MARKET SHARE CONSTRAINTS AND THE LOSS FUNCTION IN CHOICE BASED CONJOINT ANALYSIS

Transcription

1 MARKET SHARE CONSTRAINTS AND THE LOSS FUNCTION IN CHOICE BASED CONJOINT ANALYSIS Tmothy J. Glbrde Assstant Professor of Marketng 315 Mendoza College of Busness Unversty of Notre Dame Notre Dame, IN Phone: (574) Fax: (574) Peter J. Lenk Assocate Professor of Operatons and Management Scence Assocate Professor of Marketng Ross School of Busness Unversty of Mchgan Jeff D. Brazell Presdent/CEO The Modellers, LLC Salt Lake Cty, UT November 30, 2006

2 MARKET SHARE CONSTRAINTS AND THE LOSS FUNCTION IN CHOICE BASED CONJOINT ANALYSIS Abstract: Choce based conjont analyss s a popular marketng research technque to learn about consumers' preferences and to make market share forecasts under varous scenaros for product offerngs. Managers expect these forecasts to be "realstc" n terms of beng able to replcate market shares at some pre-specfed or "base case" scenaro. Frequently, there s a dscrepancy between the forecasted and base case market share. Ths paper presents a Bayesan decson theoretc approach to ncorporatng base case market shares nto conjont analyss va the loss functon. Because defnng the base case scenaro typcally nvolves a varety of management decsons, we treat the market shares as constrants on what are acceptable answers, as opposed to nformatve pror nformaton. Our approach seeks to mnmze the adjustment of parameters by usng addtve factors from a normal dstrbuton centered at 0, wth a varance as small as possble, but such that the market share constrants are satsfed. We specfy an approprate loss functon and all estmates are formally derved va mnmzng the posteror expected loss. We detal algorthms that provde posteror dstrbutons of constraned and unconstraned parameters and quanttes of nterest. The methods are demonstrated usng both multnomal logt and probt choce models wth smulated data and data from a commercal market research study. Keywords: Herarchcal Bayes, Loss Functon, Posteror Rsk, Bayesan Decson Theory, Conjont Analyss

3 MARKET SHARE CONSTRAINTS AND THE LOSS FUNCTION IN CHOICE BASED CONJOINT ANALYSIS 1. Introducton Dscrete choce conjont analyss has proven to be a useful tool for nvestgatng consumer preferences n both appled and academc studes. In appled settngs, results from a typcal conjont analyss are used by managers to decde on the most attractve combnaton of product features and prce gven the compettve offerng, or antcpated changes to the compettve set. To facltate "what-f" analyss, commercal market research frms frequently provde software packaged as a "market smulator." These smulators use the results from the conjont study together wth manager suppled nputs on product features and prcng to produce market share forecasts. When a manager enters a "base case scenaro" consstng of a set of actual product features and prces, there s usually a dscrepancy between the forecast from the smulator and the actual market share. Whle there are many reasons why the forecasted and actual market share may dffer, when ths occurs a manager may doubt the qualty of the study and/or the relablty of the forecasts. The manager really has two goals from the conjont study: to represent consumer preferences and to produce "realstc" forecasts. An nterestng queston s how analysts should use manager's "base case" scenaro expectatons. Ths paper presents a Bayesan decson theoretc approach to ncorporatng base case scenaro projectons nto dscrete choce conjont analyss va the loss functon. Forecasted market share from a conjont study may dffer from actual market share for a varety of reasons. In an early survey on the commercal use of conjont analyss, Cattn and Wttnk (1982) lst fve dffcultes n makng market share predctons usng conjont analyss. These nclude the nherent dfference between stated and revealed preferences, attrbutes present n the marketplace but excluded from the conjont study, the nablty of conjont studes to 1

4 nclude the effect of mass communcaton, dstrbuton, compettve reacton, and other effects. Orme and Johnson (2006) reterate many of these reasons n ther summary of practtoners' methods of adjustng smulated market shares to match actual market shares. It s mportant to note that Orme and Johnson do not advocate adjustng the results from the market smulators and nstead argue that managers should be educated on the reasons why smulated and actual market share may not agree. Allenby et al. (2005) revew addtonal reasons why choce experments dffer from actual market choces and outlne data collecton procedures and new types of models whch may mprove the predctve accuracy of choce models. In a Bayesan analyss, the base case market share can be modeled as part of the lkelhood functon, ncorporated nto the loss functon, or used as nformatve pror nformaton. Some researchers have suggested that stated and revealed preference data be ncorporated nto a unfed model to overcome problems wth conjont studes. Louvere (2001) notes that there s a close correspondence between stated and revealed preferences. However, t may be necessary to calbrate ether the locaton or scale parameter from analyses that only nclude stated preferences to account for the nherent dfferences between expermental and actual choce behavor. Morkawa, Ben-Akva, and McFadden (2002) detal statstcal methods of combnng stated preference and revealed preference data. When avalable, conjont analyss data should be augmented wth actual market place choces and the full set of data modeled. However, not all studes are amenable to ths soluton ether for logstcal reasons (market data s not avalable for conjont partcpants) or due to the lack of avalable models for ncorporatng aggregate market share results wth expermental studes. Further, there s no guarantee that such analyses wll necessarly produce market share projectons that match a manager's base case scenaro. 2

5 The base case scenaro and accompanyng market share used by a manager wll typcally requre several decsons. Market shares vary over tme and geography as prces change due to promoton, advertsng ntensty or message change, fluctuatons n dstrbuton, changes n compettve offerngs, etc. Managers must decde f the base case scenaro wll reflect a very specfc measure (e.g. one week, n one partcular market) or an aggregated measure (e.g. annual market share for all markets served). If the latter, then the manager must choose whether "average values" of the product attrbutes wll be used or f "representatve values" wll be selected. Selectng a pont estmate to serve as the base case scenaro s not trval. Our experence suggests that managers use a combnaton of "known facts," aggregated, and "representatve values" to arrve at the base case scenaro and accompanyng market share estmates or expectatons. From the analyst's perspectve, the manager's base case scenaro s most approprately vewed as a constrant placed on the forecasts provded by the market smulator. Bayesan decson theory mples that these constrants should be ncorporated nto the loss functon. Essentally, our approach to makng market share forecasts s to approxmate a procedure that ntegrates over the regons of the posteror dstrbuton of parameters that are consstent wth the market share constrants. We have several objectves n mnd when developng the loss functon approach. Most explctly, we requre that the market shares at the base case be suffcently close to the managerally specfed market shares such that the choce based conjont (CBC) analyss has face valdty for the marketng manager. Clearly, ths alone s not suffcent to dentfy the adjustment procedure, and one could magne any number of methods to reach ths objectve. For example, one could adjust the preferences of a subset of subjects whle leavng others untouched, or one 3

6 could modfy the coeffcents for only a subset of attrbutes. Instead of beng heavy-handed wth the adjustments, we want the fnal results to be as true as possble to the CBC data. The adjustment procedure that we develop perturbs all of the estmated parameters from the CBC data as lttle as possble n order to leave the preference structure of the CBC data relatvely ntact. Our adjustment terms are addtve factors wth mean zero, and we specfy the varance to be as small as possble whle satsfyng the constrants. An attractve feature of the proposed approach s that the estmaton of the CBC parameters and the adjustment process s decomposed nto separate operatons. The posteror dstrbuton of the CBC parameters or smulates thereof s the nput to the adjustment process. The analyst can keep the two separate and dsplay the CBC estmates before and after the adjustment process, and he or she can gve an explct representaton of the mpact of the manageral constrants. We contrast the loss functon approach wth several others that treat the base case scenaro as an nformatve pror. The frst problem one faces wth treatng the constrants as data or pror nformaton s establshng the correct weghtng between the CBC data and the base case. Because we lack revealed preference data for the subjects n the CBC data and because the base case market shares are based on a dfferent sample than the CBC study, the relatve weght to place on the pror versus the lkelhood s not dentfed. Does one treat the base case market shares for M products as M bts of nformaton, or does one treat them as mllons of transactons? The relatve weghts of the CBC and market share data can be treated as a tunng parameter, however, a sngle study s not suffcent to dentfy t. In some nstances, the loss functon and nformatve pror yeld mathematcally equvalent results. However, the loss functon approach explctly recognzes that the 4

7 adjustments are due to external crtera or goals mposed on the analyss by the manager. We beleve that ths route better fts the stuaton where the marketng researcher s attemptng to satsfy the sometmes competng goals of hs or her clent. Whle a strength of the Bayesan paradgm s ts ablty to ntroduce subjectve pror nformaton, "pror nformaton" s handled dfferently than restrctons on what consttutes an acceptable answer. Both are vald addtons to a decson problem from the standpont of the analyst. Ths paper llustrates how the Bayesan paradgm handles market share constrants and provdes practcal algorthms for mplementng them n dscrete choce conjont analyses. The remander of the paper s organzed as follows. Frst we set-up the dscrete choce analyss, revew the role of the loss functon n Bayesan decson theory, and propose a specfc loss functon that ncorporates market share constrants. Ths loss functon mnmzes changes n preferences as represented n the conjont study whle beng wthn an acceptable range of the manager's base case market share expectatons. We then contrast the loss functon approach wth several nformatve pror approaches. Smulaton and Markov-Chan Monte Carlo (MCMC) methods are detaled for conductng the analyss mpled by the loss functon and necessary for obtanng market share forecasts. These can be ncorporated nto the same computer program used to estmate the dscrete choce model and produce posteror dstrbutons of parameters wth and wth-out the market share constrant. We llustrate the proposed methods usng smulated data for both the multnomal logt and correlated probt dscrete choce models. Our smulatons show that when the market share constrants are accurate, the proposed method yelds better market share predctons for changes n product formulaton outsde the base case scenaro. We show the results from a commercal market research study and conclude wth a summary and suggestons for future research. 5

8 2. The Loss Functon and Bayesan Decson Theory 2.1 Dscrete Choce Model and Market Share Constrants Ths secton begns by statng the dscrete choce model and defnes varous terms. We envson a commercal market research study where the data conssts of a sample of respondents who have completed a CBC study and management has provded market share estmates at some base case set of product attrbutes for some set of the brands or products n the CBC study. The model set-up s as follows. N subjects evaluate J choce occasons for subject. Each choce occason or tasks conssts of M alternatves or product profles. In the CBC desgn, there are p attrbute levels, ncludng brand ntercepts. Subject s random utlty for product profle j s a gven by Y j = x' ε ~ Normal or extreme value β = θ + ν ν ~ N j p β + ε ( 0, Λ) j for subject and choce occason j where the observed attrbutes x j, the parameter vector β, and error terms ε j are p-vectors. Let W be the observed choces from the CBC. Our prmary objectve s to use the set of {β } n functons g({β }) to obtan estmates of quanttes of nterest such as predcted market share. Let X o represent the matrx of product attrbutes n the base case scenaro and S 1,,S K represent the correspondng market shares. Let S,, ˆ ˆ1 K S K represent the predcted market shares usng X o and {β }, e.g. g({β }, X o ). We defne the dscrepancy between the predcted and base case market shares as: 6

9 K k = 1 S k Sˆ k = δ (1) Our goal s to produce estmates of Ŝ k that are "close" to S k and reflect the nformaton obtaned n the conjont study. We assume that n addton to X o and S 1,,S K, management can provde δ t, a target level, or acceptable level of dscrepancy between the predcted and base case market shares. Let C represent the set of parameters used to estmate Ŝ k that satsfy the market share constrant δ δ t. We wll use the ndcator functon χ(s) = 1 to ndcate when the constrant s satsfed and χ(s) = 0 when t s not. In the stuaton that we consder n ths paper, the actual market behavors of the subjects n the CBC study are not observed. Consequently, we are unable to buld a model that relates the parameters for the revealed and stated preference data. In fact, the subjects n the CBC study may not have made a purchase n the product category durng the perod that the base market shares were computed. Another nsolvable ssue s the relatve weghtng of the market share nformaton wth the CBC data. Consder the case where a separate market based dataset, such as scanner data, s avalable. Should one base the weghts on the number of observatons n the CBC study typcally hundreds of subjects each makng dozens of choces versus the number of purchases n the market share data whch can be mllons of transactons for frequently purchased consumer goods? In ths mcro approach, the market share data, derved from a very large number of purchases, wll swamp the more lmted CBC data. Alternatvely, one could use a macro approach and treat the market shares for K products as K-1 observatons. 7

10 Then the CBC data would domnate. Intutvely, the correct weghtng s somewhere between the two extremes, but decdng where requres more than CBC data and base rates. The approach that we propose n ths paper s more drect and much smpler than developng a jont model. We propose a method to adjust the estmated parameters from the CBC study so that ther mpled market shares are close to the base market shares. The adjustment method s formally derved through Bayesan nference by ncludng the base market shares nto the loss functon. Ths adjustment perturbs the CBC estmates as lttle as possble. We llustrate the approach by consderng a sngle parameter β. A standard Bayesan analyss wll produce a posteror dstrbuton π(β W) as llustrated n Panel A of Fgure 1. The posteror s proportonal to the pror dstrbuton on β and the lkelhood of β gven the data W, or π(β W) l(β W)π(β ). Wthn π(β W) there are regons, perhaps dsjont, that satsfy some exogenous constrants as ndcated by the shaded regons n Panel A. Gven a draw β r from ths posteror dstrbuton, we generate adjustment factors {α } from a normal dstrbuton N(0, τ -1 ) wth densty j(α τ) as shown n Panel B. We consder draws of α that map nto the shaded regon of the posteror va α r + β r and satsfy the base case constrants. Ths s llustrated n Panel C of Fgure 1. Note that now the{c } such that χ(s) = 1 contans β and α. Formally, our approach condtons on the posteror dstrbuton π(β W) and the α's are dummy varables of ntegraton whch lmt the area over whch posteror analyses are conducted. The α s not a parameter n the lkelhood functon, but a devce we use to satsfy an external constrant. Note that n ths example, the regon defned by (β + α ) s dsjont, so functons such as E[(β + α )] where the expectaton s wth respect to π(β W), are not partcularly meanngful. ================ Fgure 1 =================== 8

11 Ths choce of the normal dstrbuton for the adjustment factors s not arbtrary. We set the mean to zero wth the desre to keep the adjusted estmators unbased. We use a sphercal varance structure to be ndfferent about the set of attrbutes to adjust. We use a normal dstrbuton because ts tals declne rapdly, so that the estmate for any sngle attrbute or subject wll not be greatly modfed relatve to the adjustment for other parameters. In CBC analyss we are nterested n usng parameters to estmate quanttes of nterest, such as market share. In order to enforce market share constrants, each parameter for each subject wll have ts own addtve factor. We represent functons that use these quanttes as h({β + α }) where α ~ N p (0,τ -1 I p ) and I p s a p x p dentty matrx. Clearly the addtve factors wll not be unque: for a gven set of {β } there are any number of {α } that wll satsfy the market share constrants. Forcng each α to be as close to 0 as possble by makng τ as large as possble, wll be used to help dentfy the {α }. Let B represent the set {β } and A the set {α }. We wll use the loss functon to obtan pont estmates of h(a+b). Next we revew the basc dea of loss functons and then ntroduce one that reflects our goals of usng parameters that match the market share constrants and mnmze changes to the preferences as revealed n the CBC study. 2.2 Loss Functons In decson analyss (c.f. DeGroot 1970), the loss functon L(θ,d) quantfes the loss to the decson maker of takng acton (or estmate) d when the state of nature (or parameter) s equal to θ. [Here θ s any arbtrary parameter.] Snce the state of nature s not known, the Bayesan decson maker wshes to mnmze the posteror expected loss, or rsk functon represented by: r(π,d) = L ( θ, d) π ( θ x) dθ (2) θ 9

12 where π represents the current dstrbuton of θ, n ths case the posteror. The Bayes rule s the selecton of d œ D, the set of allowable decsons, that mnmzes r(π,d). The statstcal problem of pont estmaton s one applcaton of the loss functon. In ths case, d s the partcular estmate of θ to report and use n addtonal analyss. A common loss functon s squared error loss represented as L(θ,d) = (θ - d) 2 for a scalar parameters and L(θ,d) = (θ - d)'(θ - d) for a vector of parameters. When a squared error loss functon s used and (2) s mnmzed wth respect to d, the resultng Bayes rule, or pont estmate for θ s equal to the mean of the posteror dstrbuton or: ( = (3) θ = θπ θ x ) dθ E( θ x) θ Where θ represents the Bayes rule. In fact, the posteror mean s frequently reported n both appled and academc studes usng Bayesan methods. However, the basc machnery of Bayesan decson analyss can be appled to any loss functon. The choce of a normal dstrbuton for the addtve factors, for nstance, can be motvated by consderng the loss functonal T(f) = f ( α ) ln[ f ( α )] dα, whch s mnus the entropy, of the densty f for α. Gven the constrant that the mean s zero and the precson s τ, the normal densty mnmzes the functonal T(f) (c.f. Bernardo and Smth 1994, pp ). The normal dstrbuton maxmzes the entropy for denstes on the real numbers gven constrants on the frst two moments. In other words, gven the mean and the varance, the normal densty mposes less structure on the set of A than other choces for the dstrbuton. 10

13 To enforce market share constrants on the CBC analyss, we wll use a varant of the squared error loss functon. The loss functon s represented as: L, (4) ( h d ) = [ h( A + B) d] [ h( A + B) d] for A + B C where A s the set of adjustment factors; B s the set of parameters from the CBC data; C s the constrant set defned by the market share at the base case; and d s the estmator of the functonal h. Here the requrement A + B œ C, whch means that α + β œ C for α œ A and β œ B, can be seen as lmtng the set of allowable decsons, d œ D. To obtan the rsk functon we ntegrate wth respect to all the unknown quanttes. In ths case we have the posteror dstrbuton of B and the dstrbuton of A, whch are varables unque to the loss functon. ρ( π, φ, d τ ) = whereϕ C ' [ h( A + B) d )][ h( A + B) d )] ϕ( A τ ) π ( B W ) N ( A τ ) = ϕ( α τ ) and π ( B W ) = π ( β, Kβ W ). = 1 1 N dadb (5) Note that the requrement A + B œ C s subsumed nto the area of ntegraton. To obtan the Bayes rule d*, we dfferentate (5) wth respect to d and set t equal to 0: 11

14 d * = C h( A + B) ϕ( A τ ) π ( B W ) dadb C ϕ( A τ ) π ( B W ) dadb = P( C τ ) 1 C h( A + B) ϕ( A τ ) π ( B W ) dadb d* = h( A + B) f ( A, B W, C, τ ) (6) where f ϕ( A τ ) π ( B W ) P( C τ ) ( A, B W, C, τ ) = for A + B C The support for f(a, B W, C, τ) s C and the normalzng constant s P(C τ). Thus equaton (6) states that the Bayes rule for h(a + B) s the expected value of the functon when the values of A and B are drawn from a specfc densty. Also note that the Bayes rule s condtonal on the value of τ; we wll return to ths pont shortly. The value of α + β drawn from f(a, B W, C, τ) are dependent on all other α's and β's. However, ths specfc dstrbuton s a by-product of the loss functon and arrvng at an estmate for h(a+b): the posteror dstrbuton for β and all other parameters n the model are obtaned usng standard procedures and wthout reference to α or the market share constrants. Although the base case scenaro attrbute levels X o wll be needed n order to evaluate whether A + B s n C, h(a+b) may nvolve any attrbute levels. Ths allows the analyst to do "what-f" analyss wth a set of parameters that satsfy the market share constrants (condtonal on the value of τ). However, snce the α + β are dependent on each other, averagng over the posteror draws of {β + α } for each ndvdual and usng β + α ) to predct market share wll ( not necessarly satsfy the market share constrant. Savng the sets of {β + α } n order to do "what-f" analyss represents a change n procedure for analysts accustomed to usng ndvdual 12

15 averages as nputs to market smulators. Also, as noted earler, snce the market share constrants my nduce dsjont regons n the sampled values of h(a+b), a hstogram of the values should be nspected before calculatng the expected value. The Bayes rule s condtonal on A + B œ C and the value of τ. A pror t s not clear what value of τ should be specfed by the analyst. If τ s too large then values of α from N p (0,τ - 1 I p ) may be too close to 0 n order to satsfy A + B œ C. If τ s too small then α wll have a large varance and although we may satsfy A + B œ C, A + B wll be poorly dentfed and we may nadvertently alter the ndvdual β's more than s necessary or desred. Recall that our goal s to produce market share forecasts consstent wth the base case scenaro whle alterng the β's as lttle as possble. Formally, we may select the optmal value of τ by mnmzng the Bayes rsk. The Bayes rsk as a functon of τ s obtaned by pluggng the Bayes rule, d*, back nto the rsk functon. It s represented by: r( τ ) = = P = P C [ h( A + B) d *)] [ h( A + B) d *)] ϕ( A τ ) π ( B W ) ( C τ ) [ h( A + B) d *)] [ h( A + B) d *)] f ( A, B W, C, τ ) C M ( C τ ) var[ h ( A + B) W, C, τ ] m= 1 m dadb dadb (7) Where M s the number of market shares we are estmatng. When the market share constrants are meanngful, eg. B C, ncreasng τ decreases P(C τ) because A wll be closer to 0, and wll reduce r(τ). The var[h m (A + B) W, C, τ] wll also be reduced by large values of τ snce α ~ N p (0,τ -1 I p ) and large τ mples less varance. Ths suggests that when the market share 13

16 constrants are necessary, we want τ to be as large as possble as long as A + B œ C. We show numercal results wth smulated data that renforces ths ntuton. Frst, we present several alternatve methods of ncorporatng market share constrants as nformatve prors. 3. Market Share Constrants and Informatve Prors In the loss functon approach to ncorporatng market share constrants, the rsk functon ncorporates the posteror dstrbuton of B (and all other model parameters) and a separate dstrbuton for A, the addtve factors. Formally, the analyss s broken nto two dstnct peces and wll requre draws from two dfferent dstrbutons. Ths can be represented as (focusng just on the parameter B): π(b W) l(b W)π(B) f(a, B W, C, τ) π(b W)j(A τ) for A+B œ C (8a) (8b) where (8a) s a standard herarchcal model and (8b) arses as a byproduct of estmatng the Bayes rule d* for h(a+b) from the loss functon (4). An alternatve to usng a loss functon s to treat the base case scenaro as an nformatve pror; that s, the manager's expectaton of the market share at specfed attrbute levels and the allowable dscrepancy s treated as pror "data," and not as a constrant on what consttutes a correct answer. A mathematcally equvalent way to represent (8a) and (8b) s: π(b,a W) l(b W)π(B) j(a τ) for A œ C B (9) 14

17 Note that we use the condtonng argument C B so that values of B are drawn wthout reference to A or the market share constrant. Here the dfference between the loss functon and nformatve pror approach s purely phlosophcal. An alternatve and perhaps more tradtonal way to thnk about the market share constrant s to represent (9) as π(b,a W) l(b W)π(B) j(a τ) for A+B œ C (10) and draws of B are dependent on A and the market share constrants. Margnal posteror dstrbutons of B from (8a) or (9) wll not match those from (10). Market share constrants can also be ncorporated through nformatve prors wthout usng the addtve factors A. For nstance, one can treat the forecasted market shares at X o as a "parameter" n the model and place an nformatve pror on them. Consder the followng transformatons: S ˆ k ln and k ln S k z k = ξ = for k = 1,..., K 1 (11) ˆ S K S K Now let y(x τ) ~ N K-1 (z, τ -1 I K-1 ) and I K-1 s a (K-1) x (K-1) dentty matrx. Now the pror parameter τ determnes the tradeoff between the manageral constrants and the CBC data. Very large τ wll result n posteror dstrbutons "relatvely" close to the base case, although how close wll be a functon of the data. An alternatve approach whch does not nvolve τ s to put a dstrbuton on δ from equaton (1). For nstance we mght assume that δ follows a unform 15

18 dstrbuton between (0, δ t ). Agan, n both of these approaches, draws of B are dependent on the market share constrants and margnal posteror dstrbutons wll not match those of 8(a). In the nformatve pror approaches as typfed n (9), (10), and (11), τ s a parameter n the pror dstrbutons j(a τ) or y(x τ) and should theoretcally be set before the analysts sees the data, be estmated from the data, or be determned va mnmzng the Bayes rsk. A pror, t s dffcult to preset τ. If t s estmated from the data, then for (9) and (10) the pror wll have to be very nformatve n order to dentfy the model. For (11) our experence has been that the estmated value of τ mostly reflects ts pror snce there are only K-1 peces of nformaton for ts estmate, and K s relatvely small compared to the CBC data. It s also possble to specfy a loss functon, obtan the Bayes rule, and derve the Bayes rsk for (9), (10), and (11) as a functon of τ, analogous to our development n the prevous secton. Ths may be a frutful area for addtonal research for analysts commtted to an nformatve pror approach. We favor the loss functon approach because we feel the crcumstances n most analyses are consstent wth treatng the base case market shares as constrants on the allowable set of answers. As noted earler, the decson on how to measure market share, the attrbute values at the base case scenaro, and what consttutes a "close enough" answer are management decsons. Whle Bayesan analyses are perfectly amenable to ncorporatng subjectve pror nformaton, there s a dfference between "pror nformaton" and restrctons on the analyss. Our vew s that meetng the base case scenaro wth the market smulator s an ancllary goal of the analyss. Contrast ths wth other nstances n the marketng lterature that use nformatve prors. Boatwrght, McCulloch, and Ross (1999) use truncated normal dstrbutons to ensure prce coeffcents are negatve n retal/market level sales response models. Allenby, Arora, and Gnter 16

19 (1995) enforce an a pror "more s better" orderng on part-worths n conjont analyss. In these cases, economc theory nformed the choce of pror dstrbutons. The loss functon approach has been mplemented n the statstcs lterature to capture ancllary goals of the analyses; see for nstance Lous (1984) and Ghosh (1992) who use the loss functon to "match" the posteror dstrbutons of parameters to certan emprcal dstrbutons of the data. As noted by Shen and Lous (1998), a strength of the Bayesan paradgm s ts ablty to "structure complcated models, nferental goals, and analyses" and that "methods should be lnked to an nferental goal va the loss functon." The loss functon approach we have outlned matches the baselne market share forecast wth mnmal changes to the preferences as revealed n the CBC study. The algorthm detaled n the next secton allows for straghtforward comparsons between the constraned and unconstraned analyses. 4. Estmaton Because B s ndependent of A and reles only on the CBC data, standard algorthms for estmatng herarchcal Bayesan conjont models can be modfed to obtan samples of B + A to use n estmatng E[h(A+B) W, C, τ). Alternatvely, current programs can be used to obtan samples of B, and n a separate program A can be sampled such that A+B œ C. Well known samplng and Markov-Chan Monte Carlo (MCMC) methods are used to obtan draws from f(a, B W, C, τ) from equaton (6). As s standard, we explot condtonal ndependence to draw ndvdual level parameters β and α. The generaton of the adjustment factors could ether be performed nlne wth the analyss of the CBC model or t could be performed offlne and after the MCMC for the CBC data. The nlne MCMC algorthm goes through the followng steps on each teraton: 17

20 1. For =1,,N 1a. Draw β W, X, θ, Λ Use standard methods for ether probt or MNL models 1b. Draw α β, {β - }, {α - }, X o, τ, δ t Detaled below 2. θ {β }, Λ Standard conjugate set-up 3. Λ {β },θ Standard conjugate set-up For the probt model, steps are added for data augmentaton and drawng the error-covarance matrx, see McCulloch and Ross (1994). In step 1b, the notaton {β - }and{α - } ndcates the set of parameters for all respondents other than. Thus the draw of α for person s dependent on the current value of α and β for all other respondents. The offlne algorthm treats the random draws {β g }, for subject and MCMC teraton g, from the MCMC algorthm as nput and draws a separate α g, as detaled below, from ϕ(α τ) gven α g s n C β g. A random walk Metropols-Hastngs (Chb and Greenberg, 1995) or a weghted bootstrap (Smth and Glefand, 1992) s used to draw α. Recall that A+B must satsfy A+B œ C or χ(s)=1, the market share constrant n the loss functon. For each ndvdual on each teraton, β s drawn (o) from π(β W) and that draw s used n (6) to draw α from f(a, B W, C, τ). Let α represent the draw of α from the prevous teraton. When the market share constrant equaton s evaluated (o) [equaton (1)] wth then new β, the prevous α, {β - }and{α - }, then ether χ(s) = 1 and the 18

21 constrant s satsfed or χ(s) = 0 and t s not. Snce β s drawn wthout regard to χ(s), there s no guarantee that the prevous draw of α wll satsfy the market share constrant. If the market share constrant s satsfed wthα (o), then a random walk Metropols- ( n) ( o) Hastngs (M-H) step s used to draw α. Form a canddate or new α as α = α + η where η s drawn from N p (0, zi p ) and z s a scalar chosen to ensure a 50% rejecton rate for the M-H step. Note that Λ may be used nstead of I p. The dstrbuton f(a, B W, C, τ) from equaton (6) mples (n) the followng acceptance probablty for α : τ ( n) ( n) ( n) exp α ' α χ( S, α ) 2 Mn, τ 1 exp ( o) ( o) α ' α 2 (12) (n) Note that f the new α does not satsfy χ(s) then the numerator s 0 and the old value of α s retaned. (o) If the market share constrant s not satsfed wth α, then a weghted bootstrap s used to draw a new value of α. A challenge n samplng α from f(a, B W, C, τ) s satsfyng the market share constrant. Wth an approprately selected proposal densty, the weghted bootstrap facltates ths process. Let (o) β be the value of β from the prevous teraton and ( n) ( o) ( o) draw from the current teraton. Defne µ = β ( β + α ) ; gven {β - }and{α - }, (n) β be the (n) β + µ wll satsfy the market share constrant. The target densty mpled by (6) s f(α) j(α τ)χ(s) and the proposal densty for the weghted bootstrap s g(α) ~ N(µ, τ -1 I p ). A total of R values of (r) α are drawn from g(α). For each draw r, calculate the weght w r : 19

22 w r r r ( α ' α ) τ r exp χ( S, α ) = 2 τ r r exp ( α μ)'( α μ) 2 (13) (r) The ndvdual value α s then selected wth probablty: r wr Pr( α = α ) = R (14) w r= 1 r Note that f the market share constrant s not satsfed, w r = 0 and that value ofα cannot be selected. The random walk or weghted bootstrap s completed for each on each teraton of the algorthm. The chan converges to the statonary dstrbuton mpled by the posteror dstrbuton f(a, B W, C, τ) from the loss functon. Note that the algorthm naturally produces draws of both B and B+A, whether performed nlne or offlne; ths makes comparson of analyses wth and wthout the market share constrants straghtforward. As n other smulaton based methods, pont estmates of E[h(A+B) W, C, τ) are obtaned by computng h(a+b) for each draw of A+B and averagng over the draws. Total computaton tme for the nlne algorthm, as compared to models wthout market share constrants, wll depend on the form of the error term used n the model. In order to test χ(s), market share estmates at X o wll have to be calculated. For the MNL model, choce probabltes are avalable n closed form and so there s no apprecable ncrease n computaton tme n order to calculate base case market share. However, for probt models whch are (r) 20

23 typcally estmated va data augmentaton, the necessty to calculate choce probabltes at X o wll ncrease total computaton tme. The total ncrease n computaton tme for the probt model wll depend on the specfc smulaton method used (e.g the GHK or smple frequency smulator) and the accuracy desred (e.g. the number of smulates used). Increases n computaton tme wll also be drven by the dscrepancy between the base case and unconstraned market share forecasts; the greater the dscrepancy the longer the chan necessary to reach a statonary dstrbuton of parameters that satsfy the market share constrants. There are several other practcal mplementaton ssues that are detaled wth suggestons n the appendx. Prmary among these s a method for ncreasng τ n the dstrbuton for α ~ N p (0, τ -1 I p ) as the MCMC chan progresses, subject to A+B œ C. In the next secton we show results from smulated data sets ncludng an mportance samplng scheme used to estmate the value of the Bayes rsk for dfferent values of τ. 5. Smulaton Studes Analyses wth smulated data are presented to demonstrate the effcacy of the algorthms and results when the true values of the parameters are known. Results for MNL and correlated probt models are presented. The smulated data set for each analyss conssts of 300 respondents, 12 choce sets per respondent, wth each choce set consstng of four brands. Each brand n each choce set was descrbed by two randomly generated contnuous covarates and one bnary covarate ntended to mmc a dscrete product attrbute. A standard MNL model set-up s used. Choces are restrcted to one of the four alternatves and well known algorthms are used to estmate the herarchcal Bayes MNL model. A base set of attrbutes X o was randomly generated and the market share usng the actual set of β 21

24 was measured. The market share constrants, e.g. S 1,,S K, were then set arbtrarly, but dfferent than market share usng the true values of the parameters. The model s dentfed by droppng one of the brand ntercepts and the MCMC chan was run for 30,000 teratons wth a sample of every 10 th from the last 10,000 used to descrbe posteror moments. A correlated probt model was smulated wth "dual-response" data. In the "dualresponse" format, respondents ndcate whch of the four alternatves they prefer, and then n a second queston, ndcate whether or not they would actually purchase the tem. Data augmentaton methods are used to estmate the probt model. The scale of the error term s dentfed and the error covarance matrx s estmated usng Algorthm 3 n Noble (2000). The locaton s fxed by requrng the augmented varable w * for the preferred alternatve to be greater than 0 f t would actually be chosen, and less than 0 otherwse. Full detals on estmatng the probt model va data augmentaton are avalable n McCulloch and Ross (1994), Noble (2000), and Ross, Allenby, and McCulloch (2006). Ths method of dentfyng the model allows all four brand ntercepts to be estmated. A smple "frequency smulator" wth the dentfyng restrctons and 100 smulates was used to estmate choce probabltes and market shares at X o. Other aspects of ths smulaton are smlar to the MNL model. ============= Table 1 ============ ============== Table 2 ============ Tables 1 and 2 present selected results from the smulatons wth the target level of dscrepancy δ t set at 0.10, 0.05, and In all nstances the algorthm was able to ncrease τ to the maxmum value of 100,000 (varance = τ -1 =.00001) whle meetng the market share constrants. Selected ndvdual level hstograms of β + α were nspected and t does not appear that f(a, B W, C, τ) s dsjont for these data sets. We therefore present the emprcal average of 22

25 ( β + α ) and the standard devaton averaged across ndvduals and draws from the MCMC chan. These are provded as a pont of comparson to the posteror mean of θ and λ pp, from the dstrbuton of heterogenety. Note that brand ntercept parameters are adjusted n the expected drecton gven the dfferences between the actual and constraned market share. Fgures 2 and 3 plot the ndvdual level average ( β + α ) compared to the ndvdual level average β for selected parameters and models when δ t =.02. The plots show that ndvdual level parameters are adjusted dfferently, but that adjustments are generally small and drectonally consstent. ============= Fgure 2 ============ ============== Fgure 3 ============ Taken together, these suggest that the loss functon approach s capable of meetng the market share constrants wth mnmal nfluence on the average preference structure from just the CBC data. However, the exact amount of change needed to meet the market share constrants wll depend on each data set and the dscrepancy wth the base case market share. Note also that the average values are not used n market share smulatons; they would not necessarly match the base case scenaro. The market share constrants are met by usng the addtve factors A to coordnate the draws of B + A n each teraton of the MCMC chan. Addtonal smulatons and mplcatons are explored usng the MNL model wth δ t =.02; the MNL model was chosen for analytcal convenence. Frst, the Bayes rsk was nvestgated for dfferent values of τ. Usng equaton (7), the natural log of the Bayes rsk was calculated for the functon h(b+a) that estmates the market share at the base case scenaro X o. Draws of B + A from the MCMC sampler were used together wth an mportance samplng 23

26 algorthm to estmate the log Bayes rsk along a grd of values for τ, presented as τ -1 n Table 3. Full detals on the mportance samplng algorthm are avalable from the authors. The Bayes rsk s domnated by ln[ P(C τ)] and snce the market share constrants are bndng, the smallest value of τ -1 mnmzes the log Bayes rsk. The lower half of Table 3 shows addtonal quanttes calculated usng the mportance sampler. The N ' E α α X, C, τ measures the dsperson of A 1 from 0. As expected, t reaches ts mnmum value at the mnmum value of τ -1, but the mprovement s margnal beyond τ -1 = The values of d* equal to E[h(A + B) W, C, τ] were the same up to the thrd decmal place for all the dfferent values of τ -1. ============== Table 3 ============== Snce the Bayes rsk nvolves var[h m (A + B)], then the optmal value of τ wll be dependent on the form of h(a + B), or for purposes of market forecasts, the values of attrbutes used n the market smulator. A dogmatc Bayesan wll determne the optmal value of τ for each dfferent form of h(a + B) and dfferent values of attrbutes; an mportance samplng algorthm can be used for approxmatng the Bayes rsk wth any arbtrary h(a + B). However, the above analyss suggests a more pragmatc approach of condtonng on the value of τ. When A s needed, set τ as large as possble and use draws of h(a + B) from f(a, B W, C, τ) to estmate E[h(A + B) W, C, τ]. These values can then be used drectly n a market smulator (or be used n an mportance samplng scheme to determne the optmal value of τ). Although t s possble to nvestgate the value of τ that mnmzes the Bayes rsk, the practcal benefts of dong so are unclear. In the next set of smulatons, the actual values of β were systematcally altered to reflect hypotheszed dstortons n preferences as the result of partcpatng n a dscrete choce conjont 24

27 study. Forecasted market shares usng B+A are then compared to the market share usng the actual values of β. We fnd that when the market share constrants are accurate, the loss functon approach does a remarkably good job of forecastng. The actual values of β from the orgnal MNL smulaton were modfed n three a dfferent ways. Let β represent the set of β 's used n the orgnal smulaton. The frst modfcaton was to decrease the senstvty to the 4 th product attrbute. Ths was accomplshed b by settng β = β a 0. 5 for all ; the remanng coeffcents were not changed. Attrbute 4 was desgned to mmc a prce attrbute wth a true value of θ = -1.0 n the dstrbuton of heterogenety used to generate β. Some analysts beleve that respondents aren't as senstve to prce n conjont studes as they are when makng actual choces; ths modfcaton was desgned to reflect that vew. The second modfcaton was to scale all the coeffcents by a known constant: a c β β =. In the MNL model, ths amounts to decreasng the error varance 0.75 relatve to the true model. Fnally, the fourth set of coeffcents β reflected both these bases. a Smulated choces were generated usng each of the four sets of β's: β, d b β, and β. For β, standard herarchcal Bayes methods were used to obtan samples from the d a posteror dstrbuton of ndvdual level parameters. The base case scenaro used a randomly a generated X o and the actual β to compute the market shares. The base case scenaro and the loss functon approach were then used to obtan samples of B b +A b œ C, B c +A c œ C, and B d +A d œ C. In addton to market share forecasts at the base case scenaro, smulated market shares were generated for a stuaton where the attrbutes of Brand C were changed. In the "Product Change" stuaton, the dscrete attrbute (attrbute 6) was added to Brand C and the value of attrbute 4 was c β, 25

28 a decreased by 50%. Because we use smulated data and the actual values of β are known, we can compare the results of the market smulatons to the true values. ============== Table 4 ================= Table 4 shows the market share smulatons for the varous models usng the posteror dstrbutons of β 's or B+A. In each of the three models usng B+A, the analyss s condtoned on the value of τ -1 = Smulated market shares were estmated usng a sample sze of 1,000 from the approprate dstrbutons. The dscrepancy between the actual market share and the forecasted market share s agan measured by δ. Table 4 shows that the loss functon approach not only forecasted the base case market share very accurately, but was very accurate when predctng changes to the base case scenaro. For nstance, n Panel 3 when all coeffcents were dvded by 0.75, usng the posteror of the unadjusted β the forecasted market share at the base case yelded a dscrepancy measure δ = whereas the adjusted B c +A c yelded a δ = When the attrbutes of Brand C were changed, the forecasted market share usng B c +A c was very close to the market share usng the actual β, δ = The value of δ for the modfed values of β when usng the loss functon approach n the "Product Change" stuaton ranged from to Ths compares to δ = when the CBC data were generated a usng β and standard methods of estmatng the model were used. When the manageral base case market shares are accurate, the loss functon approach s able to mprove market share predctons even for product confguratons outsde the base case. a c 26

29 6. Emprcal Example Ths secton presents the results of a commercal market research study that nvolved CBC and managerally suppled market share constrants. Due to the propretary nature of the dataset, the specfc product and the attrbutes have been dsgused. The product category nvolved a durable consumer electronc devce that s typcally used n conjuncton wth another consumer durable. The product s currently avalable n the market, but management was nterested n measurng demand for products that ncluded many new features that were recently developed. Although compettve brands are avalable n the market, the nature of how the product s purchased made ncludng brand unfeasble n the current study. The CBC study ncluded 425 respondents who each provded dual response data on 15 choce sets. Each choce set conssted of three alternatves that were unquely descrbed by 20 bnary attrbutes and the prce; as noted above, brand name was not an element n the desgn matrx. Prce was entered as the natural log of prce n the response functon. Respondents were asked to ndcate whch of the three alternatves they preferred, and then n a follow-up queston, whether they would actually purchase the alternatve f t were avalable n the market. An uncorrelated, dual response probt model was used to represent the lkelhood functon wth a standard herarchcal structure to represent heterogenety; standard conjugate, but nonnformatve prors we used to complete the herarchy. The base case scenaro provded by the study sponsors dd not nclude market share for competng brands. Because of the nature of the product category, management was only able to provde nformaton on the proporton of customers who chose a representatve base product, versus the "none" opton. The base product was descrbed by the attrbutes t ncluded and ts prce. Management provded the choce share for two a pror market segments that were 27

30 dentfed by soco-demographc varables. These varables were also avalable from the CBC study partcpants. Although the loss functon approach was developed assumng that competng brands would make-up the base case scenaro, t s straght forward to adopt t to a stuaton nvolvng a "buy/no buy" choce set wth dfferent market segments. The loss functon approach was used to obtan draws from the posteror dstrbutons of π(b W) and f(a, B W, C, τ) wth δ t = A sngle analyss that produced draws from both dstrbutons was performed. The algorthm was run for 20,000 teratons. The target δ t was met at about teraton 3,000 and τ -1 met ts pre-specfed mnmum of at about teraton 4,000. A sample of every 10 th from the last 10,000 teratons was used to compute summary statstcs. Selected ndvdual level hstograms of β + α were nspected and t does not appear that f(a, B W, C, τ) s dsjont for ths data set. Table 5 contans the summary statstcs for the constraned and unconstraned parameters. For attrbutes ncluded n the base case product profle, the constraned estmates all ncreased n mportance. For attrbutes not ncluded n the base case product profle, parameter values generally decreased or stayed the same. The constraned parameters exhbted somewhat larger measures of heterogenety. =============== Table 5 =============== =============== Table 6 =============== Changes n the parameter estmates make sense gven the dscrepancy between the base case choce share and that obtaned usng the unconstraned parameters. All forecasts are based on a sample of 1,000 draws from the ndvdual level posteror dstrbuton of β or β + α. Table 6 shows that there was a szable dfference between the forecast usng the unconstraned parameters and the manageral base case. Usng the unconstraned parameters, respondents were forecasted to be much less lkely to choose the base product. For market segment #1, the base 28

31 case was 11.4% choosng the representatve product versus a manageral expectaton of 43.6%. Thus, t makes sense that attrbutes ncluded n the base case product would ncrease n ther relatve mportance n the adjusted parameters. The loss functon approach was able to match the base case choce share to wthn the pre-specfed level of accuracy despte the relatvely large dscrepancy between t and the forecasts usng the unadjusted parameters. Fgure 4 plots the ndvdual level average ( β + α ) compared to the ndvdual level average β for selected coeffcents. Coeffcent #3 was selected because the dfference between the posteror mean of θ = and the posteror mean of ( β + α ) = from Table 5 was about average; coeffcent #5 was selected because t had the largest dfference, θ = and ( β + α ) = Table 5 and Fgure 4 show that relatvely small changes n the β s were suffcent n order to meet the choce share constrant. Although the loss functon approach s desgned to mnmze changes to the unconstraned CBC estmates, the exact amount of change needed to ndvdual level parameters wll depend on factors such as the number of attrbutes and the dscrepancy between the data and the base case. Snce the method produces estmates of both B and B+A, the analyss and changes necessary to meet the market share constrants are completely transparent to both analysts and decson makers. ========== Fgure 4 ============= Ths example shows that the loss functon approach s able to meet market share constrants wth relatvely modest adjustments to ndvdual level parameters usng real data, even when there s a bg dfference between the base case and unconstraned forecast. Further, the method s suffcently flexble to adapt to base case scenaros that dffer from the Kbrands set-up used earler to defne the loss functon approach. The algorthm performed as expected, but we antcpate addtonal research wll provde areas for mprovng ts 29