A Winning Tool for CPGThe Shapley Value game theory method

brings advantages to marketing managersevaluating product line flavor decisions.

By Michael Conklin and Stan Lipovetsky

Consider the ice cream shop ownerwho must decide which flavors tosell. The choice can be bewilder-ing, but Ihe storage space is limit-ed. This problem is no less diffi-

cult for the ice cream manufacturer who mustdecide which flavors should be produced, givenlimited plant resources. Ideally, the product man-ager would choose the flavors that maximize totalsales or profit across the entire produci line. Thedifficulty in solving the problem is that the incre-mental value of a particular product flavordepends on which other fiavors are already in theline. Having 31 fiavors may be a great marketingploy, unless all 31 are variations of strawberry.

This type of problem is one that many market-ing managers in the consumer packaged goodsindustry face on a daily basis. While adding morefiavors to product lines such as ice cream, cook-ies, juices, and carbonated soft drinks will bepotentially more appealing to consumers becauseof the new fiavors, such a move creates signifi-cant costs. Packaging, advertising, and inventorymanagement costs all rise, as the number of fla-

E x e c u t i v e S u m m a r yThe authors introduce the Shapley Value analysis, a method for eval-uating product line flavor decisions that has many advantages overthe widely used "TURF" approach. The technique can be used In anyindustry where customers choose to buy one of N offerings. The con-text of this article refers to consumer food products, but the techniquehas been used for financial services and consumer durables as well.Shapley Value provides a measure of the overall strength of each pos-sible llavor when considered within all potential product lines. Assuch, it takes into account scenarios such as out-of-stock situationsand competitive flavors. The approach also balances a flavor's abilityto appeal to a unique group of consumers with its overall appeal in thepopulation, limiting the appearance of "strange" flavors in the finalrecommended solution.

vors increases. As a result, marketing managersoften ask marketing researchers to find an optimalset of fiavors that provides the best combinationof tlavors for a fixed number of products in theline. Typically, total unduplicated reach and fre-quency (TURF) analysis is used as a tool to rec-ommend solutions to the marketer's dilemma.While TURF answers the specific question thatthe marketing manager has asked, it doesn't pro-vide information that can help with the strategicdecisions that marketing managers must make fortheir product line. Another alternative. Shapley

MARKETING RESEARCH 23

Vaiue, serves as a solid solution because it assignsa value for each flavor calculated over all possiblecombinations of flavors.

T U R F ANALYSIS

TURF, a brute-force combinatorial technique,was developed for media-mix models to find thebest combination of magazines to place ads toachieve the maximum audience reach and frequen-cy of exposure. When applied to product line deci-sions, the procedure measures the total number ofpurchasers (or purchases) for every possible com-bination of N products taken R at a time. (N is thetotal numher of flavors being considered and R isthe number desired in the final product line.) TheTURF algorithm simply cycles through each pos-sible combination of R flavors and counts howmany consumers would buy at least one of them.

TURF can be conceptualized using Venn dia-grams. Exhibit I features purchase interest inthree flavors (labeled A, B, and C). The area ofeach circle in the diagram represents the numberof people who would purchase the product. If themarketing manager can afford to market only twofiavors. then choosing flavors A and C are thebest choice. This is because A and C appeal todifferent sets of consumers. Because A and Boverlap substantially, the incremental gain fromadding C to A is much greater than any othercombination of two products.

The results from a TURF analysis are notalways as clear as in this example. In marketingapplications, a substantial overlap tends to occurin the appeal of different flavors. This makessense, because most of the product's appealresults from the base product itself, not the specif-

Exhibit I

Hypothetical interest in 3 products

ic flavor. Considering ice cream again; If I wantto buy an ice cream cone, and the ice cream shophas only one flavor. I probably still will buy itbecause 1 want ice cream. This common overlapin flavor appeal can make TURF analysis verydifficult to interpret. Many potential product linecombinations can have very similar TURF scores.Exhihit 2 features the results of a TURF analysisfrom an actual study. The table shows the totalreach of all 10 possible three flavor product linesamong five flavors that were considered. Threepotential product lines are indistinguishable, evenif we don't take into account the possibility ofsampling error. This type of behavior limits theusefulness of TURF as a decision tool.

By examining the TURF analysis results,you'll see that certain flavors tend to be in the topcombinations, while other flavors tend to he atthe bottom. This suggests a method for summa-rizing TURF results. Simply add the number oftimes a flavor shows up in the set of best combi-nations. Exhibit 2 shows all possible combina-tions of five flavors taken three at a time. In morecomplicated problems, it's not practical to list allof the possible product lines. For example.38,760 possible six flavor product lines aredrawn from 20 possible flavors. With a largenumber of possible product lines, it becomes dif-ficult to decide how many combinations to lookat to generate a summary measure.

SHAPLEY VALUE

We turn to the field of cooperative game theo-ry for a decision tool that provides clearer resultsin these situations. We can think of the productline flavor decision as a way of building coali-tions among players (flavors, in our case) in agame to maximize the total value of the coalition.In the field of cooperative games, a well-knownanalysis and decision tool for this problem isShapley Value (SV).

The SV tool was developed to provide anordering of the worth of players in a multiplayercooperative game. The key to understanding itsutility is that it represents the worth of each playerover all possible combinations of players.Extending this to the product line flavor problem.SV assigns a value for each flavor calculated overall possible combinations of flavors.

Conceptually, SV is very simple. To evaluatethe strength of any particular flavor, we comparethe average strength of all possible product linesthat include the flavor to all possible product linesthat don't have it. We can evaluate the strength ofa product line the same way we do in TURF; thatis, we look at how many consumers would find al

24 Winter 1999/$pring 2000

least one product in our line to be desirable. (Formore detail on the specifics of calculating SbapleyValues, see our reference in the AdditionalReading, page 27.)

The SV solution to the problem is closer to theactual marketing situation. In most real-world sit-uations, marketers don't have complete control ofall the variables. While a marketer may introducea product line of a specific size, there's no guar-antee that all of the flavors will be available whena consumer makes his or her choice. A specificretail outlet may choose to carry fewer flavors asa result of shelf space limitations, or varyingproduct flavors may be available as a result ofout-of-stock situations. SV includes this possibili-ty in its solution set.

The SV solution also is better because itincludes comparisons with all flavors, even ifthey're not in the fmal product line. This is anadvantage of SV. By comparing across all possi-ble combinations of flavors, we include the possi-bility of competitive action in our analysis. Themarketer wants products in his or her product linethat provide the maximum incremental value,even when considering competitive offerings. Bycalculating the SV across all possible combina-tions of potential product flavors, we assure thatwe're positioned in the best possible position,even if competitors introduce some of the flavorsthat we have declined to produce.

TURF analysis provides a way to choose aproduct line over all possible product lines of aspecific size. In contrast to TURF, the ShapleyVaiue analysis provides an ordering of the prod-uct flavors that is a "best" solution over all possi-ble combinations and sizes of product lines.Because of this averaging over ail possiblecombinations. Shapley Value provides atheoretically more consistent decision rule.

Theoretical advantages are always nice,but they're even better if they lead to practicalimprovements. While TURF analysis is a widelyused tool for analyzing product flavor decisions,it does not always lead to clear results.Interpretation of the example presented in Exhibit2 is difficult because the TURF analysis yields nosingle best choice, and all the solutions are veryclose to each other.

This can be further demonstrated by bootstrap-ping the TURF data and determining the frequen-cy with which specific "best" solutions appear.(Bootstrapping is a statistical technique in whichrepeated samples are drawn with replacementfrom the data to see how likely we could get sim-ilar results with a different sample of consumers.)Exhibit 3 features the results of 500 bootstrapsamples of the same real data set. The TURF

Exhibit 2

TURF results

H Product lineACDBCDBDEABDADEBCEACEABCABECDE

Exhibit 3

# Reached

119119119118118117115114113111

Bootstrap evaluation of TURF

Product %line

BCD

ACD

BDE

ABD

ADE

BCE

ACE

ABC

' ABE

CDE

of bootstrap iterations wiiereline is rated best by TURF

26.8

25.8

25.8

15.2

14.2

12.4

3.4

1,8

1.0

0.6

1

% Reached

42.842.842,842.442.442.141.441.040.639.9

1

1

analysis only actually effectively differentiatesthe worst possible combinations. Six of the 10possible combinations are rated best more than10% of the time. Only the four worst combina-tions are effectively ruled out. The maximumtime that any single combination was judged bestwas 26.8%, and there were numerous ties withinthe bootstrap simulations.

Calculating the Shapley Values for each of theflavors provides an ordering of D > A > B > C >E. From this, we conclude that a three-flavorproduct line should include flavors D, A, and B.In Exhibit 2, this particular combination of flavors

MARKETING RESEARCH 25

was not one of the three "best" solutions from theTURF analysis, and it also wasn't a leader in thebootstrap TURF analysis summarized in Exhibit3. This is because TURF only maximizes over athree product line while the SV is calculated overall possible product line sizes.

Furthemiore. the ordering of the flavors byShapley Value is more consistent than the TURFanalysis. Again, we demonstrate this by bootstrap-ping 500 iterations of the SV calculations. Allpaired comparisons of order are significantly dif-ferent ip<.0\). {Results of this bootstrap analysisare summarized in Exhibit 4.)

Sbapley Value can be thought of as a weightedsum of terms for each potential size of productline, from I to N products. We have discussedwhy these weighted sums across all possible prod-uct line sizes are preferable to TURF resultsbecause we cannot guarantee a specific productline size as a result of out-of-stock situations orcompetitive product introductions. However, it'salso unlikely that all potential flavors would bepresent as a result of shelf space limitations orthat only one flavor would be available from man-ufacturing inefficiencies. To address this potentialdifficulty, we added some constraints to theShapley Value calculations.

Our proposed Constrained Shapley Value(CSV) allows the product manager to specifyprobabilities for specific line sizes. Suppose amarketer is considering introducing a product linewith six flavors out of 20 potential flavors. Itseems unlikely that stores would carry only oneflavor, and perhaps equally unlikely that 20 fla-vors would be available. On the other hand, thelikelihood of exactly six flavors being available inevery store is also small. The CSV allows a prod-uct manager to gauge the strength of the specificflavors considering only likely product line sizes.The marketer could specify a 30% probability thatonly tlve flavors would be available, a 40% prob-ability of six flavors, and a 30% probability ofseven flavors. We can then perform the ShapleyValue calculations by weighting the results foreach line size by the probability. In fact, if we

Exhibit 4

Bootstrap evaluation of SV

Order comparison

D>AA>BB>CC>E

Mean ditterence0,5540.5300,3170.184

t-value7.22

8.234.192.81

assume a 100% probability of a product line ofsix flavors, the Shapley Value becomes an excel-lent summary of a TURF analysis.

The summarization of overall strength of eachOavor that is provided by Shapley Value has fur-ther advantages over the TURF approach of find-ing the "besf possible product lines. BecauseTURF finds the flavor that provides the greatestadditional reach from a given base, it tends to addproducts to the line that have unique appeal lo asmall group of customers, ln essence, TURF tendsto add unique and different flavors to your prod-uct line. Shapley Value evaluates the overallstrength of the flavor across all possible line sizes.This means that strange flavors tbat appeal to avery small group of consumers still will appearrelatively weak in an SV analysis, while they mayappear in tbe best TURF combination. However,as the size of the group that likes the "strange"flavor increases, the Shapley Value of the flavorwill increase dramatically. Essentially, theShapley Value balances between the uniquenessof the flavor's appeal and the overall breadth ofappeal. Flavors that uniquely appeal to specificmarket segments will rise to the top of theShapley Value analysis only if the size of the seg-ment is big enough.

SV's IMPLICATIONS

The Shapley Value analysis provides a pre-ferred ordering of the product line flavors, whichhas further implications for business strategy thataren't evident from a TURF analysis. Recall thatthe product flavor with the highest Shapley Valueprovides the greatest gain over all possible prod-uct combinations. This implies that it's preferableto have that flavor always in the available choiceset when consumers are purchasing. Tbe strategyimplications of this statement are that the mar-keter should spend more effort to ensure that theproduct flavor with the highest Shapley Value hasthe minimum possible out-of-stock probability.This implies management of inventory in such away as to give preference to the highest ShapleyValue flavor. Following this strategy ensures thatno matter which combination of product flavorsand competitive flavors are on the shelf al anytime, our flavors bave the highest Shapley Values,and therefore are the best situation.

These strategy implications also apply directlyto our Constrained Shapley Value. The only dif-ference is that the evaluation takes place over allpossible product combinations within all likelyproduct line sizes. With large numbers of poten-tial product flavors, this modiflcation is especiallyimportant.

26 Winf»r 1999/Sprmg2000

Finally. Shapley Value will identify flavorsthat have unique appeal to specific market seg-ments only when those segments are large enoughto matter. When TURF will usually provide a"best" solution that has some strange low appealflavor, the Shapley Value approach only willvalue such a flavor if the overall appeal is largeenough to matter. •

ADDITIONAL READING

Cohen E. (1993), "TURF Analysis," Quirk'sMarketing Research Review, (June/July). 10-3.

Conklin. M.. and Stan Lipovetsky (1998),"Choosing Product Line Variants; A GameTheory Approach," Computing Science andStatistics. S. Weisberg, ed. Fairfax Station, VA:Interface Foundation of North America Inc.

Miaoulis G.. Free V.. and H. Parsons (1990)."TURF: A New Planning Approach for ProductLine Extensions." Marketing Research,(March). 28-39.

Myerson. R.B. (1991), Game Theory: Analysis ofConflict. University Press.

Owen, G. (1982), Game Theory. Academic Press.

Roth, A.E. (1977), 'The Shapley Value as a VonNeumann-Morgan stem Utility." Econometrica,45(3). 657-64.

Roth, A.E. (ed.) (1988), The Shapley Value-Essays in Honor of Lloyd S. Shapley.Cambridge University Press.

AUTHOR INFORMATION

Michael Conklin is senior vice president ofanalytic services for Minneapolis-based CustomResearch Inc. A graduate of the Wharton School.University of Pennsylvania, he worked at GeneralElectric Co.. Stanley Tools Inc.. and Black &Decker in marketing research. He specializes inBayesian methods and the analysis of categoricaland ordinal data.

Stan Lipovelsky, a research manager forCustom Research, has an MS in theoretical phy-sics and a PhD in mathematical economics fromMoscow University. He worked at Tel Aviv Univ-ersity's Faculty of Management. His primary areasof research include multivariate statistics, muIticH-teria decision making, and marketing research.

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