combining managerial insights with predictive models when ...ibm demandtec, dunnhumby’s kss...
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Combining Managerial Insights with Predictive Models when Setting Optimal Prices A l a n L . M o n t g o m e r y P r o f e s s o r o f M a r ke t i n g , C a r n e g i e M e l l o n U n i v e r s i t y V i s i t i n g P r o f e s s o r, H K U E - m a i l : a l a n m o n t g o m e r y @ c m u . e d u A l l R i g h t s R e s e r v e d , © 2 0 1 8 A l a n M o n t g o m e r y
D o n o t d i s t r i b u t e o r r e p r o d u c e w i t h o u t A l a n M o n t g o m e r y ’s P e r m i s s i o n
Research Goal Apopularfeatureofbothoptimalpricingsolutionsinbothindustryandacademicsistoimplementbusinessruleswhichimposeconstraintsontheoptimalpricingsolution.
Theseconstraintsreflectpriorinformationonthepartoftheuseraboutthepricingsolution.However,currentapproachestreatthisinformationinanad-hocmanner.
Thisresearchshowshowbusinessrulescanbeformulatedaspriorinformationtoyieldbetterpricingdecisions.
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Example Why did Amazon charge $2m for textbook?
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Example What is the optimal price for “Asparagus Water”?
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Broader Research Question
HowtocombineDataScience/MLwithHuman/Expert/ManagerialIntuition/Judgment?◦ Onlyusethedata(objectivebutinefficient)◦ Onlyusemanagerialinsight(subjectivebutexplainable)◦ Usedataunlessconflictswithmanagerialinsight◦ Usemanagerialinsightunlessitconflictswithdata◦ Combinebothtogether(weighteachappropriately)
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Outline Currentpracticeofpriceoptimization Businessrulesaspriorinformation Creatinginformativepriorsfrombusinessrules EmpiricalExample
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Current State of Price Optimization
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Optimal Product Pricing Profits:
Optimalpricingrule:
Wherepriceelasticitymeasuresdemandresponsivenesstopricechanges:
*
1p cβ
β=
+
%%
q p qp q p
β ∂ Δ= ≈∂ Δ
( )p c qΠ = −
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Optimal Product Line Pricing TotalProfits:
Optimalpricingrule:
Wherecrosspriceelasticitiesmeasurecompetitiveeffects:
*
1ii
i iii j ji j i
j i
p cs s
ββ µ β
≠
=+ +∑
% ,%
i i iij
i i j
q p qp q p
β ∂ Δ= ≈∂ Δ
1( )
M
i i iip c q
=
Π = −∑
, i i i ii i
i j jj
p c p qsp p q
µ −= =∑
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Price Optimization in Practice Hugegrowthofpriceoptimizationinpracticeforbothretailersandmanufacturers:◦ IBMDemandTec,Dunnhumby’sKSSRetail,Oracle,PROS,SAPKhimetrics,Vendavo,VistaarTechnologies,andZilliant
GartnerMarketscope(2010)statesthat“priceoptimizationtechnologywillhaveamoredirectimpactonincreasingrevenueormarginsthananyotherCRMtechnology”.◦ TheYankeeGroupestimatedthatmorethanonebilliondollarswouldbespentonthesesystemsin2007
2012Gartnerstudy(Fletcher2012)showedgainsof2to4%ofrevenuefromusingthesesystems.◦ Anecdotalreportssuggestincreasesingrossmarginsintherangeof2-8%,retailerstypicallyhavegrossmarginsofabout25%andhaveannualrevenuesof$2.5trillion.
◦ Suggestsbenefitforretailersalonewouldbebetween$12.5band$50bannually.
Germannetal(2014)findretailersbenefitfromdeployingcustomeranalyticsmorethanfirmsinmostinudstries◦ CompredwithMediaEntertainment,Energy,Insurance,Telecom,Hospitality,Banking
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Traditional Modeling Process Themulti-productpricingdecisionprocess:
Demand Estimation Price Optimization quantity sold
past prices estimates (elasticities)
optimal prices are constrained or ignore business rules as non-binding
Demand Model linear, log-log, log-linear, etc Estimation:
ols, mle, bayesian, etc
maxp profit(p|q) s.t. fi(p|q) ≤ 0 Impose constraints post hoc
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Business Rules as Prior Information
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Business Rules Currentpricingsolutionsfrequentlyimplementconstraintsthatreflect“businessrules”,whichcodifymanagerknowledge:◦ Allowednumberandfrequencyofmarkdowns(e.g.,atleastaweekbetweentwoconsecutivemarkdowns)
◦ Min-maxdiscountlevelsormaximumlifetimediscount◦ Minimumnumberofweeksbeforeaninitialmarkdowncanoccur◦ Typesofmarkdownsallowed(e.g.,10%,25%,…)orthepermissiblesetofprices
◦ The“family”ofitemsthatmustbemarkeddowntogether
Source:ElmaghrabyandKeskinocak(2003;ManagementScience)
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Why use business rules? Answer:to“improve”thepricingsolutionandfindabetteronethanwouldbeaffordedwithouttheseconstraints Examples:◦ Strategicdecision(ElmaghrabyandKeskinocak2003)◦ Ensuredesiredpositioningoftheproduct(Hawtin2002)
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DemandTec’s Rule Relaxation Approach 1. Grouppriceadvanceordeclinerules.Theusersetsamaximum
weightedgrouppriceadvanceordeclineto10%.
2. Sizepricingrules.Theusergoeswiththedefaultthatlargeritemscostlessperequivalentunitthansmalleridenticalitems.
3. Brandpricingrules.Forsoftdrinks,theuserdesignatesthepriceofbrandAisneverlessthanthepriceofbrandB.ForjuicestheuserdesignatesthatbrandCisalwaysgreaterthanBrandD.
4. Unitpricingrules.Theusergoeswiththedefaultthattheoverallpriceoflargeritemsisgreaterthantheoverallpriceofsmalleridenticalitems.
5. Competitionrules.Theuserdesignatesthatallpricesmustbeatleast10%lessthanthepricesofthesameitemssoldbycompetitorXandarewithin2%ofthepricesofthesameitemssoldbycompetitorY.
6. Linepricerules.Theuserdesignatesthatdifferentflavorsofthesameitemarepricedthesame.
Source:Nealetal.(2010),Patent#7617119 17
Examples of Business Rules in Academic Research ◦ CorstjensandDoyle(1981)useconstraintstorepresentstorecapacity,availabilityandcontrol
◦ Montgomery(1997)constrainstheoptimalpricesubjecttoanaveragepriceand/ortotalrevenueconstraint
◦ SubramanianandSherali(2010)ensurethatnewsolutiondoesnotdeviatetoofarfromcurrent
◦ DengandYano(2006)boundpricerangestoensurereasonablesolutions◦ BitranandModschein(1997)andFengandGallego(1995)allowonlyafixednumberofpricechanges
◦ GallegoandvanRyzin(1994)priceshavetobechosenfromadiscretesetofpossibleprices
◦ ReibsteinandGatignon(1984)pricesofsmallersizescannotbegreaterthanthepriceoflargersizes
◦ Cohenetal(2015)usesrulestoavoidpromotionwearout,9¢priceendings,andconstraintsonthenumberandtimingofpromotions.
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Conjecture Businessrulesandtheconstraintsonthepricingsolutiontheyimplyareaverypopularfeatureofcurrentpricingsolutions Theproblemisthatthisisanon-Bayesiansolutionsinceconstraintsrepresentaprioriinformationaboutthesolution(eithertocompensateformodelweaknessesortodirectinformationaboutthesolution) Abetterapproach(fromadecisiontheoreticstandpoint)istofollowaconsistentlyBayesianframeworkifwetrulywishtohaveanoptimaldecision.
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Business Rules constitute Prior Information Anystatementsorconstraintsonoptimalpricesimplicitlyconstitutepriorbeliefs◦ Themanagerholdsthesebeliefsevenbeforelookingatthedata
Optimalpricesarefunctionsofpriceelasticities Therefore,manager’spriorbeliefsaboutoptimalpricesimplypriorsontheparameterspace
PriorbeliefsshouldberepresentedaspriorinformationifwefollowBayesianprinciples Therefore,businessrulesshouldbeemployedapriori
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Example Considerthefollowingsituation:◦ Apricingmanagerreceivestheoutputfromapricingoptimizationsolution,andthenremarksthatthissolutiondoesn’tlookright—theoptimalpriceshouldnotbemorethan5%higher.
Therearetwopossibilities:◦ Thepricingmanagerhasindependentinformationfromthedata/model(e.g.,wasnotinvolvedintheanalysis)andistryingtocombinetwosourcesofinformation
◦ Thepricingmanagerhasdependentinformationfromthedata/modelandistryingtoinfluencetheposteriorsolutiontoconformtohispriorbeliefs.However,thenaturalsequenceisinverted–posterior/dataàprior
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Proposed Modeling Process Themulti-productpricingdecisionprocess:
Demand Estimation Price Optimization quantity sold
past prices estimates (elasticities)
optimal prices use prior knowledge efficiently
Demand Model parametric or nonparametric Estimation: bayesian
maxp profit(p|q) No need to impose constraints since all rules already reflected in estimates
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prior information from business
rules
Creating Informative Priors from Business Rules
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Practical Problem Bayesianmodelsforceanalyststoformulatepriorsintermsofconjugate(ornon-conjugate)priorsontheparameterswhichrepresentallknowninformation. Themanagerinthepreviouscasepossessesanassessmentaboutoptimalprice—amarginalpropertyofthemodel—nottheparametersofthemodel. Ourproposalistoinferapriorfromthemarginaldistributionbyinvertingthemanager’sassessmentabouttheoptimalpriceintoanimpliedprior.
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Log-Linear Demand Example Supposequantitydemandfollowsalog-linearmodel,whereqisquantityandpisprice:
Theoptimalprice(conditioneduponparameter)andthecost(c):
Andwehaveabusinessrule:
2log( ) log( ) , ~ (0, )q p Nα β ε ε σ= + +
p* = f (β ) = β
1+ βc
c ≤ p* ≤ τ
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Change of Variable Solution Moregenerallywecanthinkofouroptimalpriceasafunctionofthepriceelasticity:
Wecantransformbetweentheoptimalpricespaceandthepriceelasticityspaceusinginversefunctions:
Theimpliedpriorcanbecomputedfromachangeofvariablesapproach,whichgivesthedistributionofoptimalprices:
( ) ( )* , where 1 1
p c f c fβ ββ ββ β
= ⋅ = ⋅ =+ +
( )* *
1 1* , where
1p p xf f xc c p x
β − −⎛ ⎞= = =⎜ ⎟ − −⎝ ⎠
* 1 * 2* 1
* * * 2( )( )p
p df p cp p p f pc dp c p c pβ β
−−⎛ ⎞⎛ ⎞ ⎛ ⎞
= ⋅ = ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠
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Decision Rules BayesianRulemaximizestheposteriorexpectationofprofits.
Thetraditionalapproach:
ThedifferencebetweenthetwoistheBayesianruleisunconstrained–italreadyincorporatestheinformation,whilethetraditionalapproachintroducestheruleposthoc.
Caution:Pricesusedinthreedistinctcontexts:◦ Observations(Data),OptimalPrice(Distribution),PricefromDecisionRule(Point)
!p = min argmax
pE π p,β( ) | D⎡⎣ ⎤⎦ ,τ⎛
⎝⎜⎞⎠⎟
!p = argmax
pE π p,β( ) | D ,R : p* ∈(c,τ )⎡⎣ ⎤⎦
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-6 -5 -4 -3 -2 -1 0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Price Elasticity Posterior Distribution
Beta
Density
Bayes
Traditional
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1.2 1.4 1.6 1.8 2.0
01
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Optimal Price Posterior Distribution
Price
Density
Bayes
Traditional
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Example Results: Elasticity and Optimal Price Estimates
Rule Price Elasticity Optimal Price Traditional -3.00 (1.00) 1.67
Bayesian -3.29 (0.79) 1.49 (0.18)
Weak Prior on Elasticity, optimal price < $2.00
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Example Results: Elasticity and Optimal Price Estimates
Rule Price Elasticity Optimal Price Traditional -3.00 (0.71) 1.50
Bayesian -3.57 (0.43) 1.40 (0.06)
Strong Prior on Elasticity, optimal price < $1.50
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Example 2: Optimal Price Prior with 3 Rules 1) Range ($2.99,$4.99), 2) 9¢ endings, 3) Near $3.19
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Example 2: Optimal Price Prior with 3 Rules 1) Range ($2.99,$4.99), 2) 9¢ endings, 3) Near $3.19
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Example 2: Optimal Price Prior with 3 Rules 1) 9¢ endings, 2) Near $2.99, 3) Range ($2.99,$4.99)
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Example 2: Optimal Price Prior with 3 Rules 1) 9¢ endings, 2) Near $2.99, 3) Range ($2.99,$4.99)
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Summary Thetraditionalapproachsplitstheproblemintoanestimationandinferencescheme.◦ Businessrulesareusedinanex-postmanner,andiftheyarenotbindingtheyareignored.
TheinferentialapproachisBayesianduringestimation.◦ Heretheoptimalpriceisconstrainedbuttheparameterestimatesarenotchanged.
TheBayesianDecisionTheoreticapproachcombinesestimationandinferenceintoasingletask.◦ Theestimatesfrombothareconsistentandefficient.Informationfrombusinessrulesareusedaprioriandoptimalpricesareunconstrained.
Substantialandpracticaldifferencesamongsttheapproaches.
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Empirical Example
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Multivariate Pricing Problem SystemofdemandequationsoverMproducts:
Themanagerwishestooptimizetotalprofitswhenfacedwithknownvariablescosts:
Optimalpricemarginssolvethefirstordercondition: Π = ( pi − ci )E qi⎡⎣ ⎤⎦
i∑
m = −diag(w) ′B( )−1w = −diag(r) ′B( )−1
r
ln(qit ) =α i + ηij
j=1
N
∑ pjt + ξi fit +ψ idit + ε it , ε it ~ N (0,σ i2 )
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Business Rules Individualpricebounds◦ Optimalpricesarewithin20%ofcurrentprices◦ Alternative:pricesabovecostandlowerthancostx150%
Enforceprice-qualitytiers◦ High-qualitytieraregreaterthannationalbrandprices◦ Nationalbrandpricesaregreaterthanstorebrandprices
p0i
* ≤ pi* ≤ p1i
*{ }
max pA
*( ) ≤ min pB*( ),max pB
*( ) ≤ min pC*( )
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Posterior Price Coefficients
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Optimal Prices Tier Product Traditional Bayesian Change
Premium TropPrem64 .0526(.0033)
.0515(.0046)
-2.1%
National TropReg64 .0408(.0027)
.0379(.0033)
-7.1%
National MinMaid64 .0409(.0028)
.0360(.0029)
-12.0%
Store Dom64 .0321(.0022)
.0276(.0029)
-14.0%
TotalProfits: $11,074(2,372)
$7,073(1,135)
-36.1%
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Findings Treatingbusinessrulesaspriorinformationhasahugeimpacton◦ Parameterestimates◦ Optimalprices◦ Expectedprofitability
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Conclusions
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Conclusion Currentoptimalpricingpractitionersandresearchersareintroducinginformationinanadhocmannerbyrelyingupon“businessrules”orconstraints. AnappropriateBayesianmethodiscorrectandefficientandshouldavoidadhoc“corrections”totheposterior.
Thisapproachiseasyandcouldleadtobetterdecisionsupportsystemsthatreflect“expert”knowledgeefficiently.
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Further Results WehaveembeddedtheoptimizationproblemwithinaMCMCalgorithm,whichgivesageneralapproachforoptimizationunderuncertainty◦ Werelaxthecertaintyofthebusinessrules(perhapsthedatacaninformwhetherthebusinessruleiscorrect)
◦ Sometimesbusinessrulesareusedtotestthemodel(e.g.,ifthemodelgivesstrangesolutionsthenitiswrong)
◦ Relaxtheassumptionaboutfunctionalformandcanlearnitfromthedatausingnonparametrictechniques
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