An alternative efficient representation of demand-based competitive

Strategic Management Journal Strat. Mgmt. J., 28: 755–766 (2007) Published online 22 March 2007 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/smj.601 Received 14 March 2006; Final revision received 10 August 2006

RESEARCH NOTES AND COMMENTARIES AN ALTERNATIVE EFFICIENT REPRESENTATION OF DEMAND-BASED COMPETITIVE ASYMMETRY WAYNE S. DESARBO* and RAJDEEP GREWAL Smeal School of Business, Pennsylvania State University, University Park, Pennsylvania, U.S.A.

Competitive asymmetry is defined in terms of the directional level of competition among brands/firms (i.e., unit of analysis), where the degree to which brand/firm A may compete with brand/firm B does not equal the degree to which brand/firm B competes with brand/firm A. Such a market structure phenomenon is quite commonplace in virtually every market, e.g., where there exists distinct market leaders and followers. DeSarbo, Grewal, and Wind recently (2006) proposed a new spatial methodology in SMJ to assess these competitive asymmetries based on information on consumer choice sets (i.e., a demand-based approach). However, the approach espoused by DeSarbo et al. results in as many competitive maps as there are brands/firms in a dataset. In this research, the authors devise a distance-based unfolding multidimensional scaling procedure for deriving joint spaces of brands/firms both as givers and takers of consumer consideration with the objective to have a more efficient representation of competitive asymmetries (i.e., one map irrespective of the number of brands/firms under study). An application is provided for an actual commercial study undertaken by a major U.S. automobile manufacturer examining the mid-size car marketplace. The strategic implications of the results are detailed. Copyright  2007 John Wiley & Sons, Ltd.

INTRODUCTION To examine competitive asymmetry in a designated market, DeSarbo, Grewal, and Wind (2006) proposed a stochastic multidimensional scaling (MDS) procedure that they calibrated on consumers’ evoked consideration set responses to derive spatial Keywords: competitive market structure; asymmetric competition; multidimensional scaling; positioning; consideration/choice sets *Correspondence to: Wayne S. DeSarbo, Smeal College of Business, Pennsylvania State University, 433 Business Building, University Park, PA 16802, U.S.A. E-mail: [email protected]

Copyright  2007 John Wiley & Sons, Ltd.

representations of ‘who competes with whom.’ This MDS procedure resulted in as many competitive maps derived as brands/firms in the dataset under investigation, and competitive asymmetry was discerned by comparing two or more competitive maps. Such comparisons across multiple competitive maps become difficult as the number of brands/firms in a dataset increase, which would be fairly common in practical applications with greater than seven or so brands/firms. As a result, strategic decisions such as repositioning brands/firms and effectively managing portfolios of brands (where one positions a portfolio of

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brands against the portfolios of primary competitors) becomes a challenging task as the manager has to be able to scrutinize information from multiple competitive maps and then mentally collate and process it to arrive at a viable decision. Indeed, we believe that from a manager’s standpoint, representing competitive asymmetries in a single map would be useful for strategic decisions such as brand repositioning and managing brand portfolios. For this purpose, we extend the important work of DeSarbo et al. (2006) with the objective of developing a more efficient (in terms of number of maps) representation of competitive asymmetries using consumer consideration set data. Specifically, we propose a new MDS technique to represent competitive asymmetry in one generic competitive map irrespective of the number of brands/firms in a dataset. We now elaborate on this technique and then illustrate the technique for an actual commercial study undertaken by a major U.S. automobile manufacturer examining the mid-size car marketplace. We end by illustrating the strategic implications for managers for this mid-size automobile dataset.

AN UNFOLDING MDS PROCEDURE FOR THE ANALYSIS OF ASYMMETRIC PROXIMITIES DATA There is a well-defined literature on MDS based procedures for the analysis of asymmetric proximities data. A variety of MDS models and associated algorithms have been introduced to uncover the underlying structure of asymmetric proximity data (e.g., Chino and Shiraiwa, 1993; Okada and Imaizumi, 1987; Saito, 1991). Many such asymmetric MDS procedures (e.g., DeSarbo and Manrai, 1992; Zielman and Heiser, 1996) typically represent the symmetric portion of the data vis-`avis a common T -dimensional space, and represent asymmetry vis-`a-vis a separate set of row and column additive constants. Others (e.g., Gowers, 1977; Harshman et al., 1982) are factor analytic in nature where a common brand space represents the symmetric portion, and a T × T asymmetric transformation matrix accounts for the non-symmetric portion. According to Okada and Imaizumi (1997), three characteristics seem most crucial in the development of asymmetric MDS procedures: (1) both Copyright  2007 John Wiley & Sons, Ltd.

symmetric and asymmetric proximity relationships are simultaneously represented in the configuration (which leads to ease of interpretation of the result; i.e., when symmetric and asymmetric relationships are not represented in the same configuration, one must juxtapose symmetric and asymmetric relationships that are represented separately and thus it is far preferable to have both types of relationships in the same configuration); (2) differences with respect to the third way of the data array (for two-mode, three-way proximity data) in both symmetric and asymmetric proximity relationships are distinguished (which seems very significant especially when such differences stem from different substantive causes); (3) the model is based on distance where both symmetric and asymmetric aspects of any pair-wise proximity is represented as distances in the configuration of objects (which is important because most researchers utilizing MDS are accustomed to a model where an object is represented as a point and the recovered distance between two objects by an inter-point distance in a configuration). Okada and Imaizumi (1997) emphasize that it is easier to interpret the result of an asymmetric MDS analysis when both symmetric and asymmetric aspects of any proximity are represented as distances in a configuration. Our approach is quite different from those posited in the literature. We wish to accommodate Okada and Imaizumi’s criteria for asymmetric proximities by allowing for two alternative representations for the stimulus objects (brands here): one as row objects and the other as column objects (thus the ‘mode’ brand appears in two ‘ways’ in the data structure). Both sets of coordinates are to be plotted in the same Euclidean space, unlike these other procedures. In addition, there is a distance (unfolding) model underlying the relationship between row and column brand representations. Given that we are usually dealing with one-mode, two-way proximity cross-sectional data, array differences are not meaningful, although we do develop the procedure for the more general case of measurements over time (i.e., three-way data). Note that traditional unfolding procedures such as ALSCAL, KYST, GENFOLD2, and PREFMAP3 are not appropriate for use for such data here because the main diagonal elements are explicitly defined in the objective function of each of these procedures. As to be seen shortly, conditional probabilities are to be utilized as the input Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

Research Notes and Commentaries asymmetric proximities where the main diagonal elements are all equal to 1.00 by definition, which is basically non-informative information. As such, this characteristic would seriously distort the resulting configuration obtained in traditional unfolding procedures whereby the row and column coordinates for the same brand would be the closest elements in the derived map, given they would have the largest proximity values. Below, we describe the technical details of our proposed MDS model (see also the Appendix) presented for the most general three-way asymmetric case. Let t = 1, . . . , T time periods (T = 1 suggests a two-way analysis as a special case); j, k = 1, . . . , B brands; r = 1, . . . , R dimensions; Pj kt = the conditional probability that brand k is considered given that brand j is already considered in time t; Xj r = the rth coordinate of brand j (row); and Ykr = the rth coordinate of brand k (column); ej kt = error. Then, the spatial model we wish to create is Qj kt ≡ f

−1

(Pj kt ) = at

R


(Xj r − Ykr )2 + bt + ej kt

r=1

(1) where f −1 (.) is a predefined inverse monotone function, Qj kt is the transformed input conditional probability, at = a time-varying multiplicative constant, bt = a time-varying additive constant, Pmmt = 1∀m = 1, . . . , B, and 0 < Pj kt < 1. Thus, we specify a type of squared Euclidean distance, simple unfolding model akin to a threeway generalization to DeSarbo and Rao’s (1986) metric GENFOLD2 procedure, which relates linearly transformed (over time) squared Euclidean distance in the derived space between row and column brands j and k to a predefined inverse transformation of the input conditional probabilities (e.g., to stabilize the variance and transform probabilities to approximately normally distributed random variables—see Fleiss, Levin, and Paik, 2003). Our goal is therefore to estimate two distinct sets of points in a common R-dimensional space—one set of points (Xj r ) representing the brands as row objects (i.e., those brands as ‘givers’) and another set of points (Ykr ) representing the same set of brands as column objects (i.e., those brands as ‘takers’). As with the vast majority of such MDS procedures involving the analysis of proximity data, a spatial structure is derived for the aggregate sample of data. Copyright  2007 John Wiley & Sons, Ltd.

757

The objective function (sum of squared errors) that is optimized is Min (X, Y , a, b)

T

t

− at

R


B




[Qj kt

j =k

(Xj r − Ykr )2 − bt ]2

(2)

r=1

An alternating least-squares procedure has been programmed to estimate X, Y , a, and b, given P and a value of the dimensionality (R), which alternates between a conjugate gradient nonlinear programming subroutine to estimate X and Y , and simple linear regression to estimate a and b by time period. Both routines conditionally minimize (2) within each iterate. The estimation procedure alternates between the conjugate gradient and linear regression procedures until convergence is reached in terms of consecutive values of (2). Rational starts employing a singular value decomposition analysis of P is initiated to enhance solution recovery. A program option allows for symmetric solutions where an X = Y constraint is explicitly enforced. Goodnessof-fit measures include expression (2), as well as a variance accounted-for (VAF) measure or R 2 between f −1 (Pj kt ) and the model predicted values. The dimensionality is selected on the basis of an inspection of these goodness-of-fit measures for sequential values of R and examining where successive increases in dimensionality lead to nonconsequential increases in fit (as in most deterministic MDS procedures). The procedure accommodates the analysis of two-way (T = 1) or three-way data (T > 1), and is sufficiently general to extend to the analysis of other types of proximity data besides conditional probabilities derived from consideration set judgments. Asymmetry is thus represented by the two locations estimated per brand where typically, for example, d(c, A) = d(a, C). Also note that, unlike correspondence analysis, distances between row and column brand points are explicitly interpretable and defined here. Given that the main diagonal in the data = 1 ∀j, k, t, the optimization occurs for j = k so that explicit row and column comparisons for the same brands are not explicitly defined. A Monte Carlo analysis was conducted with 96 synthetically created datasets experimentally designed to vary in a 24 full factorial design to varying values of T , B, R, and error Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

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levels with six replications per cell. The resulting ANOVAs and MANOVA showed consistently excellent performance of the proposed algorithm in recovering the configurations and input data, as well as explaining differences in computational effort. The Appendix presents the technical details underlying the estimation of this MDS model. Geometrically, the squared length of the line connecting any row brand point with any nonequivalent column point is inversely related to the conditional probability P (column brand | row brand). In this same spatial representation, one can also construct α percent iso-consideration regions for any row brand point to indicate which column brand points actively compete with it in the minds of consumers. With values of Xj r , a and b estimated and fixed, we seek to find these values of Ykr∗ such that    R ∗ 
Qj kt − bt =  (Xj r − Ykr∗ )2 (3) at r=1 where Qj∗kt = f −1 (α). As displayed in Equation 3, this takes the form of a circle of fixed radius around referent brand Xj r (in two dimensions; a sphere in three dimensions). Again, such iso-consideration contours can be generated for any/all brands in the data. Clearly, with advances in graphics technology, such iso-consideration contours can be displayed simultaneously in a single map using different colors and display options. Thus, unlike the DeSarbo et al. (2006) spatial MDS methodology, which generates B maps (one map per brand in the study), the proposed unfolding based MDS procedure only generates a single map which proves much more efficient, especially in applications involving portfolios of brands offered by manufacturers.

A COMMERCIAL APPLICATION: THE MID-SIZE AUTOMOBILE MARKET Study background A major U.S. automobile manufacturer sponsored a marketing research project in the late 1980s to examine the mid-size automobile market. Personal interviews were obtained with N = 289 consumers who stated that they were intending to purchase a mid-size automobile within the next 6 months (thus other non-mid-size automobiles were not Copyright  2007 John Wiley & Sons, Ltd.

under consideration). These respondents were initially pre-screened on the basis of demographic and present car ownership information to represent members of the target market segment of interest. The study was conducted in a number of automobile clinics occurring at four different geographical locations in the United States (T = 1 time period here). One section of the questionnaire asked the respondent to check off from a list of 10 mid-size cars, specified a priori by this manufacturer sponsor and thought to compete in the same target market segment at that time (based on their prior research), which brands they would consider purchasing as a replacement vehicle after recalling their perceptions of expected benefits and costs of each brand. Corresponding intention-tobuy preference scores were also collected for each brand from these same consumers. The manufacturer and data collection vendor were not concerned with the considerations of other vehicles outside of this particular product segment. The 10 nameplates tested were (firms that manufacture them are in parentheses): Buick Century (GM), Taurus (Ford), Olds Cutlass Supreme (GM), Thunderbird (Ford), Chevrolet Celebrity (GM), Accord (Honda), Pontiac Grand AM (GM), Chevrolet Corsica (GM), Tempo (Ford), and Camry (Toyota). The vast majority of respondents’ elicited consideration/choice sets in the range of two to six automobiles from the list of 10. See DeSarbo and Jedidi (1995) for further study details. Akin to DeSarbo et al. (2006), we present in Table 1 the joint probabilities (P (Yij = 1) and P (Yik = 1)) of co-considering brands (i, j ) in part (a), and the computed matrix of conditional probabilities (P (Yij = 1)|P (Yik = 1)) or ‘cohits’, i.e., the (i, j ) entry designates the percentage of the column brand selected in the same consideration set as the row brand, in part (b). These matrices were computed directly from the joint frequency distribution derived from the sample consideration data for all pairs of brands. In other words, part (a) of this table indicates the probability of jointly considering brands i and j together, while part (b) indicates that given that the row brand was in a consideration set, what is the probability the column brand was also selected? The first matrix in (a) is symmetric, while the conditional probabilities in (b) are asymmetric. The main diagonals of the matrix in Table 1(a) render the probability that the particular brand was considered. As shown in these main diagonals of Table 1(a), there are clear Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

Research Notes and Commentaries Table 1.

759

Joint and conditional probabilities for the mid-size automobile study

(a) Joint probabilities

Century Taurus Cutlass T-Bird Celebrity Accord Grand-Am Corsica Tempo Camry

Century

Taurus

Cutlass

T-Bird

Celebrity

Accord

Grand-Am

Corsica

Tempo

Camry

0.294 0.131 0.201 0.107 0.176 0.111 0.090 0.042 0.038 0.128

0.131 0.464 0.173 0.260 0.270 0.221 0.093 0.062 0.121 0.242

0.201 0.173 0.498 0.280 0.287 0.215 0.190 0.062 0.038 0.232

0.107 0.260 0.280 0.516 0.266 0.235 0.180 0.059 0.066 0.215

0.176 0.270 0.287 0.266 0.585 0.280 0.156 0.097 0.093 0.280

0.111 0.221 0.215 0.235 0.280 0.516 0.111 0.052 0.045 0.377

0.090 0.093 0.190 0.180 0.156 0.111 0.294 0.042 0.038 0.118

0.042 0.062 0.062 0.059 0.097 0.052 0.042 0.128 0.035 0.062

0.038 0.121 0.038 0.066 0.093 0.045 0.038 0.035 0.149 0.076

0.128 0.242 0.232 0.215 0.280 0.377 0.118 0.062 0.076 0.516

(b) Conditional probabilities Century

Taurus

Cutlass

T-Bird

Celebrity

Accord

Grand-Am

Corsica

Tempo

Camry

Century Taurus Cutlass T-Bird Celebrity Accord Grand-Am Corsica Tempo Camry

1.000 0.284 0.403 0.208 0.302 0.215 0.306 0.324 0.256 0.248

0.447 1.000 0.347 0.503 0.462 0.430 0.318 0.486 0.814 0.470

0.682 0.373 1.000 0.544 0.491 0.416 0.647 0.486 0.256 0.450

0.365 0.560 0.563 1.000 0.456 0.456 0.612 0.459 0.442 0.416

0.600 0.582 0.576 0.517 1.000 0.544 0.529 0.757 0.628 0.544

0.376 0.478 0.431 0.456 0.479 1.000 0.376 0.405 0.302 0.732

0.306 0.201 0.382 0.349 0.266 0.215 1.000 0.324 0.256 0.228

0.141 0.134 0.125 0.114 0.166 0.101 0.141 1.000 0.233 0.121

0.129 0.261 0.076 0.128 0.160 0.087 0.129 0.270 1.000 0.148

0.435 0.522 0.465 0.416 0.479 0.732 0.400 0.486 0.512 1.000

Vincibility: Potency: Choice share:

0.387 0.283 0.294

0.377 0.475 0.464

0.374 0.483 0.498

0.359 0.481 0.516

0.362 0.586 0.585

0.355 0.448 0.516

0.384 0.281 0.294

0.444 0.142 0.128

0.411 0.154 0.149

0.373 0.494 0.516

winner and loser brands in this category among these car intenders. The Chevy Celebrity, Toyota Camry, Honda Accord, Ford Taurus and T-Bird, and Olds Cutlass collect the highest overall consideration probabilities, while the Ford Tempo and Chevy Corsica (small, cheaper cars in this class) possess the lowest consideration probabilities. Table 1(b) represents the asymmetric component of this dataset. One can get a preliminary indication of the asymmetric row vs. column main effects by computing simple averages of the rows and columns (ignoring the main diagonal element). Akin to the notions of ‘clout’ and ‘vulnerability’ in dealing with summaries of calculated market mix elasticities of Kamakura and Russell (1989), we compute adjusted row and column averages also presented in Table 1(b), and label them as vincibility (row averages) and potency (column averages) (cf. DeSarbo et al., 2006; Peterson, Balasubramanian, and Bronnenberg, 1997; Siddarth, Bucklin, Copyright  2007 John Wiley & Sons, Ltd.

and Morrison, 1995). One can see pronounced differences in the corresponding row vs. column averages for the more popular and least popular mid-size cars. For example, the potency scores for the Chevy Celebrity, Honda Accord, and Toyota Camry are substantially larger than their corresponding vincibility scores, indicating solid market positions. The opposite trend occurs for the Chevy Corsica and Ford Tempo, where the vincibility scores are near three times as large as their corresponding potency scores. Note, much of this particular phenomenon can be explained as a function of the empirical choice shares, also shown at the bottom of this Table 1(b), which correlate highly with potency (0.991) and vincibility (−0.866). Further insight into the nature and magnitude of these asymmetries can be witnessed by examining the off-diagonal entries in Table 1(b). For example, the Buick Century (row)–Chevy Celebrity (column) value (0.600) is nearly twice as large as the Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

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Chevy Celebrity (row)–Buick Century (column) value (0.302). In fact, there are a number of substantial differences seen across many comparisons made with (j, k) vs. (k, j ) brand comparisons. Clearly, as the number of brands increase, it is virtually impossible to grasp the nature and underlying structure of such asymmetric matrices of conditional probabilities. Cell-by-cell comparisons become increasingly infeasible, and our goal is to attempt to better summarize these complex relationships spatially vis-`a-vis application of the proposed MDS procedure. In addition, it is virtually impossible to visually discern the dimensional structure underlying the numbers presented in Table 1(b), especially for large B.

The proposed MDS model results Our proposed asymmetric MDS model was estimated in R = 1, 2, 3, and 4 dimensions using the conditional probabilities presented in Table 1(b) as input. Here, we convert the input probabilities to √ angles using an ( arcsin)−1 reverse scale transformation as recommended specifically in Fleiss et al. (2003) for the analysis of proportions. In Table 2, we present the associated goodness-of-fit heuristics associated with these solutions. As shown (and using scree plots), these measures appear to level off at R = 3 dimensions. We also compared this R = 3 solution with a much simpler nested one of assuming symmetry (i.e., X = Y) for this dataset. As aptly shown at the bottom of Table 2, this symmetric solution is easily rejected in favor of the asymmetric solution as ascertained by a comparison of the two goodness-of-fit heuristics. In particular, the asymmetric R = 3 solution produces a 40.1 percent lower error sums of squares compared to the symmetric R = 3∗ symmetric solution. Table 2. Goodness-of-fit dimensions

measures

by

number

of

R

Sum of squares

VAF

1 2 3 4 3∗

10861.81 8937.67 5949.33 4991.72 9934.49

0.692 0.757 0.831 0.859 0.717

3∗ denotes the symmetric three-dimensional solution as derived from the proposed MDS procedure. Copyright  2007 John Wiley & Sons, Ltd.

In order to help interpret the resulting three dimensions for the asymmetric solution, we computed correlations between four designated attributes reflecting manufacturer and size (dummy variables for GM, Ford, Imports, and Size [large vs. small]). Table 3 presents these correlations for both row and column sets of coordinates. As seen in Table 3, there are low intercorrelations between each set of estimated dimensions within row and column sets, indicating that we have extracted somewhat distinct dimensions. The first dimension appears to distinguish the two imported brands (Honda Accord and Toyota Camry) from the domestic Ford and Chevy brands. The second dimension separates the Ford brands from the Chevy brands. Finally, the third dimension appears to be related to size. Thus, we can label the three derived dimensions as: (1) Import–Domestic, (2) GM vs. Ford, and (3) Size. Figure 1 displays each of the three dimensions, one at a time, across the page for both row (bold) and column (italic) brand positions. We see an excellent intermixture of row and column brand points and no evidence of a degenerate solution (i.e., where row and column points are grossly separated from each other; c.f. DeSarbo and Rao, 1986) which typically plagues traditional unfolding models. Figure 1 also provides a dimensional decomposition of the structure captured by this spatial model. As an example, let us look at the Pontiac Grand-AM in the three unidimensional spaces provided in Figure 1. Concerning the Import–Domestic dimension (1), when Grand-AM is considered (bold), only the T-Bird is co-considered (italic) on the basis of this dimension, as indicated by its rather unique position near the bottom of the first dimension. However, with respect to dimension (2), the Celebrity and Accord appear very competitive with the GrandAM. On dimension (3), only the T-Bird appears to compete with the Grand-AM. When we now examine the opposite (italic) set of coordinates, we see that on dimension (1) Grand-AM threatens no one. Yet, with respect to dimension (2), we see that the Grand-AM threatens the Corsica, Cutlass, and Camry. On dimension (3), Grand-Am threatens the T-Bird and Cutlass. Thus, we can utilize the derived MDS results to position by dimension competitive threats and opportunities asymmetrically. That is, not only do we obtain a dimensional reduction to spatially represent the data structure Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

Research Notes and Commentaries Table 3.

761

Correlations of row and column brand coordinates with brand features

Row brand coordinates correlations

I II III GM Ford Imports Size

I

II

III

GM

Ford

Imports

Size

1.000 −0.017 0.090 −0.450 −0.154 0.739 0.096

−0.017 1.000 0.090 0.655 −0.792 0.088 −0.051

0.090 0.090 1.000 0.325 −0.062 −0.334 −0.168

−0.450 0.655 0.325 1.000 −0.655 −0.500 0.000

−0.154 −0.792 −0.062 −0.655 1.000 −0.327 0.117

0.739 0.088 −0.334 −0.500 −0.327 1.000 −0.134

0.096 −0.051 −0.168 0.000 0.117 −0.134 1.000

Column brand coordinates correlations

I II III GM Ford Imports Size

I

II

III

GM

Ford

Imports

Size

1.000 −0.230 −0.048 −0.668 0.177 0.633 0.277

−0.230 1.000 −0.535 0.695 −0.720 −0.044 0.318

−0.048 −0.535 1.000 −0.107 0.216 −0.114 −0.779

−0.668 0.695 −0.107 1.000 −0.655 −0.500 0.000

0.177 −0.720 0.216 −0.655 1.000 −0.327 0.117

0.633 −0.044 −0.114 −0.500 −0.327 1.000 −0.134

0.277 0.318 −0.779 0.000 0.117 −0.134 1.000

2

2 Toyota Camry

2

Buick Century

Honda Accord

Chevy Corsica

Buick Century

1.5

1.5

Ford Tempo 1.5 Chevy Celebrity

Honda Accord

Toyota Camry

Olds Cutlass Supreme Ford Taurus

1

1 Olds Cutlass Supreme

Chevy Celebrity Ford Taurus

Chevy Celebrity

0.5

Chevy Corsica 0.5 Pontiac Grand AM Chevy Celebrity Toyota Camry

Chevy Corsica 0 Buick Century Ford Tempo

1 Honda Accord Pontiac Grand AM

0.5

Ford Taurus Chevy Corsica Buick Century

Ford Tempo

Buick Century Ford Taurus Chevy Celebrity

0Honda Accord Buick Century

-0.5 Olds Cutlass Supreme

Chevy Corsica

Chevy Corsica -1 Ford T-Bird

0 Ford Tempo Toyota Camry

Toyota Camry

Chevy Celebrity -0.5 Ford Taurus Olds Cutlass Supreme

Honda Accord Olds Cutlass Supreme -0.5

Ford T-Bird

Pontiac Grand AM -1.5

-1

Pontiac Grand AM Toyota Camry

Ford T-Bird Ford Taurus Ford T-Bird Olds Cutlass Supreme -1

Ford T-Bird -1.5

-2

Pontiac Grand AM

Ford T-Bird Pontiac Grand AM Honda Accord

Ford Tempo Ford Tempo

-2.5 -1

-0.5

0

0.5

1

Dimension 1

Figure 1. Copyright  2007 John Wiley & Sons, Ltd.

-2 -1

-0.5

0

0.5

1

-1.5 -1

-0.5

Dimension 2

0

0.5

1

Dimension 3

The three-dimensional asymmetric solution Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

762

W. S. DeSarbo and R. Grewal 2.5

2

2 Ford T-Bird

Honda Accord

Ford Taurus

2

1.5

1.5 Ford T-Bird

Toyota Camry

Pontiac Grand AM Ford Tempo

1.5

1

1

0.5

Olds Cutlass Supreme 1

0.5 Honda Accord

0.5

0Honda Accord Toyota Camry Chevy Corsica Pontiac Grand AM

0

0 Toyota Camry

-0.5 Chevy Celebrity

-0.5Buick Century

Ford Taurus

Ford Taurus

Chevy Celebrity Ford T-Bird Chevy Corsica Olds -0.5 Cutlass Supreme Ford Tempo Buick Century Pontiac Grand AM

Chevy Corsica -1

-1 Chevy Celebrity Ford Tempo

Olds Cutlass Supreme -1

-1

-0.5

0

0.5

1

Buick Century -1.5 -1 -0.5 0

-1.5 -1

-0.5

0

0.5

1

Dimension 3

The three-dimensional symmetric solution

in Table 1, but we are also able to decompose the specific nature of the bases of asymmetry by dimension. Such insights are clearly not obtainable from any visual inspection of Table 1(b), nor from any analysis of the empirical choice shares at the bottom of that same table. We also present the symmetric R = 3 solution in Figure 2 as a comparison where X = Y and only one set of brands are represented in the corresponding map. Here, dimension (1) represents the foreign vs. domestic dimension witnessed in the asymmetric solution. Dimension (2) resembles the Ford vs. GM dimension also seen in the asymmetric solution. However, dimension (3) does not relate to size, but rather to preference intensity when we correlate the average preference scores for the sample with this dimension. Missing from Copyright  2007 John Wiley & Sons, Ltd.

1

Dimension 2

Dimension 1

Figure 2.

0.5

the plot is any indication of the asymmetry of the data, as well as any differences in potency vs. vulnerability. The assumption in this symmetric solution is that the relationships between pairs of brands j and k are equal with respect to givers and takers of choice shares—an assumption that is rejected given the fact that the asymmetric R = 3 solution decreased the error sums of squares by 40.1 percent. The predicted table of conditional probabilities akin to Table 1(b) would be symmetric under this model, a result that clearly runs counter to the empirical asymmetry observed in Table 1(b). Finally, Figure 3 presents the entire threedimensional asymmetric configuration together in one plot. Here, we also constructed the 50 percent predicted competitive sphere (in three dimensions) Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

Research Notes and Commentaries

763

2 Ford Tempo

1.5

Chevy Corsica

Chevy Celebrity 1

Ford Taurus

Dimension 3

0.5

Chevy Corsica

Ford Taurus

Buick Century

Ford Tempo

0

Buick Century

Chevy Celebrity

Toyota Camry

-0.5

Honda Accord

Olds Cutlass Supreme Pontiac Grand AM

Ford T-Bird Toyota Camry

-1

Ford T-Bird Olds Cutlass Supreme Pontiac Grand AM

-1.5

Honda Accord

-2

-2.5 -2 -1 0 nsion

Dime

Figure 3.

1 2

2

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

Dimension 1

The estimated three-dimensional asymmetric joint space solution

centered around the Grand-AM (bold) as an illustration of the methodology.1 Here, all the brands in italics are indicative of serious competitive threats to Grand-AM as the sphere indicated the predicted area of at least 50 percent conditional consideration probability. That is, given Grand-AM is in the consumer’s consideration set, the sphere constructed here indicates those alternative brands that are also co-considered at least 50 percent of the time. As shown in Figure 2, the T-bird and Cutlass are predicted to be most competitive with the Grand-AM.

DISCUSSION Building on the important work of DeSarbo et al. (2006), we recognized that it is critical to discern 1 Here again, for ease of exposition we display competition for Grand-AM. Clearly, in an interactive graphic display on a computer screen one could easily display several such spheres simultaneously or in a user-friendly sequential manner that can be customized for the task at hand.

Copyright  2007 John Wiley & Sons, Ltd.

competitive asymmetry and have sought to devise a distance-based unfolding type MDS model to efficiently represent asymmetric competition. We demonstrate the proposed methodology with data on consumer choice sets regarding 10 brands of mid-size automobiles. The results for mid-size automobile data show that the asymmetric competition model performs better than the model that assumes symmetric competition. The asymmetric unfolding MDS plots (Figures 1 and 3) along with vincibility and potency indicators (Table 1b) provide a much richer picture of competition among brands than the corresponding symmetric analysis does (Figure 2). The vincibility indicators range from a low of 0.355 for Honda Accord to a high of 0.444 for Chevy Corsica. These values suggest that on average firms are fairly similar on vincibility. There is greater variation in the potency indicators, which varies from a low of 0.142 for Chevy Corsica to a high of 0.586 for Chevy Celebrity. Potency is the ability of the focal brand to make it to the consideration sets when other brands Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

764

W. S. DeSarbo and R. Grewal

are already in the consideration set. Thus, these numbers discriminate between strong brands such as Chevy Celebrity and Toyota Camry and weak brands such as Ford Tempo and Chevy Corsica (perhaps one reason why these two later brands no longer are currently offered in the market). The implications of the research for several strategic decisions, such as positioning of new brands, brand repositioning, and brand portfolio management, are considerable, as we develop below for one of the brands in the study. Let us consider Figure 3 and the example of Pontiac Grand-AM to examine brand positioning and repositioning issues. The primary competitive threat to the Grand-AM comes from GM’s Cutlass and Ford’s T-Bird. Clearly, GM has to worry about cannibalization as it would not be a good idea to introduce a new brand positioned closed to Grand-AM. In terms of competition between Grand AM and T-Bird, it is evident from Figure 1 that on Dimension 1 T-Bird competes more intensely with Grand-AM than Grand-AM does with T-Bird. The two brands have very little competition on Dimension 2, while on Dimension 3 the intensity of competition between the two brands seems to be fairly symmetric. Thus, given the cannibalization issues and competitive weakness of Grand-AM vis-`a-vis T-Bird, it is in GM’s interest to either reposition Grand-AM or Cutlass to compete effectively with T-Bird. To stress the original motivation for this research, the procedure espoused by DeSarbo et al. (2006) would have required managers to wade their way through 10 maps to discern the insights that the proposed procedure provides in a single map. The benefits for today’s time-deprived managers of an efficient representation of competitive asymmetries could be substantial. For example, from a brand portfolio perspective, the mid-size automobile dataset consists of three GM brands, three Ford brands, two Chevy brands, one Honda brand, and one Toyota brand. Based upon Figure 1, it is evident that aside from Ford Taurus, the three GM brands (Century, Cutlass, Grand-AM)2 and the remaining two Ford brands (T-Bird and Tempo) compete intensely with each other on the first dimension. On Dimension 2, the GM brands are 2 GM no longer manufactured Olds mobile products after the 2003 models.

Copyright  2007 John Wiley & Sons, Ltd.

on the top half and the Ford brands are on the bottom half, signaling little competition between the two firms on this dimension. On the third dimension, the GM and Ford brands compete with each other, with one exception. The exception is that Ford Tempo does not compete with the GM brands, but the GM brands do compete with Ford Tempo (a manifestation of competitive asymmetry and a competitive weakness of the Ford Tempo). Here again, the advantages of the proposed efficient approach over DeSarbo et al.’s (2006) methodology for brand portfolio management are evident as portfolio management would require managers to scrutinize several competitive maps. Indeed, we believe that our research complements that of DeSarbo et al. (2006) in proving insights for managers concerning several strategic decisions such as brand position and brand portfolio management.

ACKNOWLEDGEMENTS We wish to thank the editor and two anonymous reviewers for their constructive comments, which helped to improve the contribution of this manuscript for SMJ.

REFERENCES Chino N, Shiraiwa K. 1993. A geometrical structures of some non-distance models for asymmetric MDS. Behaviormetrika 20(1): 35–47. DeSarbo WS, Jedidi K. 1995. The spatial representation of heterogeneous consideration sets. Marketing Science 14(3): 326–342. DeSarbo WS, Manrai AK. 1992. A multidimensional scaling methodology for the analysis of asymmetric proximity data in marketing research. Marketing Science 11(1): 1–20. DeSarbo WS, Rao VR. 1986. ‘GENFOLD2: a new constrained unfolding model for product positioning. Marketing Science 5(1): 1–19. DeSarbo WS, Grewal R, Wind J. 2006. Who competes with whom? A demand-based perspective on identifying and representing asymmetric competition. Strategic Management Journal 27(2): 101–129. Fleiss JL, Levin B, Paik MC. 2003. Statistical Methods for Rates and Proportions. Wiley: New York. Gowers JC. 1977. The analysis of asymmetry and orthogonality. In Recent Developments in Statistics, Barra JR, Brodeau F, Romier G, van Cutsen B (Eds). North-Holland: Amsterdam; 109–123. Harshman RA, Green PE, Wind Y, Mundy ME. 1982. A model for the analysis of asymmetric data in marketing research. Marketing Science 1(2): 205–242. Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

Research Notes and Commentaries Kamakura W, Russell GJ. 1989. A probabilistic choice model of market segmentation and elasticity structure. Journal of Marketing Research 26(4): 379–390. Okada A, Imaizumi T. 1987. Nonmetric multidimensional scaling of asymmetric proximities. Behaviormetrika, 14(21): 81–96. Okada A, Imaizumi T. 1997. Asymmetric multidimensional scaling of two-mode three-way proximities. Journal of Classification 14(2): 195–224. Peterson RA, Balasubramanian S, Bronnenberg BJ. 1997. Exploring the implications of the Internet for consumer marketing. Journal of the Academy of Marketing Science 25: (Fall): 329–346. Saito T. 1991. Analysis of asymmetric proximity matrix by a model of distance and additive terms. Behaviormetrika 18(29): 45–60. Siddarth S, Bucklin RE, Morrison DG. 1995. Making the cut: modeling and analyzing choice set restriction in scanner panel data. Journal of Marketing Research 32(3): 255–266. Zielman B, Heiser WJ. 1996. Models for asymmetric proximities. British Journal of Mathematical and Statistical Psychology 49: 127–146.

APPENDIX: A TECHNICAL DESCRIPTION OF THE ESTIMATION ALGORITHM The objective of the estimation procedure is to estimate X, Y , a, b, given P and a value of the dimensionality (R) so as to minimize the error sums of squares in (2): =

T





R


(Xj r − Ykr ) − bt

.

(A.1)

r=1

The alternating least-squares procedure utilized to iteratively estimate the model parameters is summarized below: A. Generate starting values. T  Set I T = 0. Calculate Qj∗k = T1 Qj kt and pert=1 

form a singular value decomposition of Q∗ Q∗ in R dimensions to obtain a B × R matrix of brand coordinates: Z. Now, define X (0) = Z + ε1

(A.2)

= Z + ε2

(A.3)

Y

(0)

Copyright  2007 John Wiley & Sons, Ltd.

B. Set IT = IT Y 1. Calculate: Mj k =

R


(Xj r − Ykr )2

(A.4)

r=1

Qj∗∗kt =

Qj kt − bt at

(A.5)

for all t = 1, . . . , T . Then, perform T simple regressions to update values of at and bt : IT

bˆt = (V  V )−1 V  S t (A.6) aˆ tI T where V = (1, W ); W = vec(M), where M = ((Mj k )); S t = vec(Q∗∗ ), for the tth time period; t Q∗∗ = ((Qj∗∗kt )). t C. Estimate X .IT / . Compute the partial derivatives: T

2 2

where ε1 , ε2 are each B × R matrices whose elements are randomly generated from an N(0,1) distribution. Set a = 1, b = 0.




∂ =2 (Qj kt − at (Xj r − Ykr )2 ∂Xj r t=1 k=j r=1

Qj kt

j =k

t=1

−at




B

765

R

B

− bt )(2at (Xj r − Ykr )) =4

T
B


(Qj kt − at

t=1 k=j

R


(Xj t − Ykr )2

r=1

− bt )(at (Xj r − Ykr )).

(A.7)

A conjugate gradient estimation procedure using a quadratic line search procedure is utilized to estimate X I T within MIT iterations (user specified). D. Estimate Y .IT / . Here we calculate the partial derivatives:


∂ = −4 (Qj kt − at (Xj r − Ykr )2 ∂Ykr t=1 j =k r=1 T

R

B

− bt )(at (Xj t − Ykt ))

(A.8)

Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj

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W. S. DeSarbo and R. Grewal

and we utilize a similar conjugate gradient procedure with quadratic line search to estimate Y (I T ) within MIT iterations.

B. As with the vast majority of deterministic MDS procedures, dimensionality selection is determined by a scree plot of VAF(R) vs. R and subsequent interpretation.

E. Test for convergence. We compute a variance accounted for statistic (VAF) or R 2 between Qj kt and Qˆ j kt , where Qˆ j kt = aˆ t

R


(Xˆ j r − Yˆkr )2 + bˆt

(A.9)

r=1

as T
B



VAF = 1 −

(Qj kt − Qˆ j kt )2

t=1 j =k T
B



(A.10) (Qj kt − Qj kt )2

F. Modeling issues. All MDS procedures suffer from various solution indeterminacies and the present procedure is also affected. In particular, there are R indeterminacies with respect to the origin of (X, Y ) since centering does not affect Euclidean squared distance. In addition, one can orthogonally rotate X and Y by the same non-singular orthogonal rotation matrix (R(R − 1)/2 indeterminacies) and not affect such squared distances. Finally, in the two-way case (T = 1), at is not identifiable since the square root of its value can be embedded in X and Y directly.

t−1 j =k

where Q..t =


B
j =k

Qj kt /(B 2 − B).

(A.11)

Convergence is attained if VAFI T − VAFI T −1 ≤ 0.0001. If convergence is not attained, go to step

Copyright  2007 John Wiley & Sons, Ltd.

Strat. Mgmt. J., 28: 755–766 (2007) DOI: 10.1002/smj