a network equilibrium framework for internet advertising ... · highlights of this research ¾it is...
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A Network Equilibrium FrameworkA Network Equilibrium Frameworkfor Internet Advertising:for Internet Advertising:
Models, Quantitative Analysis, andModels, Quantitative Analysis, andAlgorithmsAlgorithms
Lan Zhao Lan Zhao and Anna Nagurneyand Anna Nagurney
IFORS - Hong KongIFORS - Hong KongJune 25-28, 2006June 25-28, 2006
Lan ZhaoLan ZhaoDepartment of Mathematics and Computer SciencesDepartment of Mathematics and Computer Sciences
SUNY/College at Old Westbury, NY11568SUNY/College at Old Westbury, NY11568
Anna NagurneyAnna NagurneyRadcliffe Institute FellowRadcliffe Institute Fellow
Radcliffe Institute for Advanced StudyRadcliffe Institute for Advanced Study34 Concord Avenue34 Concord AvenueHarvard UniversityHarvard University
Cambridge, MA02138Cambridge, MA02138
Department of Finance and Operations ManagementDepartment of Finance and Operations ManagementIsenberg School of ManagementIsenberg School of Management
University of MassachusettsUniversity of MassachusettsAmherst. MA01003Amherst. MA01003
Highlights of this ResearchHighlights of this Research
It is the first attempt to formulate theIt is the first attempt to formulate thecompetitive Internet marketing strategiescompetitive Internet marketing strategiesas a network equilibrium problem, whichas a network equilibrium problem, whichallows us to take advantage of networkallows us to take advantage of networktheory in terms of qualitative analysistheory in terms of qualitative analysisand computations.and computations.
Highlights of this ResearchHighlights of this Research
A Variational Inequality model is alsoA Variational Inequality model is alsoestablished for the equilibrium ofestablished for the equilibrium ofcompetitive Internet marketing strategies.competitive Internet marketing strategies.
Highlights of this ResearchHighlights of this Research
The size of a firm’s online budget shouldThe size of a firm’s online budget shouldnot be a pre-fixed number, but rather, thenot be a pre-fixed number, but rather, thesize should be elastically adjusted with thesize should be elastically adjusted with theonline marginal responses which isonline marginal responses which is --affected by the online advertising efforts; --affected by the online advertising efforts; --affected by the inherent nature of the --affected by the inherent nature of theinternet medium.internet medium.
Highlights of this ResearchHighlights of this Research
A numerical example is demonstrated,A numerical example is demonstrated,which shows that online budget is anwhich shows that online budget is anincreasing function of online marginalincreasing function of online marginalresponse (to online advertising efforts).response (to online advertising efforts).
Highlights of this ResearchHighlights of this Research
The existence and uniqueness conditionsThe existence and uniqueness conditionsof the online marketing equilibrium areof the online marketing equilibrium areestablished.established.
Highlights of this ResearchHighlights of this Research
An algorithm that takes the advantage ofAn algorithm that takes the advantage ofthe network structure is proposed for thethe network structure is proposed for theequilibrium solution.equilibrium solution.--size of the Internet marketing budget is--size of the Internet marketing budget iscalculatedcalculated--allocation of the total budget to each of--allocation of the total budget to each ofInternet websites is calculatedInternet websites is calculated
Highlights of this ResearchHighlights of this Research
A numerical example is provided to testA numerical example is provided to testthe algorithmthe algorithm
Network Modeling has beenNetwork Modeling has beenUsed in NumerousUsed in Numerous
Applications and Disciplines:Applications and Disciplines:Transportation Science and LogisticsTransportation Science and LogisticsTelecommunications and Computer ScienceTelecommunications and Computer ScienceRegional Science and EconomicsRegional Science and EconomicsEngineeringEngineeringFinanceFinanceOperations Research and ManagementOperations Research and ManagementScienceScience
Literature ReferredLiterature Referred
Advertising Competition Under ConsumersAdvertising Competition Under ConsumersInertia,Inertia, Banerjee, B and S. Bandyopadhyay Banerjee, B and S. Bandyopadhyay(2003), Marketing Science 22, 131-144.(2003), Marketing Science 22, 131-144.
Modeling the Clickstream: Implications forModeling the Clickstream: Implications forWeb-based Advertising Efforts, Web-based Advertising Efforts, P. ChatterjeeP. ChatterjeeD.L. Hoffman, and T. P. Novak (2003),D.L. Hoffman, and T. P. Novak (2003),Marketing Science 22, 520-541.Marketing Science 22, 520-541.
Literature ReferredLiterature Referred
The General Multimodal NetworkThe General Multimodal NetworkEquilibrium Problem with Elastic Demand,Equilibrium Problem with Elastic Demand,S. DafermosS. Dafermos, , 19821982, Networks 12, 57-72., Networks 12, 57-72.An Iterative Scheme for VariationalAn Iterative Scheme for VariationalInequalities, Inequalities, S. Dafermos, 1983,S. Dafermos, 1983,Mathematics Programming 28, 174-185Mathematics Programming 28, 174-185..
Literature ReferredLiterature Referred
Network Economics: A VariationalNetwork Economics: A VariationalInequalities and Their Applications, Inequalities and Their Applications, A.A.Nagurney, 1993,Nagurney, 1993, KluwerKluwer AcademicAcademicPublishersPublishers..A Network Modeling Approach for theA Network Modeling Approach for theOptimization of Internet-Based AdvertisingOptimization of Internet-Based AdvertisingStrategies and Pricing with a QuantitativeStrategies and Pricing with a QuantitativeExplanation of Two Paradoxes, L. ZhaoExplanation of Two Paradoxes, L. Zhaoand A. Nagurney, 2005, Netnomics, inand A. Nagurney, 2005, Netnomics, inpress.press.
AssumptionsAssumptionsThere are N firms advertising in all mediums:There are N firms advertising in all mediums:one off-line medium and M online mediums.one off-line medium and M online mediums.
The off-line response is an increasing,The off-line response is an increasing,concave function of off-line advertisingconcave function of off-line advertisingspending.spending.
)( onono frr =
AssumptionsAssumptionsOnline response (Amount of click-through) isOnline response (Amount of click-through) isan increasing, concave function of the onlinean increasing, concave function of the onlineadvertisement spending.advertisement spending.Amount of click-through in website i is alsoAmount of click-through in website i is alsoimpacted by advertisement spending on otherimpacted by advertisement spending on otherwebsites.websites.
)( wnwnw frr =
Online Advertising BudgetOnline Advertising Budget
to decide online/offline budget allocation, a firmto decide online/offline budget allocation, a firmneeds to solveneeds to solve , ,
where where CCnn is firm is firm’’s total budget:s total budget:
0,:..
)(,
≥≤+
+
nwno
nnwno
nwnoff
ffCffts
rrMaxnwno
Online Advertising BudgetOnline Advertising BudgetSolving the Max problem, we mathematicallySolving the Max problem, we mathematicallyprove that budget is an increasing function ofprove that budget is an increasing function ofmarginal response (to marketing investment):marginal response (to marketing investment):
-- if additional online investment would -- if additional online investment would yield more response than offline yield more response than offline investment, then the firm is willing to investment, then the firm is willing to increaseincreaseonline investment and reduce online investment and reduce offlineofflineinvestment.investment.
)( nwnn bb η=
ExampleExample
11502
1500004
21004
1000002
2
2
++−=
++−=
ooo
www
ffr
ffr
ExampleExample
The firm is toThe firm is to
0,900:..
)(,
≥≤+
+
wo
wo
woff
ffffts
rrMaxwo
Example -- ResultExample -- Result
$0$00.0080000.0080000.0080.008$100$100$800$800
$100$1000.0026670.0026670.0120.012$200$200$700$700
$200$200-0.002667-0.0026670.0160.016$300$300$600$600
$300$300-0.008000-0.0080000.0200.020$400$400$500$500
adjustmadjustmentent
Off lineOff linemarginmarginηηoo
OnlineOnlinemarginmarginηηww
Initial offInitial offline exp.line exp.ffoo
Initial onInitial online exp.line exp.ffww
The Network Equilibrium ModelThe Network Equilibrium ModelBasic AssumptionsBasic Assumptions-- There are n=1,2,…,N firms compete in-- There are n=1,2,…,N firms compete inm=1,2,…,M internet websites.m=1,2,…,M internet websites.-- each firm is to maximize its own aggregate ad-- each firm is to maximize its own aggregate adresults (amount of total click-through).results (amount of total click-through).--amount of click-through in website m for firm n--amount of click-through in website m for firm nis a function of f, where f is a vector of adis a function of f, where f is a vector of adexpenditure of all firms on all websites.expenditure of all firms on all websites.--firm’s internet ad budget is an increasing--firm’s internet ad budget is an increasingfunction of marginal click-through.function of marginal click-through.
The Network Equilibrium ModelThe Network Equilibrium ModelEach firm is to:Each firm is to:
n=1,2,…,Nn=1,2,…,NMmf
bfts
frMax
mn
nn
M
mmn
M
mmnff MNn
,...,2,1,0
)(:..
)(
1
1,...,1
=≥
≤∑
∑
=
=
η
The Network Equilibrium ModelThe Network Equilibrium ModelAfter applying Kuhn-Tucker conditionsAfter applying Kuhn-Tucker conditionsequilibrium conditions are obtained:equilibrium conditions are obtained:
∑=
=+
>≤>=
=≤>=
∂∂
M
mnnsmn
nsnn
nsnn
mnnn
mnnn
mn
n
bff
fbfb
fifbfifb
ffr
1***
,0**),(,0**),(
0
,0**),(,0**),(*)(
λλ
λλ
The Equivalent Network ModelThe Equivalent Network Model
The dotted link is the dummy link that absorbs theThe dotted link is the dummy link that absorbs thebudget surplus fbudget surplus fns.ns.
},0|{
0*)(*)(1
1CfffSf
fffBn
ii =≥=∈∀
≤−•∇
∑+
=
The Variational InequalityThe Variational InequalityFormulationFormulation
The Variational InequalityThe Variational InequalityFormulationFormulation
LetLet
∑ =+∈=Κ
=−=
==∂∂
=
++ },),(|),{(
),...,1),(()(
),...,1;,...,1,)(()(
nnsmnNMN
nn
mn
n
bffRbfbf
Nnbb
NnMmffrfu
λλ
The Variational InequalityThe Variational InequalityFormulationFormulation
Then, equilibrium online budget b* and itsThen, equilibrium online budget b* and itsallocation f* is a solution ofallocation f* is a solution of
Κ∈∀≤−−−
),(0*)*)((*)*)((
bfbbbfffu λ
This variational inequality can be solved byThis variational inequality can be solved byan iterative scheme where the function inan iterative scheme where the function ineach of the sub problems is separable andeach of the sub problems is separable andquadratic (see Dafermos and Sparrow (1969),quadratic (see Dafermos and Sparrow (1969),Dafermos (1980), Zhao and Dafermos (1991),Dafermos (1980), Zhao and Dafermos (1991),Nagurney (1999)).Nagurney (1999)).
The solution (f*,b*) determines the size of theThe solution (f*,b*) determines the size of theonline budget and its allocation.online budget and its allocation.
Existence and Uniqueness of theExistence and Uniqueness of theSolutionSolution
If vector function (-u(f), If vector function (-u(f), λλ(b)) is strongly(b)) is stronglymonotone on K, thenmonotone on K, then
Equilibrium of the competition exists.Equilibrium of the competition exists.The equilibrium is unique.The equilibrium is unique.The algorithm for finding the equilibrium isThe algorithm for finding the equilibrium isconvergent.convergent.
Sufficient Conditions forSufficient Conditions forMonotonicityMonotonicity
The matrices of second derivatives of -r(f)The matrices of second derivatives of -r(f)and the first derivatives of and the first derivatives of λλ(b) are positive(b) are positivedefinite.definite.
ExampleExampleThere are two firms competing over threeThere are two firms competing over three
websites:websites:
;452805.14
1002
322131
222121
121111
++−=+−−=
+−−=
ffuffu
ffu
ExampleExample
108,105
;9025803905.0
2211
313232
212222
111212
−=−=
++−=+−−=+−−=
bb
ffuffuffu
λλ
SolutionSolution
(15.38,11.92)(15.38,11.92)(12.31,3.08,0.00,8.48,0.52,2(12.31,3.08,0.00,8.48,0.52,2.93).93)
2323
…………………………
(15.53,12.03)(15.53,12.03)(12.30,3.23,0.00,5.93,3.89,2(12.30,3.23,0.00,5.93,3.89,2.21).21)
22
(16.76,13.53)(16.76,13.53)(10.54,6.21,0.00,4.25,7.60,1(10.54,6.21,0.00,4.25,7.60,1.68).68)
11
(39,00,35.00)(39,00,35.00)(14.00,12.00,13.00,12.00,20(14.00,12.00,13.00,12.00,20.00,3.00).00,3.00)
00
bbnnffnnnn
Future ResearchFuture Research
Modeling the impact of asymmetric informationModeling the impact of asymmetric informationon optimal marketing strategies.on optimal marketing strategies.
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