a network equilibrium framework for internet advertising ... · highlights of this research ¾it is...

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.


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.


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.


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).


The existence and uniqueness conditionsThe existence and uniqueness conditionsof the online marketing equilibrium areof the online marketing equilibrium areestablished.established.


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


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.


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..


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


LetLet

∑ =+∈=Κ

=−=

==∂∂

=

++ },),(|),{(

),...,1),(()(

),...,1;,...,1,)(()(

nnsmnNMN

nn

mn

n

bffRbfbf

Nnbb

NnMmffrfu

λλ


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.

For more information see:

http://supernet.som.umass.edu

Thank you!!!Thank you!!!

a network equilibrium framework for internet advertising ... · highlights of this research ¾it is...

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