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 = −

9

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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

23

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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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