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Chapter 6
Competitive Maps

We are concerned in this chapter with the conversion of analytical results into decision-related factors. Any such discussion must center on the use of elasticities. A marketing plan should contain an instrument-by-instrument account of the actions to be taken - what prices to charge and what discounts to offer, what print ads to commission, how much to allocate to radio and television, what coupons to offer as a manufacturer and what coupons to cosponsor with retailers. To make these decisions on a sensible basis one needs to estimate the expected market response to changes in each of these elements of the marketing mix. Elasticities provide that instrument-by-instrument account of expected market response. The marketing literature provides many examples. Research on optimal advertising expendituresDorfman, Robert A. & Peter D. Steiner [1954], ``Optimal Advertising and Optimal Quality,'' American Economic Review, 44 (December), 826-36. Clarke, Darral G. [1973], ``Sales-Advertising Cross-Elasticities and Advertising Competition,'' Journal of Marketing Research, 10 (August), 250-61. Bultez, Alain V. & Philippe A. Naert [1979], ``Does Lag Structure Really Matter in Optimal Advertising Expenditures?'' Management Science, 25, 5 (May), 454-65. Magat, Wesley A., John M. McCann & Richard C. Morey [1986], ``When Does Lag Structure Really Matter in Optimal Advertising Expenditures?'' Management Science, 32, 2 (February), 182-93.provides many insightful special cases of the general, elasticity-based rule that you allocate resources to advertising in proportion to the marginal effectiveness of advertising in generating contributions to profits. Similarly, the literature on optimal price,Bass, Frank M. & Alain V. Bultez [1982], ``A Note On Optimal Strategic Pricing of Technological Innovations,'' Marketing Science, 1, 4 (Fall), 371-78. Kalish, Shlomo [1983], ``Monopolistic Pricing With Dynamic Demand and Production Cost,'' Marketing Science, 2, 2 (Spring), 135-59. Rao, Ram C. & Frank M. Bass [1985], ``Competition, Strategy, and Price Dynamics: A Theoretical and Empirical Investigation,'' Journal of Marketing Research, XXII (August), 283-96.and general issues of optimal marketing mix and marketing effectivenessLambin, Jean-Jacques, Philippe A. Naert, & Alain V. Bultez [1975], ``Optimal Market Behavior in Oligopoly,'' European Economic Review, 6, 105-28. Karnani, Aneel [1983], ``Minimum Market Share,'' Marketing Science, 2, 1 (Winter), 75-93. Morey, Richard C. & John M. McCann [1983], ``Estimating the Confidence Interval for the Optimal Marketing Mix: An Application to Lead Generation,'' Marketing Science, 2, 2 (Spring), 193-202.use elasticities in describing optimal decisions.

The problem is that much of this literature deals with an elasticity as if it were a fixed quantity or at best a random variable whose expected value completely summarizes it. Even if some of these studies oversimplify the case for expository purposes, managers seem most often interested in the single number that represent their brand's price elasticity or advertising elasticity.

Studies of the relation between price and product life cycleParsons, Leonard J. [1975], ``The Product Life Cycle and Time-Varying Advertising Elasticities,'' Journal of Marketing Research, XVI (November), 439-52. Simon, Hermann [1979], ``Dynamics of Price Elasticities and Brand Life Cycles: An Empirical Study,'' Journal of Marketing Research, XVI (November), 439-452. Shoemaker, Robert W. [1986], ``Comments on Dynamics of Price Elasticities and Brand Life Cycles: An Empirical Study,'' Journal of Marketing Research, XXIII (February), 78-82.deal with expected changes in elasticities as a function of the developmental stage of a product. Such efforts are steps in the right direction, for they recognize that elasticities are not fixed and invariant. This is a view we support. First, we have seen in Figures 2.1 and 3.3 four functional forms describing the changes in elasticities with changing market conditions. Second, we believe that managers should be interested in the whole changing pattern of elasticities and cross elasticities in a market. So instead of a single estimate of own-price elasticity, we advocate dealing with a matrix of price elasticities in each time and place.

In this chapter we present ways to visualize and understand the complex pattern of elasticities. A special case of multimode factor analysis portrays the systematic structure driving changes in these asymmetric cross elasticities. Continuing the coffee-market example introduced in the last chapter, we will see that the analysis of the variation in elasticities over retail outlets and weeks reveals competitive patterns showing that this market shifts during sales for the major brands. Analysis of the brand domain results in a map with each brand appearing twice. One set of brand positions portrays how brands exert influence over the competition. The other set of points portrays how brands are influenced by others. The interset distances (angles) provide direct measures of competitive pressures.

Our approach is founded on an understanding of how elasticities reflect the competitive structure in a market. Elasticities serve as measures of competition - indicators of market structure. We combine the visual emphasis coming out of psychometrics with the mathematical emphasis of economics.

The traditional psychometric approach to market-structure analysis has developed mainly without reference to or use of elasticities. This approach uses multidimensional scaling (MDS) to map perceived similarities or preferences among the brands, or to model consumer choice as some function of how far each brand is from the most preferred position in a brand map.For a summary of this research see Cooper, Lee G. [1983], ``A Review of Multidimensional Scaling in Marketing Research,'' Applied Psychological Measurement, 7 (Fall), 427-50.People are asked one of the most neutral questions in all of social science (e.g., ``How similar are the brands in each pair?''), and MDS provides from the answers powerful visual methods for portraying the dimensions underlying consumer perceptions. In new-product research, MDS provides a basis for understanding how consumers might react to new offerings. But, for the management of mature brands, particularly frequently purchased branded goods (FPBGs), this approach's power of discovery contributes to an important representational problem. Depending on the context, substitutable and complementary products could both appear close together in the perceptual space. For example, hot dogs and Coke (complements) could be near each other in one perceptual space, while Coke and Pepsi (substitutes) could be close together in another perceptual space.Factor analysis of consumer rating scales has less difficulty here than does MDS, since the attributes of substitutes should be much more highly correlated than the attributes of complements - leading to similar locations for substitutes, but quite dissimilar positions for complements. ince substitutes are competitors and complements are not, we would want these very different products to take very distinct positions in any visual representation. While careful consideration of market boundaries is helpful, the ambiguity between the treatment of substitutes versus complements can diminish the utility of traditional MDS for brand management.

Approaches to market structure using panel data may be able to overcome this ambiguity. The record of interpurchase intervals available in panel data can reveal information about substitutes versus complements. In the extreme case, co-purchase of two brands on a single buying occasion indicates complementarity, while switching between brands with equal interpurchase times indicates substitutability.This can be muddled, however, since panel records are mostly of household purchases. Co-purchase may simply indicate that different members of the household like different brands. So-called super-position processes are discussed by Kahn, Barbara E., Donald G. Morrison & Gordon P. Wright [1986], ``Aggregating Individual Purchases to the Household Level,'' Marketing Science, 5, 3 (Summer), 260-68.Fraser and BradfordFraser, Cynthia & John W. Bradford [1983], ``Competitive Market Structure Analysis: Principal Partitioning of Revealed Substitutabilities,'' Journal of Consumer Research , 10 (June), 15-30. sed this kind of information in a panel-based index of revealed substitutability which they decomposed using principal components. But their method is more of a market partitioning than a competitive mapping. There is also a whole stream of research based on panel or discrete-choice data beginning with Lehmann'sLehmann, Don R. [1972], ``Judged Similarity and Brand-Switching Data as Similarity Measures,''Journal of Marketing Research , 9, 331-4.use of brand-switching data as similarity measures in MDS. The Hendry Model,Kalwani, Manohar U., & Donald G. Morrison [1977], ``Parsimonious Description of the Hendry System,'' Management Science , 23 (January), 467-77.the wandering vector model,Carroll, J. Douglas [1980], ``Models and Methods for Multidimensional Analysis of Preferential Choice (or Other Dominance) Data,'' in Ernst D. Lantermann & Hubert Feger (editors), Similarity and Choice , Bern: Hans Huber Publishers, 234-89. DeSoete, Geert & J. Douglas Carroll [1983], ``A Maximum Likelihood Method of Fitting the Wandering Vector Model,'' Psychometrika, 48, 4 (December), 553-66. Genfold2DeSarbo, Wayne S. & Vitalla R. Rao [1984], ``GENFOLD2: A Set of Models and Algorithms for GENeral UnFOLDing Analysis of Preference/Dominance Data,'' Journal of Classification, 1, 2, 147-86. Moore & Winer`sMoore, William L.& Russell S. Winer [1987], ``A Panel-Data Based Method for Merging Joint Space and Market Response Function Estimation,'' Marketing Science, 6, 1 (Winter), 25-42.use of panel data in Levine'sLevine, Joel H. [1979], ``Joint-Space Analysis of `Pick-Any' Data: Analysis of Choices from an Unconstrained Set of Alternatives,'' Psychometrika , 44, 85-92., and the powerful maximum-likelihood procedures in Elrod's Choice MapElrod, Terry [1987], ``Choice Map: Inferring Brand Attributes and Heterogeneous Consumer Preferences From Panel Data,'' Marketing Science, forthcoming.can be thought of as part of the general effort to develop market-structure maps from disaggregate choice data. Moore & Winer [1987] distinguish their effort by using a multiple-equation system to integrate panel data with market-level data, but only Fraser & Bradford [1983] specifically address the potential of panel data to resolve the representational ambiguity involving substitutes and complements.

The modeling efforts using aggregate (store-level) data have had difficulty dealing with both complements and substitutes. ShuganShugan, Steven M. [1986], ``Brand Positioning Maps From Price/Share Data,'' University of Chicago, Graduate School of Business, Revised, July. Shugan, Steven M. [1987], ``Estimating Brand Positioning Maps From Supermarket Scanning Data,'' Journal of Marketing Research, XXIV (February), 1-18.has developed methods to represent the market structure specifically implied by the demand function in the Defender model.Hauser, John R. & Steven M. Shugan [1983], ``Defensive Marketing Strategies,'' Marketing Science, 2, 4 (Fall), 319-360.This market-structure map contains price-scaled dimensions. The elasticities implied by the Defender model can be computed as simple relations among the angles brands make with these per-dollar dimensions. While brand positions have the advantage of relating directly to the rich strategic implications of the Defender model, choice sets have to be very carefully defined to screen out complements. Otherwise brands may be forced to have negative coordinates on some dimension. There is still uncertainty about the meaning of a negative coordinate on a per-dollar dimension. VanhonackerVanhonacker, Wilfried [1984], ``Structuring and Analyzing Brand Competition Using Scanner Data,'' Columbia University, Graduate School of Business, April.is one of the few to use elasticities to map market structure. He has worked on methods which result in two separate structural maps - one for negative cross elasticities and one for positive cross elasticities. But how we integrate information across these two maps is, as yet, unresolved.

Yet the full set of elasticities provides a very natural and conceptually appealing basis for portraying market and competitive structure. The brands could be represented as vectors from the origin. The stronger the cross elasticity between two brands, the more correlated those brand vectors should be. The more complementary two brands are in the market, the more opposite they should be in the map. If two brands do not compete at all (zero cross elasticity), their vectors should be at right angles (orthogonal). Thus the patterns of substitutability, complementarity, and independence could be represented in a single map. These are the properties of a map we would get by viewing cross elasticities as cosines (or scalar products) between brand positions in a multidimensional space. What we gain is a way of visualizing a whole matrix of elasticities - a competitive map.

6.1  *Asymmetric Three-Mode Factor Analysis

Consider a three-way array of the cross-elasticities as depicted in Figure 6.1. If we think of it as a loaf of sandwich bread, each slice summarizes the elasticities in a time period. The entries in each row of a slice of this loaf summarize how all the brands' prices (promotions) affect that row brand's sales. The entries in a column from any slice summarize how one brand's prices (promotions) affect all brands' sales. Then the elements in an elasticities matrix are represented by the inner (scalar) product of (1) a row from a matrix R reflecting how receptive or vulnerable each brand is to being influenced by some small number of underlying market-place forces, and (2) the elements in a row from a matrix C reflecting how much clout each brand has in the market. We can think of a series of basic factors that reflect the clout or power of brands in a market. How influential each basic source of power is in the overall clout of a particular brand would be reflected in the scores of that brand in the C matrix. Similarly, we can think of a series of basic factors that reflect vulnerability in a market. How influential each basic source of vulnerability is in the overall vulnerability of a particular brand would be reflected in the scores of that brand in the R matrix. A cross elasticity is the inner (scalar) product of the clout of one brand times the receptivity/vulnerability of the other brand.

After developing this representation for a single time period, it is generalized by assuming that the competitive patterns underlying time periods are related by simple nonsingular transformations (i.e., that the dimensions of a common space can be differentially reweighted and differentially correlated to approximate the pattern of influences in any single time period). Establishing a common origin and units of measure for the R and C matrices allow plotting in a joint space.

Figure 6.1: Three-Mode Array of Elasticities

Equation (6.1) represents the cross elasticities E in a particular time period t as the scalar product of a row space R (reflecting scores for brands on receptivity/vulnerability factors) for time period t and a column space C (reflecting scores for brands on clout factors) for time period t plus a matrix D of discrepancies (lack of fit).

Similar entries in the row space for t indicate similarities between brands in the way they are influenced by competitive pressures - receptivity or vulnerability. Similar entries in the column space for t indicate similarities between brands in how they exert influence on others - clout. So while the inner (scalar) product across R and C reflects cross elasticities, the inner product within R reflects similarity in the pattern of how brands are influenced (i.e., receptivity or vulnerability), and the inner product within C indicates similarity in the pattern of how brands exert influence on other brands (i.e., clout).

We can think of the row space for a particular period t as being related to a common row space R . The dimensions of the space for a particular period could be a simple reweighting (shrinking or stretching) of the dimensions of the common space and/or the dimensions of the common space might have to be differentially correlated to reflect what is going on in a particular period. The combinations of shrinking or stretching of each dimension and differential correlation between the dimensions of the common space to reflect a particular period can be summarized in a nonsingular transformation Q for relating the particular period t to the common row space R .

Similarly, we can think of the column space for t as a nonsingular transformation U of common column space C:

TuckerTucker, Ledyard R [1969], ``Some Relations Between Multi-Mode Factor Analysis and Linear Models for Individual Differences in Choice, Judgmental and Performance Data,'' Paper presented to the Psychometric Society Symposium: Multi-Mode Factor Analysis and Models for Individual Differences in Psychological Phenomena, April 18.developed this basic formulation as an extension of his pioneering work on three-mode factor analysis.Tucker, Ledyard R [1963], ``Implications of Factor Analysis of Three-Way Matrices for Measurement of Change,'' in Chester W. Harris (editor), Problems in Measuring Change, Madison: University of Wisconsin Press, 122-37. Tucker, Ledyard R [1966], ``Some Mathematical Notes On Three-Mode Factor Analysis,'' Psychometrika, 31, 4 (December), 279-311. Closely related developments are reported in Tucker, Ledyard R [1972], ``Relations Between Multidimensional Scaling and Three-Mode Factor Analysis,'' Psychometrika, 37, 1 (March), 3-27, and illustrated in Cooper, Lee G. [1973], ``A Multivariate Investigation of Preferences,'' Multivariate Behavioral Research, 8 (April), 253-72. In these latter two articles symmetric, brand-by-brand arrays make up each layer of the three-mode matrix, making the analysis a very general model for individual differences in multidimensional scaling. In the current context, individual differences are replaced by differences in the competitive mix from one time to another, and the symmetric measures of brand similarity are replaced by asymmetric measures of brand competition.Elasticities in each period are represented, in terms of the common row and column spaces, as the sum (over time-factors l = 1, ...,L) of the triple matrix-product - the row space times the appropriate layer of the core matrix G^(l) times the common column space. Each term in the triple product is weighted by a coefficient w_tl showing the association of each time period with each time-factor (like factor scores for time periods).

A joint space represents brand competition on each time-factor.See Kroonenberg, Pieter M. [1983], Three-Mode Principal Components Analysis: Theory and Applications, Leiden, The Netherlands: DSWO Press, 164-67. ach layer of the core matrix is diagonalized using singular-value decomposition:

where V^(l) contains the left principal vectors of a particular layer of the core matrix, Y^(l) contains the right principal vectors, and G^(l)2 is a diagonal matrix of singular values.

In this joint space, R^(l) reflects the similarities in how brands are influenced, C^(l) reflects the similarities in how brands exert influence, and the proximity (cosine between the vectors) of row and column points reflects how much the brands compete. From the joint-space coordinates, we approximate the elasticities corresponding to any particular week or any simulated pattern of marketing activity, using:

Thus if the analysis of differences in competitive contexts reveals particularly interesting patterns, we could approximate the elasticities which reflect those competitive conditions. Repeating the analysis on just the approximated elasticities for some special condition, we create a competitive map specific to this context. With only one layer this is a two-mode analysis which amounts to a singular-value decomposition of the E matrix, in which the variance is split between the left principal vectors and the right principal vectors. The result is the asymmetric three-mode equivalent of idealized-individual analysis developed by Tucker and MessickTucker, Ledyard R & Samuel Messick [1963], ``An Individual Differences Model for Multidimensional Scaling,'' Psychometrika, 28, 4 (December), 333-67.for the individual-differences model for multidimensional scaling. For any idealized competitive pattern t^* in which the elasticities have been approximated by equation (6.8), we get the simplest representation:

This provides a very direct visualization of the idealized elasticities because the inner product of the coordinates in C^(t*) for brand j and the coordinates in R^(t*) for brand i will reproduce the (ij) entry in [E\tilde]^(t*). These idealized competitive patterns are isolated and interpreted in the illustration that follows.

6.2  Portraying the Coffee Market

The price parameters from the market-share model for the Coffee-Market Example were used to generate market-share-price cross elasticities for each grocery chain in each week. The average elasticities are shown in Table 6.1. The greatest price elasticity is for Chock Full O'Nuts (-4.71). The clear policy of this brand is to maintain a high shelf price and generate sales through frequent promotions. Over 80% of Chock Full O'Nuts sales in these two cities are on price promotions. Chock Full O'Nuts maintains the third largest market share with this policy. The substantial price elasticities for Folgers, Maxwell House, and All Other Branded result from a similar policy, but with less frequent and less predictable price-promotions. Master Blend, Hills Bros., and Yuban have elasticities more like the private label brands (i.e., PL 1, PL 2, PL 3, and AOPL). Since the private-label brands have so little to offer other than price, we might expect them to have greater price elasticities. But with an every-day-low-price strategy these brands do not generate enough variation in price to achieve the elasticities of the more frequently promoted brands. As we see in the subsequent analyses, the average elasticities in Table 6.1 reflect an aggregation of widely differing competitive conditions. There are shelf-price elasticities which are quite different from the promotion-price elasticities one obtains during sales for the three major brands in these markets.

Table 1: Average Market-Share Elasticities of Price

6.2.1  Signalling Competitive Change

Three-mode factor analysis is a technique for structured exploration. Although elasticities help researchers understand the raw data, the average elasticities are too aggregate to reflect the diversity of the competitive environment. The first task of the analysis is to signal when particular competitive events are part of a systematic pattern. Knowing that say five particular weeks in particular grocery chains constitute a pattern, we can go back to the original data to seek the meaning of that pattern in the antecedent conditions (e.g., these are weeks of deep price cuts for Folgers). This reduces the noise, so that signals are more easily detected. Next, the three-mode analysis determines the competitive building blocks for the elasticities. There is a pair of matrices R and C for each time factor (l = 1, 2, ¼, L) in the matrix W = { w_tl }_{T ×L}. Equation (6.8) approximates the elasticities for any particular competitive pattern as a linear combination of the building blocks, where the entries in a row of W serve as the linear combining weights. While each pair of matrices R and C can be interpreted, we find it best to form the linear combinations implied by particularly interesting competitive patterns and interpret the competitive maps resulting from the approximated elasticities. The building blocks may include more dimensions than are operative in any particular competitive pattern, and thus may be more difficult to interpret.

The three-mode factor analysis in this illustration was implemented using the matrix-algebra routines in SAS^(R) . The limit of 32,767 elements in the maximum array size in SAS^(R) 's PROC MATRIX meant that only three of the seven grocery chains reporting each week could be analyzed.The PROC MATRIX routine is available from the authors. Real applications of much larger size are currently feasible. First, SAS IML apparently removes the size restriction. Second, a general three-mode program, developed by Pieter Kroonenberg, is available for a small fee from the Department of Data Theory, University of Lieden, P.O. Box 9507, 2300 RA Leiden, The Netherlands. This limitation resulted in exclusion of Private Label 3 and All Other Private Labels from the subsequent analyses, because these brands were not distributed in Chains 1 - 3.

We choose the number of dimensions to investigate by inspecting a plot showing how the proportion of variance explained by each factor trails off as the number of factors increases. Variance is information and we look for the last relatively large drop in variance - indicating all subsequent factors have relatively little information. Figure 6.2 shows this plot for the structure over chains and weeks. The factor structure over the 52 weeks for the three chains shows that the last large drop is between factors 4 and 5. Consequently we retain four dimensions, accounting for almost 93% of the variation over chains and weeks. The dominant first factor accounts for over 74% of the variance, but we must look for the last large drop, not the first. The next three factors account for 9%, 6.5%, and 3% respectively, while none of the remaining factors accounts for even 1%.Choosing factors in this way is of course judgmental, but has a long tradition in psychometrics. The idea is that there are three classes of factors in any data set - major-domain factors, minor, but systematic factors and error or random factors. While statistical criteria attempt to exclude the random factors, psychometricians traditionally have used factor analysis to isolate only the major domain. While the minor factors are present and systematic, exploratory factor analysis gives clearest focus on the major domain. Confirmatory factor analysis and latent-variable causal modeling, use statistical criteria for choosing the dimensionality, and are consequently more useful in better developed domains, where the minor-systematic influences are well known.

Figure 6.2: The Number of Factors Over Chains and Weeks

Figures 6.3 and 6.4 plot the weights w_tl showing the influence of the time-factors (l = 1 - 4) on the weeks t - with Figure 6.3 depicting the first two factors over time, and Figure 6.4 portraying factors 3 and 4 over time. The symbols \bigcirc (circles) and \Diamond (diamonds) correspond to grocery chains 2 and 3, respectively. The ^[¯] (boxes) some of which have letters inside, represent Chain 1. To help reflect the third dimension

Figure 6.4: Competitive Structure Over Stores - Factors 3 & 4

in these figures the size of the symbol decreases the farther away the observation is from the ``Week'' axis. The coefficients for the grocery chains are indicators of systematic structure of events in these weeks. Each of these four factors corresponds to a fundamental building block which collectively can represent any pattern of competition in the data.

The goal is to interpret the patterns of competition. First we note that it is easy to see that the grocery chains are quite distinct, reflecting substantial differences in promotion policies among chains. It is easier to summarize the differences over chains after we understand the pattern within a chain. Let us look at Chain 1. We are directed in this inquiry by the fact the original measures are price elasticities. Do the weeks that stand out correspond to recognizable price events ? The weeks marked by blank boxes (^[¯] ) indicate Shelf Prices - weeks in which there are no price promotions in Chain 1. ``F,'' ``M,'' and ``C'' indicate big price cuts for Folgers, Maxwell House, and Chock Full O'Nuts, respectively. Note that the weeks in Chain 1 which have high weights on the first factor reflect shelf-price competition - weeks in which no major brand is being promoted. These weeks had an average loading of about .11 on Factor 1 and -.08 on Factor 2, but very little weight on Factor 3 (-.02) or Factor 4 (-.04). These linear-combining weights were used to develop approximate or idealized shelf-price elasticities from the basic building blocks. These shelf-price elasticities were mapped and are discussed below.

The weeks with Folgers on sale have somewhat less weight on Factor 1 than the Shelf-Price weeks. Sale weeks for Chock Full O'Nuts have the weights closest to zero on Factor 2 (see Figure 6.3). Folgers sale weeks have slightly positive weights on Factor 3, while Chock Full O'Nuts sale weeks have the greatest negative weight (see Figure 6.4). This pattern is reversed in Factor 4, with Folgers sale weeks have the most negative weights and Chock Full O'Nuts sale weeks having strong positive weights. The positive weights on dimension 3 for Folgers are neutralized when Folgers and Maxwell House promote simultaneously. These weeks, marked ``FM'' in Figure 6.4, look much more like shelf-price weeks, indicating head-to-head promotions partially cancel each other. In general, Maxwell House sale weeks have less weight on the first factor than the shelf-price weeks, but are otherwise relatively indistinguishable from the pattern for shelf-price weeks.

Now that we see that the patterns within a chain differentiate brand promotions from shelf-price conditions, we can look at the patterns across chains. The factor structure clearly differentiates Chain 2 as having less weight on the first, dominant factor (see Figure 6.3). Since the highest weights on this first factor reflect shelf-price conditions in Chain 1, we shouldn't be surprised to find out that Chain 2, which has the lowest weight on this factor, has the fewest weeks of shelf-price competition. Chain 2 tends to run features for majors for up to eight weeks in a row, and run features for more than one major brand head-to-head - leaving the fewest weeks without promotions.Chain 2 had over 36 deal-weeks for the three largest brands combined, compared to 19 deal-weeks for Chain 1 and 24 deal-weeks for Chain 3 for these brands.Chain 3 has the highest positive weights on the third factor (see Figure 6.4). This seems to correspond to the policy of having very high feature prices for the major brands.Chain 3 tends to feature Folgers or Maxwell House at about $2.32, while the corresponding price-on-feature is about $1.91 and $2.10 for Chains 1 and 2, respectively.So it seems that these differences in promotion policies are summarized in the relatively distinct locales for each grocery chain in this factor space. We can further note that within each general chain location, the brand promotions seem to have similar impacts. For instance, in Figures 6.3 and 6.4, the shelf-price weeks have the highest position on Factor 1 relative to the position of each chain, Folgers or Maxwell House promotions have the lowest positions on Factor 1 and the highest positions on Factor 3, while Chock Full O'Nuts promotions have the most positive weights on Factors 2 and 4 and the most negative weights on Factor 3. Similar to the interpretation of any factor structure, we must note the events which stand out and use the associated market conditions and other information to guide us in finding meaningful patterns. In this sense any factor structure is an information space into which the available data are reflected in the search for meaning. First we reflected chain number, and then promotions for different brands, and so on until we understand the pattern. For all the power of the analysis, it still takes the ability of the analyst to recognize substantive patterns.

When we isolate a pattern of interest - a particular week or average of similar weeks, we simply note its coordinates. From these we can use equation (6.9) to create idealized elasticities corresponding to the competitive pattern. So far we have focused on Chain 1 and noted the patterns associated with shelf-price weeks, Folgers-on-sale weeks, Maxwell House-on-sale week, and Chock Full O'Nuts-on-sale weeks. RossRoss, John [1966], ``A Remark on Tucker and Messick's Points of View Analysis,'' Psychometrika, 31, 1 (March), 27-31.advises that idealized individuals be placed very near the positions of real individuals to minimize the possibility that averaging several locations could create unreal dimensional structures. Following this advice, the idealized shelf-price elasticities correspond to the coordinates in the W matrix for Chain 1 Week 2, the idealized Folgers-on-sale elasticities come from the coordinates for Chain 1 Week 1, the Maxwell House-on-sale elasticities come from Chain 1 Week 11, and the Chock Full O'Nuts-on-sale elasticities come from Chain 1 Week 25. The linear-combining weights for these patterns are shown in Table 6.2. The idealized elasticities for these four competitive patterns appear in the appendix (see Table 6.7). The competitive maps are developed and interpreted in the next section.

Table 6.2: Coordinates of the Idealized Competitive Conditions

6.2.2  Competitive Maps: The Structure Over Brands

The common scaling space developed by a three-mode factor analysis of asymmetric cross elasticities, as well as the two-mode representations of idealized competitive patterns, both provide maps of competitive interactions, rather than necessarily portraying attribute relations among the brands. These are competitive maps, rather than product perceptual spaces. In this illustration all the maps relate to price as an attribute, since price elasticities are used to develop the maps. In applied contexts the maps derived from all other promotional instruments would be investigated.

A competitive map involves two sets of points plotted in the same space, corresponding to the two processes reflected in the elasticities. Elasticities show how a percent change in the price of a brand, j , translates into change in the market share of brand i . The first process involved deals with how much clout brand j has. We can think of brands which just seem more able to influence others, or brands which pressure no others. Correspondingly, the first set of points - symbolized by circles \bigcirc - represents the way brands exert influence on one another. Similar positions for two brands indicate they exert a similar pattern of influence on the market place. The second process deals with how able some brands are to resist the advances of competitors, while others seem quite vulnerable. And so the second set of points - symbolized by squares ^[¯] - represents the way brands are influenced by competitive pressures. Similar positions for two brands indicate they are similarly vulnerablility to pressures from other brands.

Most joint-space, multidimensional-scaling methods deal with different rows than columns (e.g., with brands for columns and consumer preferences for row the joint space would locate ideal points in a brand map). But this multidimensional-scaling method has some very special properties. Because we are using ratio-scale quantities (i.e., elasticities have a meaningful zero-point and unit of measure), the origin of the space has great importance. Most MDS methods are based on interpoint distances and the origin is arbitrarily placed at the centroid of all the brands - merely for convenience since the interpoint distances are unaffected by a translation of origin. But this model doesn't work with estimates of interpoint distances. Brands are vectors from a fixed origin. For the \bigcirc brands, the distance of a brand from the origin of the space is a measure of how much clout the brand has.Formally it is a function of the sum of squares of the cross elasticities of other brands' shares with respect to this brand's price.For the ^[¯] brands, the distance from the origin is a measure of how vulnerability or receptive a brand's sales are to price competition.Formally it is a function of the sum of squares of the cross elasticities of this brand's share with respect to the other brands' prices. Elasticities reflect percentage changes. So if a brand with small share can lose a large percent of its share to other brands, it can appear far from the origin and be very vulnerable in percentage terms.Hence, two \bigcirc brands on the same vector from the origin exert the same pattern of pressure on the other brands, but differ in the amount of clout each possesses. Two ^[¯] brands on the same vector from the origin are pressured by the same competitors, but could be differentially vulnerable or receptive. A \bigcirc brand on the same vector as a ^[¯] brand would exert its greatest cross-elastic pressure on that ^[¯] brand. The cross elasticity falls off as the cosine between the angles of the brands drops toward zero (brands at right angles). Brands on opposite sides of the origin (angles greater than 90^°) reflect complementary cross elasticities, rather than competitive pressures.

In the ground, caffeinated coffee market under study there are four major dimensions describing the relations among brands on each time-factor. The dimensionality is chosen, as before, by inspecting the variance accounted for by each dimension. In the appendix to this chapter Table 6.3 shows the coordinates of the brands in the common row scaling space and the common column scaling space, Table 6.4 lists the variance on each of the common dimensions for the row space and the column space, Table 6.5 displays the four planes in the core matrix G, and Table 6.6 lists the coordinates of the joint-space, building blocks corresponding to each core plane. Even though these building blocks are four-dimensional, the linear combinations representing each of these four special cases are three-dimensional. Any particular competitive pattern need not involve all the basic factors.

Figure 6.5 portrays the competitive map accounting for 98.7% of the idealized shelf-price elasticities in Table 6.7. The size of the symbol for each brand represents the distance from the fixed origin of this space. This reflects the clout or receptivity/vulnerability of the brand. We see that Folgers and Maxwell House exert a similar pattern of pressure, with Maxwell House having more clout at shelf prices. They are both aligned to exert the greatest pressure on the premium brands in the All-Other-Branded category, which are quite vulnerable to their attack. Even though ^[¯] Chock Full O'Nuts is separated from \bigcirc Folgers and Maxwell House by a sizeable angle, its extreme receptivity translates into its being strongly pressured by both Folgers and Maxwell House. The almost 180^° angle between Folgers \bigcirc and Folgers ^[¯] indicates that Folgers helps itself quite directly with price cuts. The most extreme example of this involves Chock Full O'Nuts which has a great deal of clout and is very vulnerable. The pattern in Figure 6.5 shows Chock Full O'Nuts competing much more with Folgers than with Maxwell House, while being very receptive to its own price moves. \bigcirc Master Blend is positioned to exert its greatest pressure on Folgers at shelf prices. But Folgers is not in the best position to return the pressure on either Regular Maxwell House or Master Blend.

Yuban (unlabelled in the figure) resides at the fixed origin of this space, and the private-label brands PL 1 and PL 2 sit very near the origin, exhibiting no role in the competitive interplay. This is expected since the shelf-price map reflects an idealization of condition in Chain 1 which does not distribute Yuban or its own private label. If we were to create idealized elasticities closer to the position of a shelf-price week in Chain 2, which distributes this brand, it might behave more like the other premium brands in the AOB category. If we were to create idealized elasticities closer to the position of a shelf-price week in Chain 3, we

would see a greater role for the private labels.It has already been noted that Chain 3 has a much higher feature price for the major brands than the other chains. Combined with the very frequent in-store displays and lower shelf prices for its private-label brand, the policy seems to be to draw shoppers in with a major-brand feature and encourage them to switch through the displays at the coffee aisle. hen Folgers goes on sale the pattern in Figure 6.6 is operative (accounting for 99.8% of the corresponding idealized elasticities in Table 6.4). First, note that Folgers has less clout on sale. Shelf-price elasticities are like potential energy. When the brand actually goes on sale some of this energy is dissipated, in this case by being translated into sales. The reduction in the angle between \bigcirc and ^[¯] for Folgers on sale indicates at least a small dissipation of Folgers' influence over its own market share. An approximately 90^° between \bigcirc and ^[¯] for Folgers would indicate that Folgers cannot help itself by further price reductions. On sale, Folgers is substantially less vulnerable to both Chock Full O'Nuts and AOB. Folgers can still attack the All-Other-Branded category and Chock Full O'Nuts, but the reduced shares for these brands during a Folgers promotion provide less incentive to Folgers.

When Maxwell House goes on sale Figure 6.7 depicts the action. As clearly indicated Maxwell House puts the greatest pressure on All Other Branded. While the premium brands which make up the AOB category possess considerable clout of their own, and are aligned to be able to help themselves, they are not particularly well aligned to return the pressure on Maxwell House. Only Chase & Sanborne is aligned for counter attack and potent enough to be a threat. Chock Full O'Nuts is a potent force under these market conditions, but is aligned to impact Folgers much more than it can impact Maxwell House. Note that in the one week of coincident promotions for Maxwell House and Chock Full O'Nuts (Week 24), the coefficients look like those for Chock Full O'Nuts sale weeks, rather than other Maxwell House sale weeks (see Figures 6.3 and 6.4). Brand managers for Maxwell House might do well to incorporate this information into planning the timing of their promotional events.

When Chock Full O'Nuts goes on sale (see Figure 6.8) it exerts a great deal of pressure directly on Folgers, which is vulnerable to the attack. Chock Full O'Nuts also pressures Hills Bros. and Master Blend. Maxwell House seems not very vulnerable to Chock Full O'Nuts on sale. Chock Full O'Nuts is not vulnerable to counter attack, although Chase & Sanborne's alignment makes it the most potent threat. Almost all the brands have near-zero loadings on the second dimension - making this

map almost two-dimensional.

Overall, these patterns show substantial asymmetries which would not have been revealed by any other market-structure map.

There are very simple relations between the maps and the idealized elasticities in Tables 6.7. As implied by equation (6.9), one need only multiply the clout coordinate of brand j times the receptivity coordinate of brand i, and sum over dimensions to produce the elasticity of brand i's market share with respect to brand j's price. For those who are more comfortable with maps than with matrices, these maps provide a visual representation of the richly asymmetric competitive patterns resulting from price changes in the coffee market. As in other categories of frequently purchase branded goods, price is used as a major weapon of promotional strategy. What one can read from these maps is what brands constitute the major threats to others with their price policy and where the major opportunities for competitive advantage may reside. These maps offer many signals which are new and very different from the market-structure maps of the past. Only a very few of these signals have been mentioned in this illustration. The full meaning of these signals is better interpreted by managers and management scientists involved in these markets than by academic researchers involved in methods development.

6.3  *Elasticities and Market Structure

The value of much of the developments so far rests on the propriety of using cross elasticities to reflect market structure. There are at least three deficiencies to elasticities as measures of competition:Thanks are due to an anonymous reviewer for Management Science who pointed out these potential problems.1) they are static measures as they assume no competitive reaction to change in a marketing-mix variable, 2) because they are static measures, they do not account well for structural change in markets, and 3) they can be difficult to measure when price changes are infrequent or are of low magnitude.

First, both historic lags and competitive reactions can be included in elasticity calculations. This was indicated by Hanssens,Hanssens, Dominique M. [1980], ``Market Response, Competitive Behavior, and Time Series Analysis,'' Journal of Marketing Research, XXVII (November), 470-85.although misspecification of his equation (2) precluded computation. Lagged influences on brand i 's market share can be represented as e_ijt*_t^(k). This is the influence that brand j 's price (k^th marketing instrument) in historic time-period t^* has on brand i 's market share in period t. But for a competitive reaction to influence a current elasticity, a combination of events must occur. There must be an action involving marketing instrument k¢ by some brand i¢ in historic period t¢ which produces a significant price reaction by brand j in some historic period t^*, and there must be a nonzero elasticity for the effect of brand j 's price in period t^* on brand i 's market share in period t. We can represent the reaction elasticity as e<sub>k¢i¢t¢[`R]kjt</sub>*, where the subscripts before [`R] indicate the antecedents producing the reaction, while the subscripts after [`R] indicate where the reaction occurs. Then the market-share cross elasticity is represented as:

where h is the maximum relevant historic lag. Note that if either e_{k¢i¢t¢[`R]kjt}* or e_ijt*_t^(k) is zero, the entire term makes no contribution to e_ijt^(k) . In the current illustration there were no significant lagged effects (or cross effects) on market share, nor were there any significant competitive reactions. Given the highly disaggregate nature of the modeling effort (i.e., modeling marketing shares for brands in each grocery chain each week) and the irregular timing of major promotions, the absence of such effects is not surprising.

Second, exploration of the time mode (chain-weeks in the current illustration) can help minimize the limitations of elasticities in reflecting structural changes. While a regular lag structure may not be evident, one of the attractive features of the three-mode factor analysis is in its ability to highlight structural events which occur at irregular intervals over the study period. In the coffee market, promotions for major brands signalled the big structural changes. These will most likely occur at irregular intervals to minimize competitive reaction as well as consumers delaying purchase in the certain anticipation of a sale for their favorite brand.

Third, if there is too little variation in a marketing instrument, elasticities can be hard to estimate accurately. In the retail coffee market there are very frequent price promotions, features, displays, and considerable couponing activity. About 50% of all sales are made on a promotion of some kind. But in warehouse-withdrawal data, more temporally or regionally aggregated data, or in categories with less frequent retail promotions (e.g., bar soap), lack of variation would be more of a concern.

The benefits of this style of analysis become clearer when we consider the task of intelligently using scanner data for brand planning. We could plot sales, prices, features, displays, and coupons for each brand, each chain and each week. But the points of information become so numerous that without further guidance, the ability to assimilate soon suffers. In the current illustration this would entail a scatter plot for each brand in each chain over 52 weeks or a pie chart summarizing each week in each chain over all the brands. Market-response models provide an enormous concentration of information. But how do we assimilate the implications of a market-response model? Simulations and forecasts are very valuable, but they reverse the concentration of information achieved by the market-response model. For each simulation run we must track the competitive strategies of all brands as well as all the outcomes - estimates of sales and profits for all manufacturer and all retailers.

The parameters of the market-response model can be a source of insight. We could even factor the matrix B using the model in equation (6.9). While we might obtain some sense of the structure of competitive forces, we would have no idea of how that structure changes with changes in competitive patterns. The notion of reflecting changes is one of the most basic and appealing features of elasticities.

Elasticities can provide quantitative understanding of a market. Like simulations, however, elasticities reverse the information concentration achieved by market-response models. Using asymmetric three-mode factor analysis summarizes the 22,464 elasticities (3 Chains × 52 Weeks × [12 × 12] Brands) into only two plots for the factors differentiating grocery chains over weeks, and a plot for each of the idealized competitive situations. The plots representing the structure over time can be helpful in planning the timing of promotions, tracking promotional effectiveness, and detecting promotional wear-out. Looking at over-time patterns helps counteract some of the limitations imposed by the static nature of each week's elasticity estimates. In the analysis of these data aggregated to the weekly level over grocery chains, these plots have revealed the market-wide expansion of elasticities in key pay-weeks. It is during the key-pay week that all Federal checks (Social Security, Aid to Families with Dependent Children, welfare, government pensions, etc.) arrive. These Federal checks are often cashed at supermarkets and then banked in the form of food purchases for the month. These first-week customers, typically being of limited means, are some of the most price-sensitive shoppers and are purchasing a disproportionate share of their monthly needs in this week.Special thanks is due to J. Dennis Bender and John Totten for helping me understand the meaning and significance of this finding.Other applications could reveal temporal or seasonal patterns of interest. This format signals what are the systematic structural events in a dataset which otherwise might be too large to explore. The structure over brands is contained in figures describing the idealized competitive contexts which characterize this market. Shelf-price competition, and the structure of competition during sales for each of the three largest-selling brands, are differentiated in a manner which could never be detected from the average elasticities. The planning exercise in Chapter 7 shows how the competitive maps can be used to help focus the path of inquiry and limit the number of simulations.

6.4  *Interpretive Aids for Competitive Maps

The limitations of this approach stem mainly from its being descriptive, rather than prescriptive. The dimensions of a competitive map describe the terrain. While there is no guarantee that the map can easily be labelled in terms of brand attributes, the structure over brands could perhaps be made more useful if ideal points or property vectors were located in the space. Ideal points describe the most preferred position in a perceptual map. Property vector describe directions such as the direction of increasing economy or sportiness often seen in maps of the car market. Two special logit modelsLee G. Cooper & Masao Nakanishi [1983b], ``Two Logit Models for External Analysis of Preferences,'' Psychometrika, 48, 4, 607-20. ere designed to do this for competitive maps. These models provide an external analysis of preference, perceptions, or brand attributes. Internal analysis of preferences attempts to develop both the brand map and ideal points from ratings or ranking of consumer preferences. In external analysis of preferences we must have a pre-existing brand map, and we simply wish to estimate the most popular region(s) of the map. The competitive map contains the scale values for the brand - fulfilling the need for a pre-existing brand map, while sales data provide the relative choice frequencies required for locating the ideal points for each week.The relative choice frequencies were originally supposed to come from paired-comparison judgments gathered in an experimental setting, but we can substitute the number of choices (sales) for on brand compared to the number of choices (sales) for the other brand for the required paired comparisons as long as most of the buyers purchase a single unit (pound of coffee, for example) at a time.

The basic idea of this style of analysis is that preference increases as brands get closer to an ideal point in a map. We can consider a distance function d_i^* which reflects how far brand i is from the ideal location.

where:

X_hi = the known coordinates of brand i on the given dimensions h = 1, 2, ¼, H of a map

X_h^* = the unknown coordinates of the ideal point on dimension h, and

b_h = the unknown weights reflecting the importance of dimension h in capturing the preferences.

The statistical problem is to be able to estimate the ideal coordinates X_h^* and the dimensional importance weights b_h from the relative choice frequencies. Let [`f]_ij be the expected relative frequency of choice of brand i over brand j .

The model for [`f]_ij is given by:

where d_i^* is as previously defined and e_i is a log-normally distributed specification-error term, unique to the alternative.The function in equation (6.11) is formally derivable from the generalized extreme value distribution for a random utility model. See appendix 2.9.3 and Yellott, J. I. [1977], ``The Relationship Between Luce's Choice Axiom, Thurstone's Theory of Comparative Judgments, and the Double Exponential Distribution,'' Journal of Mathematical Psychology , 15, 109-44, for further details.This model actually allows for three different views of the relations between ideal points and choices. If all the weights b_h are negative, we have the standard ideal-point model in which preference declines smoothly in any direction away from the ideal point. This is portrayed in Figure 6.9 with the ideal point at (0,0). But sometimes preferences are more readily characterized by what we don't like than what we do. In such cases all the weights b_h are positive and we can locate an anti-ideal point, the least desirable point. As is pictured in Figure 6.10 preferences increase as we diverge in any direction away from the anti-ideal point at (0,0). If some of the weights b_h are positive and others are negative, preferences are represented by a saddle point, as is shown in Figure 6.11. The classic example of saddle points involves tea drinks. Some people like iced tea and some like hot tea. But on the temperature dimension there is an anti-ideal point at tepid tea. On the sweetness dimension some people like two lumps of sugar and their preference declines if either more or less sugar is used. The combination of an anti-ideal dimension of temperature and the ideal dimension of sweetness creates a saddle point.

Figure 6.9: Preference and the Ideal Point

Figure 6.10: Preference and the Anti-Ideal Point

Figure 6.11: Preference and the Saddle Point

In addition to ideal dimensions and anti-ideal dimensions there are many dimensions on which only the most preferred direction is known. Price is an obvious example. The ideal price is almost always lower than any offered price. We know the direction of increasing preference but cannot isolate an ideal point. Such special cases are known as vector models.The ideals in a vector model do not have to be infinite. Any time the ideal is outside the configuration of points in a map a vector model may be more apt than an ideal point model.Corresponding to the logit ideal point model in (6.11) we have the following logit vector model.

where a_h is the coordinate of the preference vector on dimensions h (the greater the coordinate the more influence that dimension has on preference).

Both the logit ideal-point model and the vector model can be estimated by regression techniques. Taking the logit of the expected, relative choice frequencies, reveals a linear form

We can see how to estimate both the importance weight and the ideal coordinate on each dimension by noting

where the importance of each dimension is reflected in the parameter associated with the difference in squared scale values (b_h2 = b_h), and the ideal coordinate on each dimension can be found by solving a simple function involving the parameter associated with the difference in scale values (b_h1 = -2b_h X_h^*).

Taking the logit of the observed, relative choice frequencies reveals the estimation form of the ideal-point model as:

where u_ij is a stochastic-disturbance term which represents the combined influences of specification errors e_i and e_j and sampling error for the departure of the observed, relative choice frequency, f_ij , from its expectation, [`f]_ij .The nonspherical error-covariance matrix requires generalized least-square estimation procedures which are fully presented in Cooper & Nakanishi [1983b].

The logit vector model is estimated as:

The difference between these two models is simply that the ideal-point model has one additional parameter associated with the differences in the squares of the scale values. Thus it is very straightforward to test if the logit ideal-point model or the logit vector model is most appropriate. Estimation and reduced-model tests are discussed in the original article.

There are several uses for this style of analysis. First, sales data from each time period or region can be transformed into relative choice frequencies and used to locate ideal points. If panel data are available, we could locate ideal points for each segment. In surveys consumers could be asked to pick any of a variety of attributes which are descriptive of each brand. The relative choice frequencies could come from such pick-any data as well as from more traditional, paired comparisons. The added benefit is that we can readily assess if an attribute vector or an ideal point is most appropriate for each property fit into the competitive map.The ideal points or vector could be positioned with respect to either the scale values representing the clout of each brand or its vulnerability. For most applications it is probably appropriate to use the scale values representing clout. For sales data we would then see how clout translates into sales.

Although we may always attempt to imbed property vectors in the space to help interpret the dimensions, the fit may not be good even in a good map. A map does not tell us how to reposition a brand to avoid competitive pressures. It merely reflects what those pressures are.

Kamakura and SrivastavaKamakura, Wagner A. & Rajendra K. Srivastava [1986], ``An Ideal-Point Probabilistic Choice Model for Heterogeneous Preferences,'' Marketing Science , 5, 3 (Summer), 199-218.develop an alternative to the market-share model and the logit ideal-point model. They first create a probit version of the Cooper and Nakanishi logit ideal-point model (which they call the CN model), and then further generalize that model to estimate the distribution of ideal points in any choice situation. In so doing they end up with a model which can predict choice probabilities (market shares) in a way not bound by Luce's IIA assumption. Their models can be thought of as asymmetric, probit-based alternatives to asymmetric market-share models.

In comparing these efforts there are two technical issues and one general issue. Kamakura and Srivastava [1986] claim the CN model is a Luce-type model, subject to the problems inherent in the IIA assumption. When the CN model is used as they use it, this is true. But used as part of the system of models for competitive analysis, it is not true. In any system of models the problem of differential substitutability need only be solved once. In our case, the asymmetric market-share model solves this problem by estimating specific cross-competitive effects and by modeling the distinctiveness of marketing activities with zeta-scores or exp(z-scores). Both the competitive maps and the ideal points imbedded into them are not limited by Luce-type assumptions. The second technical point concerns the Kamakura and Srivastava claim that the CN model estimates only a single ideal point. The CN model was developed to estimate an ideal point in each time period, each consumer segment, each region, or each other delineation of choice situations. The example included in the original article (Cooper & Nakanishi [1983b]) specifically estimated ideal points for differing orders-of-presentation, just to illustrate this potential. In our current example there could be 156 ideal points - one for each store in each week. The idea is that, in coordination with market-share models and competitive maps, we could see how the competitive structure changes over time or competitive conditions and how those changes were translated into sales (or shares). Kamakura and Srivastava estimate the distribution of ideal points in any one context, but their efforts would have to be extended to estimate the distribution of ideal points in possibly many different competitive contexts.

The general issue involves the prediction of choice probabilities or market shares. Kamakura and Srivastava indicate that their methods were developed especially for predicting shares of choices. It is an empirical matter, but hard to imagine that a mapping procedure could predict market shares better than a market-share model. In our multimodel system there is specialization of purpose. The forecasting is done by the market-share model. Visualizations of competitive structures are accomplished with maps, while interpretation and opportunity analysis may be aided by ideal points. Thus no single model in the system is required to do more than what it does best. The Kamakura and Srivastava model seems to provide representations useful in product positioning or repositioning, which competitive maps are not designed to do. But forecasting market shares does not seem to be their model's strength.

The market-share models, the procedures for estimating elasticities from those models, three-mode factor analysis for representing the structure underlying those elasticities both over time and over brands, and the ideal-point model are a major part of the system of models for competitive analysis. Rather than proposing specific behavioral models of consumer response, these models provide the structural relations between the entities in a market information system. Although this system of models can be used for many purposes, the scientific goal is to provide a more systematic basis for using market information, while the practical goal is to provide a graphic understanding of the competitive structure and dynamics of a marketplace. Chapter 7 discusses how these methods can be used to provide a firmer empirical base for brand planning.

6.5  *Appendix for Chapter 6

This appendix presents the tables which resulted from the asymmetric three-mode factor analysis of the coffee-market example. While the technical development in the body of the chapter focused on the basic matrix concepts involved in the analysis, this appendix will deal more with the estimation methods. In particular the focus will be on the eigenvalue-eigenvector approach, in the belief it is easier to understand the mechanics of three-mode analysis from this perspective. Once the problem to be solved is understood it is easier to accept that alternating least-squares procedures (see Kroonenberg [1983]) can provide an overall least-squares solution to the problem, while the eigenvalue-eigenvector approach provides just an approximate least-squares solution (albeit a good approximation).

To understand the three views of the data we obtain from three-mode analysis it helps to review briefly some properties of two-mode tables. Consider a matrix _nX_p with n rows for individuals and p columns for variables . Let us assume X is of rank r £ min(n,p) . We can always represent X as a triple product.

where _nW_r is an orthonormal matrix containing coefficients relating the individuals to the factors , _r L_r is a diagonal matrix containing the square roots of the variance of each factor on the diagonal, and _pU_r is an orthonormal matrix containing coefficients which relate the variables to the factors. We speak of factors loosely, since this basic-structure model relates more to a principal-axes model than the formal factor-analytic model. Since W is orthonormal we know

If we form the scalar products showing the associations between variables we get

Since U is orthonormal we know

If we form the scalar products showing the associations between individuals we get

In this form we should be able to recognize W as the left-principal vectors (also known as eigenvectors or characteristic vectors), and U as the right principal vectors and the diagonal entries in L² as the eigenvalues (also known as latent roots or characteristic roots). We have long had the numerical algorithms to solve for the eigenvalues and eigenvectors of the symmetric matrices X¢X and XX¢ . Algorithms which solve directly for W , L and U are called singular-value-decomposition algorithms, where L contains the singular values (square roots of the eigenvalues).

This system sets up a series of orthogonal (independent) axes. The axes are linear combinations of the original variables. The first axis contains the greatest variance (information) which any single axis can hold; the next contains the greatest variance in any direction independent of the first axis. The third axis contains the greatest amount of variance in any direction that is independent of the first two, etc. The value of this kind of a representation becomes more apparent when we realize that if we discard the smallest dimension, we retain the most information we can pack into an (r -1) -dimensional representation. The best two-dimensional approximation to the information in the original matrix comes from retaining the first two principal axes. The concept of approximating the information in a matrix, by a lower-dimensional system has been around for over 50 years.Eckart, Carl & Gale Young [1936], ``The Approximation of One Matrix by Another of Lower Rank,'' Psychometrika , 1, 211-18.It underlies much of the thinking which led Tucker to develop three-mode factor analysis.

The common scaling space for the columns can be found from a singular value decomposition of a matrix E formed by vertically concatenating the elasticity matrices for each chain-week:

We need only the right principal vectors of this matrix which are the same as the eigenvectors of _jE^¢ E_j . As previously stated, these vectors will reflect the similarity among the columns of an elasticities matrix - the structure underlying how brands exert influence on the marketplace. These vectors appear in Table 6.3 as the Column Space.

Table 6.3: Common Scaling Space

The corresponding eigenvalues (shown in Table 6.4 as the Clout Factors) are used to select the dimensionality of the common scaling space for columns. Note that the last large drop is from the 7.4% of the variance in the fourth factor to the 4.1% of the variance contained in the fifth factor, so that four factors are retained for the common scaling space matrix C introduced in equation (6.4).

Table 6.4: Dimensionality of Common Scaling Space

The common scaling space for the rows can be found from a singular value decomposition of a matrix E formed by horizontally concatenating the elasticity matrices for each chain-week:

We need only the left principal vectors of this matrix which are the same as the eigenvectors of _iE E_i^¢ . These vectors will reflect the similarity among the rows of an elasticities matrix - the structure underlying how brands are influenced by marketplace forces. These vectors appear in Table 6.3 as the Row Space.

The corresponding eigenvalues (shown in Table 6.4 as the Vulnerability Factors) are used to select the dimensionality of the common scaling space for rows. It is possible, though inconvenient, that the dimensionality of the row space and that of the column space could differ. In fact, in this case there is some evidence that there might be a fifth factor among the rows. For, though there is a striking alignment of the variance controlled by the first four factors in the two spaces, the fifth factor among the rows is somewhat larger than that for the columns. It is a minor point, indeed, but choosing a common dimensionality for both the row space and column space is advisable. In this case we decided to select four factors for each space.

So far we have seen that there are two ways to organize the data, each corresponding to a different view of the brands. There is yet another way to view the data. Each matrix E^(t) could be strung out into a column vector, e_t , containing m² elements. These column vectors could be horizontally concatenated into a matrix E_t

We may represent E_t by

where W contains the right principal vectors of E_t - the elements w_tl introduced in equation (6.8). The right principal vectors of this matrix are the same as the eigenvectors of _tE^¢E_t . These vectors will reflect the similarities among stores and weeks - the structural forces leading to the representations in Figures 6.3 and 6.4.

These three different views of the data are tied together by the core matrix, G , introduced in equation (6.4). The core planes can be found one at a time, by taking each column of S and reorganizing it into an m ×m matrix S^(l), essentially reversing the process by which E^(t) was strung out into e_t. The l^th layer or the core matrix can be found from

The core matrix is presented in Table 6.5

In the general three-mode factor-analysis model, the core matrix is very important in understanding how factors on one mode relate to factors on the other modes. We simplify the issue in this special case by diagonalizing each core plane as shown in equation (6.5), and forming the matrices _iR_q^(l) and _qC_j^(l) developed in equations (6.6) and (6.7) and displayed for the coffee-market example in Table 6.6.

These joint-space coefficients are the building blocks from which the idealized elasticities are created from equation (6.8) corresponding to particular competitive patterns of interest. The idealized elasticities for the four conditions highlighted in the coffee-market example are presented in Table 6.7.

As a result of this presentation we hope it will be clearer from where the spatial representations come. While the eigenvalue-eigenvector development can help provide such insight, numerically it creates only an approximate least-squares solution to the systems of equations. Kroonenberg [1983] shows how all the components can be estimated using an alternating least-squares (ALS) algorithm. Whether this numerical refinement makes a practical difference will have to be determined in future research.

Table 6.5: The Core Matrix for the Coffee-Market Example

Table 6.6: Joint-Space Coefficients for Chain-Week Factors

Table 6.7: Elasticities for Idealized Competitive Conditions