrelevant alternatives (IIA) property. We will briefly discuss the same before our discussion of non-IIA models of brand choice, which do not suffer from the curse of the IIA property. 2.1. Independent random utility models of brand choice behavior

5

i.e., the utility used in choice evaluation is the compensatory preference scale value, Vii, plus error e i. The above model structure, under the assumptions that a consumer selects a brand with highest utility and the errors e / a r e independently distributed with the type I extreme value distributions or the Weibull distribution, the probability of choosing brand i from the set S is: P ( i / S ) = Prob(U/>~ Uj, j" ~ S and j 4: i).

(4)

for P(e i <~e) = exp( - exp( - e)); it can be shown In the multi-attribute utility approach, the competing brands are usually depicted by their location in a multidimensional space whose axes are related to the attributes of those brands. In contrast to economists' use of a space spanned by physical attributes, marketers (Eliashberg and Manrai, 1992; Manrai and Manrai, 1994) have typically employed a perceptual space of reduced dimensionality. There are a variety of procedures available for determining the perceptual attribute space. The most commonly used approaches are multidimensional scaling of proximity type data (DeSarbo and Martrai, 1992; DeSarbo et al., 1994) or factor analysis of attribute ratings (see for example, Manrai and Manrai, 1993). We define a space spanned by perceptual attributes, as folows: Let S = {1,2,... ,N} be a set of competing brands, with brand i having coordinates X i = (Xn,Xi2 . . . . . XiK) in the K-dimensional perceptual attributes space. A linear compensatory preference model would suggest that utility V/ is a weighted additive function of attribute levels: Vi = ~ l X i l + ~2xi2 + . . .

+~KXiK,

(2)

i.e., V/= ~ ' X i, where ~' = (/31,/32. . . . . /3K) is a vec L tor of attribute importance weights. The V's denote preference scale values or strict utilities which summarize the attractiveness of competing brands. McFadden (1976) provides us with a model of choice, independent multinomial logit (MNL), in the random utility framework that uses the compensatory model such as given in (2), with an additively separable linear form. The model assumes that U/= V/+ e i

and

V/= ~'Xi,

(3)

e Vi

P ( i / S ) = EeV~ J

e t3'xi

EetJ,xj,

j~S.

(5)

J

This model is easy to estimate and it may be the simplicity of the model that is the reason for its many diverse applications in the marketing field (see for example, Malhotra, 1984). Incidently, when the random variables in Eq. (3) are independent, normally distributed, and have equal variance, we get an independent probit model (IPM) (Currim, 1982). Under similar assumptions, the economists have referred to the Luce model, also as a random utility model. Note that in the MNL model given in Eq. (5), the preference scale values of a brand depends solely on the attributes of that brand and not on the competing brands. In other words, Luce's (1959) independence from irrelevant alternatives, the IIA axiom, holds, i.e., P { i / S } / P { j / S } is a constant for all choice sets S such that i,j ~ S. A link between the IIA axiom and the random utility model in Eq. (4) is also demonstrated in the literature. The main advantage of the IIA axiom is in economies in data processing and computation by permitting a researcher to analyze samples of brands from a large set of competing brands. On the negative side, it implies a uniform pattern of response to changes in the attributes of one brand which are inconsistent with heterogeneous patterns of similarities often found in the marketing context (Currim, 1982). The IIA property of these models, " I I A models" hereafter, makes them context independent, i.e., they ignore the effect of similarities among competing brands on the probability of choice. This

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may not be a realistic assumption of consumer behavior in several marketing contexts (Green and Srinivasan, 1990). Manrai and Sinha (1989) provide an intuitively appealing and simple example to demonstrate that the MNL model fails to consider similarity or substitutability between different brands. This failure of the MNL model due to its IIA property is not only counter-intuitive but it has been challenged on empirical grounds by several psychologists (see for example, Tversky, 1972). 2.2. Non-IIA models of brand choice

Researchers in various disciplines, including economics, marketing, psychology, and transportation management have proposed a variety of non-IIA models of brand choice to accommodate brand interdependence. The common goal of the non-IIA models is to circumvent the problems due to the IIA property of the MNL model. This goal has often been achieved at some cost in computational complexity. McFadden (1976, 1981) provides an excellent description of research seeking variants of the MNL model form to capitalize on its computational advantage while permitting a more flexible pattern of substitutions. The non-IIA models typically replace the assumption of independence among errors of the MNL model and allow a more general pattern of correlations in the error structure. 2.2.1. Extensions o f the M N L model to non-IIA models One of the first attempts to extend the MNL model was the DOGIT model (Gaudry and Dagenais, 1979). The DOGIT model employs an additional parameter to capture similarity between pairs of brands. T h e probability of choice of a brand i in set S of competing brands is given by:

eV~+ 0 i ~ e b P(i/S) =

i , [ 1 + ~ 0 J ) Eev~ j

i,j ~ S,

where 0i is a non-negative parameter reflecting the degree to which strict utility of brand i would be affected by the introduction of a new brand to the choice set. It is called DOGIT because it is a generalized logit model that "dodges" the IIA assumption. DOGIT met with only partial success because of (i) the fact that the substitution parameter 0 is not modeled as a direct function of the similarity of the brand attributes and (ii) its computational complexity. Meyer and Eagle (1982) and others suggested similar extensions of the logit model by using an additional term for similarity between brands in the MNL model and thus simultaneously providing for the effects of utility and substitutability. The following equation captures the basic idea commonly shared by these models although each is slightly different in its specific formulation: eZi0i P ( i / S ) = }.,eVJOj,

i,j~S,

(7)

J

where 0i, bounded between 0 and 1, is an inverse measure of similarity of brand i to all other brands in the choice set. Here 0i is either treated as a single scaling constant to be empirically determined from choice data, or parameterized separately, with 0i being obtained from attributes measures.

2.2.2. The generalized extreme value (GEV) model McFadden (1981) captures brand interdependence by postulating hierarchical relationships which imply more substitution among some pairs of brands than others. The resulting generalized extreme value (GEV) model can be derived from a theory of stochastic utility maximization (see McFadden, 1978, for the proof of a theorem that captures this relationship). The GEV model uses the generalized extreme-value distribution, which is defined as:

(6) = exp[-

G(exp(--61)

, . . . . e x ' p ( -- 6 N ) ) ] ,

A.K. Manrai /European Journal of Operational Research 82 (1995) 1-17 where G is a nonnegative homogeneous function of degree 1. The G E V model assumes that a consumer partitions the set of competing brands S into r subsets represented by Ss, s = 1,2 . . . . . r, of similar brands such that the I I A property of the M N L model holds within these sets but not between these sets. The G E V model structure suggests the probability of selection of a brand i of a subset t is given by Pti = Pi/t "P t, eVa/(1 --Tt)

Pi/t

~ eVo/(1-r,)'

i , j ~ S t,

(8)

J at

e,= j~.aj

j~S,,

Nj -70 ~ e Vu/(1 i=1 ~2 e vii/O-w) i=1 s = l . . . . ,r,

(9)

where Vti is the utility of brand i in subset t as in Eq. (2) and Ys is an inverse measure of the correlation among unobserved components o f utility within subset S,, s = 1,2,... ,r.

2. 2. 3. Nested multinomial logit ( N M N L ) model T h e idea of the nested multinomial logit (NMNL) model is related to the G E V model. The formal development of the N M N L model is presented in M c F a d d e n (1981). The N M N L model presupposes the decision process to have a hierarchical or tree structure because of which it may also be classified as an attribute-based processing model. T h e N M N L model assigns the hierarchical transition probabilities to be M N L with scales which capture the inclusive values of the branches under each node of the decision tree. M c F a d d e n (1978) showed that N M N L can also be derived from the theory of stochastic utility m a x i m i z a t i o n . As an example, consider the problem of brand choice, with the choice of a store indexed by t = 1,2 . . . . ,T and brands within a store as i = 1,2 . . . . . N t in store t. The consumer will have a

7

utility Uti for alternative (t,i) that is a function of the attributes of this alternative, i.e., attributes of the store t and brand i, as well as the consumer's characteristics. We assume:

Uti =Vti -[- Eti

(10)

and

Vti = ~'Xti q- oltVt,

(11)

where Xti is the vector of attributes that vary with both store and brand (discounted price of a brand in a store) and Yt is the vector of attributes that vary only with store (e.g., location of a store). The probability Pti that the (t,i)th alternative is selected is given by Pti = Pi / t "P t,

Pi/t = ca'X"/ez',

(12)

Pt = e'~'r'+/' eWr'+r', /s=l

(13)

where I t = log(EN~le~.'x) is defined as inclusive value. This model can be easily extended to accommodate a tree with several branches.

2.2.4. Multinomial probit model An alternative to the extensions of the logit type models discussed above is the multinomial probit (MNP) model. The M N P model is a generalization of the independent probit model (IPM) which was briefly described in Section 2.1. In the M N P model, the errors of the r a n d o m utility model in Eq. (3) have a multivariate normal distribution which permits more flexible patterns of similarity or interdependence among competing brands. The M N P model, in its direct form or in its r a n d o m coefficient form ( H a u s m a n and Wise, 1978), works with a general or factor analytic covariance structure to accommodate the problems caused by the I I A property of the M N L model. The M N P models focus on the covariance between r a n d o m utilities arising out of variation of the p r e f e r e n c e vector ( H a u s m a n and Wise, 1978) or due to a distribution on consumers' ideal points. The M N P models tend to be computationally b u r d e n s o m e for choice sets o f larger than 3 or 4 brands because the computations involve

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A.K. Manrai/European Journal of Operational Research 82 (1995) 1-17

evaluating multiple integrals as shown in the example below. Consider the case of three brands with the following utility structure: U 1 = V 1 q- e l ,

(14)

U 2 = V 2 q- E2,

(15)

U3 = V3 + %,

(16)

probit, and hybrid model behavior as special cases. For an N-brand problem, the most general specification of the error is:

102-..ON] [ 02

N

Assuming errors to jointly follow a trivariate normal distribution with mean vector zero and covariance matrix given as below:

IO'? 0-12 0-13] "~ = [0-12

L0-13

0-2

0.23

0-23

0-2

(17)

The probability that brand i will be selected is given by:

erob(Ul > U2,Wa> W3) =

erob(E 2 - E-1 < V 1 - V 2 , E 3 - E 1 < V 1 - V 3 ) . (18)

The probability of selection of brand 1 requires evaluation of a double integral as shown in the following equation: P ( 1 / { 1,2,3})

= fvSwzfv_l-w~qb(E2-q,e3-el) d.2_qd.3_.l, (19) where ~ b ( % , el, a3 - %) has a bivariate normal distribution with covariance matrix /~ given below and mean vector zero.

ON+I

"'"

02N-1

02N-1

"'"

ON(N+I)/2

(21)

2.2.5. Generalized logit models Dalal and Klein (1988) propose a rather flexible class of generalized logit models (GLM). The members of the G L M family are non-IIA models, which are shown to approximate any choice model obtained from the principle of utility maximization. G L M assumes that conditional on a consumer type "c", the utilities, U1. . . . . UN for S = {1,2,...,N} brands are independent across brands, and their distribution depends on the brands by powers Wl(C), . . . , WN(C) of a fixed unidimensional distribution function F. T h e authors point out that, unconditioned on "c", U1, .... Uu are in general dependent, if the UJc has some probability distribution function, F~'j(C)(u/) at a point u/. H e r e F can have any convenient form of a unidimensional distribution function. Assuming Q as the probability distribution of the consumer type "c", the G L M family is formalized as follows: P(U1 ~ Ul . . . . ,UN ~ gN) N

"4- 0"2 -- 20"12

0"? -- 0"13 -- 0"12 + 0"23 ]

=f

I-IFW~(C)(uj) dQ(c).

(22)

j=l

(20) The generalized multinomial probit model (GMNP) (Currim, 1982) captures interdependence by estimating the covariance among random utilities for competing brands based on the most general specification of ~ which allows for negative exponential, extreme value, independent

2.2.6. Multiplicative competitive interaction (or attraction) model Another descendent of the Luce choice theory is the multiplicative competitive interaction (MCI) model (Cooper and Nakanishi, 1988) or an "At-

A . K Manrai ~European Journal of Operational Research 82 (1995) 1-17

traction model". It has the following general mathematical structure:

Ai MSi=

U

'

(23)

l'-i K

4

= I-I

k=l

where MS i is the market Share of brand i; A i is the attraction of brand i; N is the number of brands; K is the number of attributes; Xki is the value of the kth attribute for brand i; fk is a monotone transformation function on Xki; and /3k are parameters to be estimated. A commonly accepted functional form of the attraction effect is: K

Ai = e~ I'-I X~k" ei,

(24)

k=l

where ol, is a brand-specific parameter and ei is an error term. It can be relatively easily shown by using the above specification of the attraction effect in the MCI model and taking logs of both sides that the following relationship holds: log ~ =

= ol* + -

fiklOg = Iog(

k_____~/ + e*,

(25)

#e),

where MS, X, and g are the geometric means of MS/, Xki, and e i respectively. The MCI model, in which the log of the error term is assumed to be normally distributed, also permits more general conditions on the covariance matrix of errors than would be strictly imposed b y the IIA property. Batsell and Polking (1985) propose a new class of market share models which are demonstrated to be equivalent to the MCI models.

2.3. Estimation and applications of the utility maximizing choice models Estimation of random utility models is usually done by maximum likelihood estimation (MLE)

9

procedures requiring nonlinear search. Typically, it is not a problem for Luce-type logit models or the direct variants of the MNL model (see Malhotra, 1984, and many others). Recently, the logit model is extended to predict not only brand choice but also brand quantity and purchase timing decisions (Chintagunta, 1993). Chintagunta (1994) and many other researchers also model consumer heterogeneity in logit models. Kalwani et al. (1994) provide fairer benchmarks for comparing choice models incorporating consumer heterogeneity. Swait and Louviere (1993) discuss the role of the scale parameter in the estimation and comparison of MNL models while Horowitz and Louviere (1993) provide a new testing procedure for comparing observed choices with those predicted by probabilistic brand choice models. Dalai and Klein (1988) provide a MLE procedure to estimate the GLM and show that estimation is not a problem even when dealing with large numbers of brands a n d / o r attributes. In case of MCI models, MLE procedures yield consistent estimates, but they are not minimum variance estimates. Cooper and Nakanishi (1988) have demonstrated that a log transformation of MCI models renders them linear in parameters, and they suggest the use of least-square parameter estimation. Batsell and Polking (1985) provide empirical applications. The MNP, GMNP, GEV, and NMNL are known to present numerical analysis problems of convergence and high computation time because of the high degree of nonlinearity involved in these models. Notwithstanding the difficulties, these models have been applied in a variety of empirical situations. Hausman and Wise (1978) have demonstrated the application of MNP with correlated errors and compared its performance with MNL in the context of travel mode choice decisions of commuters to the central business district of Washington, DC. Their example has three competing transportation modes. Currim (1982) shows an empirical application using five transportation alternatives available to the residents of San Francisco Bay A rea. Currim (I982) provides comparative predictions f r o m GMNP, which is MNP with correlated errors, and other non-IIA (e.g., extreme value model) as well as

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IIA models (e.g., M N L model). Kannan and Wright (1991) present a nested logit approach to modeling and testing market structures. The details of alternative estimation methods, i.e., sequential versus simultaneous estimation of N M N L models, are presented in Henshner (1986). The N M N L model has been found to be computationally more attractive for large problems while the G E V model has not met with as much practical success as NMNL. Both the N M N L and G E V models require the a priori specifications of as, pects and hierarchical tree-like structures. Other approaches which do not need such a priori specifications, e.g., GMNP, require an estimation of a large number of parameters which increases as a quadratic function of the number of competing alternatives or brands in the choice set. Chintagunta (1992) provides an efficient way to estimate a multinomial probit model using the method of simulated moments. The G M N P model and the Batsell and Polking model run into difficulty in predicting choice shares for new product concepts not included in the estimation sample.

3. Brand choice models driven by the psychological principle of attribute-based sequential elimination A common assumption m a d e by "attributebased processing" models is that brands are collections of measurable aspects/attributes (Restle, 1961) and that a consumer uses information relating to aspects of brands selectively and sequentially to eliminate brands from the choice set until only one brand remains in the set. Restle (1961) formulated a binary choice model, BCM, on this basis. In BCM the probability of choice of brand 1 in a binary set {1,2} is given by: P ( 1/{ 1,2} ) = M(-A1 ¢3Xz) + M ( A 1 A A 2 ) ' (26) where M is a measure function that transforms features into a scaler, A 1 and A 2 a r e feature sets of brands 1 and 2, and (A 1 ~ X z) and (~zT1 0 A 2) are the distinct features of brands 1 and 2 respec-

tively. Restle provides the process insight that evaluations in a choice situation are based on distinct features of brands and that common or shared features do not play a role in choice decision. Tversky's (1972) Elimination-By-Aspects (EBA) also makes use of the representation of brands as collection of aspects, but it generalizes BCM to choice situations with more than two competing brands. We shall discuss the E B A model in the following subsection.

3.1. Elimination-By-Aspects (EBA) model of brand choice Tversky (1972) proposed EBA, a probabilistic model of choice, based on covert elimination process. E B A accounts for observed dependencies among competing brands and thus represents a generalization of Luce's (1959) model. In EBA, at each stage of the choice process, an aspect or feature is selected with a probability proportional tO its utility or weight. All the brands that do not include the selected aspect are eliminated from the choice set. The process continues until all brands but one are eliminated. Implicit in this covert elimination process is the importance placed by E B A on distinctive aspects possessed by a brand. As in BCM, in the EBA model the aspects shared by all the brands do not influence the probability of choice of a brand. The EBA model may be formalized using the following mathematical notation. As before, let S = {1,2 . . . . N} be the set of competing brands faced by a consumer. Let Sa, $2, S 3 denote the nonempty subsets of S. Let P(1/S1, ~) be the probability of selecting brand 1 from the subset S 1 which share aspect a. Next, let S~ be a set of aspects (unlike S 1 which is a set of brands). Additionally, let 1' be a set of aspects of brand 1, such that, 1' = {a,/~ . . . . } and u(a) be a scale value defining utility or weight of aspect a. EBA is defined in terms of the following recursive formulation:

~_~ u(a)'P(1/S1) P(1/S) = , ~ r

(27) E

'

A.K. Manrai/European Journal of OperationalResearch 82 (1995) 1-17 where u(/3) is the utility or weight of feature /3 which may or may not be possessed by brand 1. In contrast to the logit-like models, where choice probability is expressed in terms of externally predetermined perceived attribute values of brands, note that in the E B A model it is not necessary, prior to the selection o f a brand, for a researcher to measure the aspects which are used to eliminate brands. Rather, one could do an internal analysis, where the goal is to derive a set of psychological scales for brands which best fits the observed choice data (Cooper and Nakanishi, 1988). 3.2. EBA-like models o f brand choice Despite the intuitive appeal and elegance of this psychological processing model of choice, E B A has not been widely used by marketers for consumer choice modeling. E B A is less suitable because it requires estimation of a large number of parameters, up to 2 N - 3 parameters for a choice set containing N brands. Furthermore, no special-purpose parameter estimation technique software has yet emerged. Early on, Gensch and Svestka (1979) presented an algorithm f o r an EBA-like sequential choice model, called HIRAC, for analyzing individual choice decisions. H I R A C is a semi-order lexicographic m o d e l that empirically generates aggregate estimates of a set of threshold tolerances for attributes. In an empirical application presented by the authors of the model, H I R A C and M N L were found to be similar in predictive accuracy but yielded very different types of diagnostic information. Gensch and Svestka (1984) later introduced a maximumlikelihood-hierarch model ( M L H ) w h i c h has both theoretical and computational advantages over HIRAC. Researchers in marketing and psychology have developed some creative alternative formulations by producing several other EBA-like models. The same are discussed in this section. 3.2.1. P R E T R E E model o f brand choice Tversky and Sattath (1979) addressed the problem of large numbers of parameters in E B A by proposing elimination by tree (EBT), which requires only a subset of brands sharing aspects

11

to form a hierarchical structure on that set. They also proposed the hierarchical elimination by aspects (HEBA) model as a choice process in which a single prespecified sequence of binary choices is used. Despite the differences in the hypothesized choice process, E B T and H E B A are mathematically equivalent. Both these models are referred to by the generic term preference trees or PRET R E E . In P R E T R E E the number of parameters to be estimated is reduced to (2N - 2) for a set of N competing brands. In P R E T R E E a consumer selects a branch from the tree (with probability proportional to its length) and eliminates all brands that do not include that branch. The same process is applied to each selected branch until only one brand remains. A detailed example and f i n e r details of the mathematical structure of this model is discussed by DeSarbo et al. (1993). An extensive and excellent overview of several other non-spatial tree models which are beyond the scope of this review is also provided in DeSarbo et al. (1993). 3.2.2. Elimination-by-cutoffs model of brand choice The aspect or feature representations are not restricted to binary variables and they may also apply to ordinal or cardinal variables (i.e., dimensions). Rotondo (1986) suggested that price is inherently a continuous variable and must be treated as such within the E B A modeling framework. H e points out a limiting constraint on his approach to incorporate price in EBA, which may not allow other continuous variables to be incorporated in the E B A model. Manrai and Sinha (1989) developed the elimination-by-cutoffs (EBC) model as an extension of the E B A model in a continuous multi-attribute space. EBC is an individual-level model of brand choice that uses ratings on multiple perceptual attributes derived from the location of competing brands in a perceptual map and yields choice probabilities in the E B A framework, In the EBC model, utility functions are first defined over each perceptual dimension. The competing brands, S = {1,2,... N}, a r e then represented in this utility space. Thus, a brand i is represented in the utility space as the vector (u~ . . . . u~) where u~ is the utility of brand i along perceptual dimension k, for k = 1,2,... K.

A.K. Manrai /European Journal of Operational Research 82 (1995) 1-17

12

Let the sorted utilities of the N brands in the choice set along a dimension k be u~lk,u~ 2k. . . . . . urkN~, where rag is the index of the least preferred brand along dimension k, rzk is the brand with the next higher utility and so on. The differences u~2k - u~lk, u~3k- u~:k, ... between adjacent utility levels are the utility quanta:

qjk=Ur~--U~-lS ',

/ = 2 . . . . . N,

k = l . . . . . K. The set of brands that share a utility quantum is defined to be:

BqJ~ = (rjk,rj+l, k . . . . . rNk),

(28)

i.e., all brands whose utilities are, in part, comprised of quantum qjk" The recursive mathematical formulation of the choice model is: K

P(i/S) = E

N

qjk

E ~

"P(i/Bqs~)

(29)

k=lj=l

and

U= E Eqjk, k

j

where P ( i / * ) i s the probability of choice of brand i from set *, B qj, is the subset of brands in the choice set which share quantum qjk along the k t h dimension, and qjk is the utility of the jth quantum along the k t h dimension. The formulation is recursive in that P(i/BqJO may itself require further evaluation until a single brand remains. Manrai and Sinha (1989) give an empirical application of EBC using three brands in a two-dimensional perceptual space. Using continuous dimensions and not aspects to represent competing brands, Gensch and Ghose (1992) propose a new procedure called elimination-by-dimensions (EBD). They formalize a procedure of matching factors obtained via factor analysis of attribute ratings to a preference tree. The preference tree is obtained via ADD T R E E , which is in turn driven by a brand similarity matrix obtained from a matrix of average factor scores. They apply the EBD procedure to industrial and consumer product situations with four competing alternative suppliers/brands and ten factors or perceptual dimensions in the indus-

trial situation; and nine factors in the consumer product situation.

3.3. Estimation and applications of attribute-based processing models As discussed earlier, the requirement of estimating a large number of parameters in E B A has prevented its empirical applications. P R E T R E E or the H E B A model solves this problem as it requires estimation of only ( 2 N - 2) parameters as against (2 N - 3) parameters required in EBA. H e r e N is the number of brands in the choice set. It is shown in the literature that P R E T R E E is an alternative to the N M N L model presented in Section 2.2.3. McFadden (1981) provided empirical evidence to demonstrate that the H E B A (or P R E T R E E ) and N M N L models give virtually the same fits to data. In marketing, several researchers have used the P R E T R E E model for a variety of purposes. For instance, Moore et al. (1986) found that the preference tree analysis eliminates many of the shortcomings in traditional hierarchical clustering methods which are based on brand switching data. Glazer et al. (1991) evaluated the performance of P R E T R E E in the context of constrained choice. Despite computational tractability, EBC and EBD have met with limited success, thus far, when judged by the number of published field applications of these models, although authors of these models have provided empirical applications in their respective papers. The N M N L model is computationally tractable for large problems. In applications with several explanatory variables, it is better to use the N M N L model rather than the P R E T R E E model.

4. Two-stage brand choice models

The models reviewed above are the single-stage models of brand choice. The two-stage models combine the principle of attribute-based sequential elimination (typically employed in stage 1 of these models) and the principle of utility maximization (typically used in stage 2 of these mod-

A.I~ Manrai /European Journal of Operational Research 82 (1995) 1-17

els). T h e r e is some evidence in the marketing literature that consumers use different strategies at various stages of the decision process in a manner that suggests the two-stage approach proposed in Gensch (1987). The first stage in this choice process uses the maximum-likelihoodhierarch model (MLH) developed by Gensch and Svestka (1984) to reduce the number of brands in the feasible set down to the final choice set. The second stage in Gensch's (1987) model selects a single brand from the final choice set using a logit approach. The two-stage model is estimated sequentially using real world data in an industrial context. The predictive accuracy of the two-stage model compared favorably to the single-stage models. Roberts and Urban (1988) model individual brand choice probability, given category purchase and consideration of the brand. They incorporate the effect of uncertainty on multi-attribute preference models, using a decision analysis framework. They also model the diffusion effects by suggesting that a consumer's belief about attribute levels and uncertainty changes as he or she gains more information about the brand. The effect of new information is modeled in a Bayesian updating framework. R e c e n t l y several researchers have attempted to make these dynamic models of brand choice behavior more parsimonious in parameters to allow for the aggregate effects of changes that may take place at the individual level (see for example, Roberts and Urban, 1988). Fader and McAlister (1990) proposed an individual level EBA-like model which is really a two-stage model of brand choice. In the first stage of Fader and McAlister's model, brands are screened using one aspect e.g., promotion, and then in the second stage they use a M N L type model to estimate the probability of choice of a brand from the reduced choice set. They applied this modified approach to scanner panel coffee data. Siddarth et al. (1993) proposed a dynamic heuristic procedure in which the choice sets are estimated using an Empirical Bayes method employing consumer purchase histories in conjunction with the M N L model. They use brand loyalty as the single screening variable at the first stage of consideration set formation. The procedure is

13

applied to the liquid laundry detergent category and they provide interesting and revealing managerial insights, such as, competitive positioning, cannibalization potential, and long-run advantages of sales promotion. Andrews and Srinivasan (1994) present a dynamic consideration set formation model which represents a brand-based processing approach. This model is complementary to the static cost-benefit models developed by Roberts and Lattin (1991) and others. The dynamic model does not require direct consumer reports of consideration. The consideration stage of the model is driven by the independent availability Iogit (IAL) model. In IAL the probability that any given brand i is available, Di, is estimated directly. Given the estimated availability probabilities for each brand, the probability that a certain choice set C will occur can be calculated as: I-[ D~ ]-I (1 - D , )

P(C) =

,~c

i~c

1- I--[(I-D/) i

(30)

The probability that brand i is chosen from choice set C is given by:

P(i) = ~ P ( i / C ) " P ( C ) ,

(31)

c

where C is one of the possible 2 s - 1 nonempty sets. The probability of selecting brand i from set C, that is, P(i/C), is computed in the second stage using a M N L model. Andrews and Srinivasan (1995) avoid the need for estimates of each consumer's U* (threshold utility) by assuming a distribution of the U* values. With the assumption of a normal distribution for U*, the probability that consumer z will consider brand i on some purchase occasion is the probability that its consideration utility will exceed the threshold level required for consideration, i.e., Oiz = P r o b ( U ik > U * ) = qb([J'Xiz),

(32)

where cb(-) is the standard normal distribution function, V = 13'Xiz, Xiz is a vector containing brand i's characteristics as experienced by consumer z, and /3 is a vector of importance weights or consideration parameters. The equations (30),

A . K Manrai / European Journal of Operational Research 82 (1995) 1-17

14

(31), and (32) jointly determine the full model proposed in Andrews and Srinivasan (1995). They show improved predictive accuracy of their dynamic consideration set model over the MNL model using scanner data for ketchup and yogurt. Recently, Andrews and Manrai (1994) proposed a new tw0-stage hybrid model of brand consideration and choice, Consideration-ByAspects (CBA), in which the first stage is modeled using an EBA-like model to reduce the initial set of brands to a smaller choice set and then MNL is used in the second stage to compute probability of choice of a single brand from the choice set. The models proposed by Fader and McAlister (1990) and Siddarth et al. (1993) are nested within the CBA proposal. The probability of choice of brand i under CBA is given as below:

P(i/S)MNL + E

UcP(i/C)MNL

C:i~C

P(i/S)cBA=

1 + ~U c C

(33) and exp(ffXi)

P( i/C)MNL = E exp(18'Xq)'

(34)

q~C

where S, as before, is a set of N competing brands; Uc is the utility of all aspects shared by brands in a consideration set C; P(i/C)MN L is the probability of selecting brand i from consideration set C, as predicted by the MNL model; P(i/S)ivn,~L is the probability of selecting brand i from the entire set S, as predicted by the MNL model; X/ is a vector of perceptual attributes of brand i; and 18' is a vector of importance weights to be empirically determined.

5. Conclusions and directions for future research

The two-stage choice models that postulate a different type of decision processing in each stage is the state of the art in modeling brand choice behavior. In these models;the first stage eliminates certain brands to reduce the size of the full

feasible set of brands to a smaller consideration set. This stage is often guided by the principle of attribute-based sequential elimination or "attribute-based processing". It is followed by a second stage of "brand-based processing" driven by the principle of utility maximization. The roots of the two-stage models are, therefore, well grounded in the theoretical literature in economics, marketing, and psychology. Almost all of the empirical evidence available supports the fact that the predictive accuracy of two-stage choice models is at least as good as, and many a times superior to, the single-stage models discussed in Sections 2 and 3 (Manrai and Andrews, 1994; Gensch, 1987; Siddarth et al., 1993). The two-stage models, therefore, offer intuitive, integrative, theoretically rigorous, and empirically advantageous models of brand choice behavior. There are obviously, significant and commendable developments in the field of mathematical modeling of brand choice behavior as documented in this paper. However, more research effort is required to quantify and mathematically formalize: (i) various situation based simplifying heuristics and (ii) biases (such as anchoring, availability and representativeness) that have attracted marketing researchers in the area of consumer behavior (e.g., Payne et al., 1988) and psychology (Kahneman and Tversky, 1979). Furthermore, most of the models driven by the attribute-based processing principle (e.g., EBA) have no error theory associated with them. The testing and research work as presented in Johnson (1988), relating to the relevance of hierarchical models to choice among non-comparable brands, Currim and Sarin (1989) on the expected utility theory and prospect theory, Currim (1982) on comparing probit models with various singlestage choice models, and Manrai and Andrews, (1994) focusing on comparative testing of some recent two-stage models, ought to continue to be able to specify sharper boundary conditions under which a certain model of brand choice behavior outperforms the competing models. Several brand choice models (e.g., CBA, EBC, EBD and GLM) work with the assumption of independence among utilities of brand attributes. A more general theoretical framework is needed

A.K. Manrai /European Journal of Operational Research 82 (1995) 1-17

to accommodate the possibility of interacting utility functions. These models also assume a vector model of preferences. It may also be desirable to extend these models to work with ideal-point models using single peak nonmonotonic utility functions. Furthermore, these models are developed in a static framework. The dynamic updating of attribute utilities (Roberts and Urban, 1988) would lead to dynamic and dependent forms of these models. We see the potential for the incorporation of variety seeking and loyalty in the dynamic models. For example, from a variety seeking perspective, the selection of a brand could be modeled to reduce the utility of brands with the consumed attributes and augment the utilities of brands that are distinct. The approaches presented by Manrai and Sinha (1989) and many others use a perceptual map together with choice data to parameterize the EBC (or some other model). A n alternative would be to use choice data alone to construct a multidimensional representation of brands as discussed by DeSarbo et al. (1994). Such maps may be useful to brand managers in formulating marketing strategies. As stated earlier, the EBA-like models (e.g., EBC) may be enriched by adding a random element to the evaluation of perceptual attribute levels. Such models may lead to better predictions by using a Monte Carlo simulation procedure. Each trial of the simulation could sample from the distribution of the brand locations in the perceptual map. These would then be transformed into choice probabilities using the EBC model. More research is also needed to develop formal models of simplifying heuristics used by consumers to select brands under varying conditions of information availability, i.e., too much information versus lack of adequate information (see for example, Keller and Staelin, 1989, and others). Clearly, many phenomena need to be better modeled in the area of b r a n d choice behavior. Although, developments in the area of mathematical modeling of brand choice behavior have a rich and complex history, the future offers many promising research opportunities to develop more robust, flexible, and dynamic models for better understanding of the brand choice behavior, prediction of consumer choice in varying circum-

15

stances, and formulation of marketing strategy in increasingly complex and global markets.

Acknowledgements The author greatly appreciates the comments and suggestions made by Jehoshua Eliashberg of University of Pennsylvania and William Gehrlein of University of Delaware. The general guidelines and suggestions provided by the editor, Alan Mercer were also very helpful in preparing this paper.

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