A Network Equilibrium Framework for Internet Advertising: Models, Quantitative Analysis, and Algorithms

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1 A Network Equilibrium Framework or Internet Advertising: Models, Quantitative Analysis, and Algorithms Lan Zhao and Anna Nagurney IFORS - Hong Kong June 25-28, 2006

2 Lan Zhao Department o Mathematics and Computer Sciences SUNY/College at Old Westbury, NY11568 Anna Nagurney Radclie Institute Fellow Radclie Institute or Advanced Study 34 Concord Avenue Harvard University Cambridge, MA02138 Department o Finance and Operations Management Isenberg School o Management University o Massachusetts Amherst. MA01003

Concord Avenue Harvard University Cambridge, MA02138 Department o Finance and

3 Highlights o this Research It is the irst attempt to ormulate the competitive Internet marketing strategies as a network equilibrium problem, which allows us to take advantage o network theory in terms o qualitative analysis and computations.

network equilibrium problem, which allows us to take

4 Highlights o this Research A Variational Inequality model is also established or the equilibrium o competitive Internet marketing strategies.

5 Highlights o this Research The size o a irm s online budget should not be a pre-ixed number, but rather, the size should be elastically adjusted with the online marginal responses which is --aected by the online advertising eorts; --aected by the inherent nature o the internet medium.

adjusted with the online marginal responses which is --aected by the

6 Highlights o this Research A numerical example is demonstrated, which shows that online budget is an increasing unction o online marginal response (to online advertising eorts).

7 Highlights o this Research The existence and uniqueness conditions o the online marketing equilibrium are established.

8 Highlights o this Research An algorithm that takes the advantage o the network structure is proposed or the equilibrium solution. --size o the Internet marketing budget is calculated --allocation o the total budget to each o Internet websites is calculated

9 Highlights o this Research A numerical example is provided to test the algorithm

10 Network Modeling has been Used in Numerous Applications and Disciplines: Transportation Science and Logistics Telecommunications and Computer Science Regional Science and Economics Engineering Finance Operations Research and Management Science

Telecommunications and Computer Science Regional Science and

11 Literature Reerred Advertising Competition Under Consumers Inertia, Banerjee, B and S. Bandyopadhyay (2003), Marketing Science 22, Modeling the Clickstream: Implications or Web-based Advertising Eorts, P. Chatterjee D.L. Homan, and T. P. Novak (2003), Marketing Science 22,

Modeling the Clickstream: Implications or Web-based Advertising Eorts, P.

12 Literature Reerred The General Multimodal Network Equilibrium Problem with Elastic Demand, S. Daermos, 1982,, Networks 12, An Iterative Scheme or Variational Inequalities, S. Daermos, 1983, Mathematics Programming 28,

13 Literature Reerred Network Economics: A Variational Inequalities and Their Applications, A. Nagurney, 1993, Kluwer Academic Publishers. A Network Modeling Approach or the Optimization o Internet-Based Advertising Strategies and Pricing with a Quantitative Explanation o Two Paradoxes, L. Zhao and A. Nagurney, 2005, Netnomics, in press.

A Network Modeling Approach or the Optimization o Internet-Based Advertising

14 Assumptions There are N irms advertising in all mediums: one o-line medium and M online mediums. The o-line response is an increasing, concave unction o o-line advertising spending. r = r ( ) no no o

15 Assumptions Online response (Amount o click-through) is an increasing, concave unction o the online advertisement spending. Amount o click-through in website i is also impacted by advertisement spending on other websites. r = nw r nw ( w )

16 Online Advertising Budget to decide online/oline budget allocation, a irm needs to solve, Max no, nw ( r no + r nw ) s. t.: no + nw C n no, nw 0 where C n is irm s s total budget:

17 Online Advertising Budget Solving the Max problem, we mathematically prove that budget is an increasing unction o marginal response (to marketing investment): b = b ( η ) n n nw -- i additional online investment would yield more response than oline investment, then the irm is willing to increase online investment and reduce oline investment.

( η ) n n nw -- i additional online investment would yield more response than oline

18 Example r w 2 = w w + 2 r o 4 = o o + 1

19 Example The irm is to Max o, w ( r o + r w ) s. t.: o + w 900 o, w 0

20 Example -- Result Initial on line exp. Initial o line exp. Online margin O line margin adjustm ent w $500 o $400 η w η o $300 $600 $ $200 $700 $ $100 $800 $ $0

0.020 η o -0.008000 $300 $600 $300 0.016-0.

21 The Network Equilibrium Model Basic Assumptions -- There are n=1,2,,n irms compete in m=1,2,,m internet websites. -- each irm is to maximize its own aggregate ad results (amount o total click-through). --amount o click-through in website m or irm n is a unction o, where is a vector o ad expenditure o all irms on all websites. --irm s internet ad budget is an increasing unction o marginal click-through.

22 The Network Equilibrium Model Each irm is to: n=1,2,,n Max 1n,..., s. t.: mn MN M m= 1 0, M m= 1 m r mn mn ( b n ) ( η ) = 1,2,..., n M

23 The Network Equilibrium Model Ater applying Kuhn-Tucker conditions equilibrium conditions are obtained: 0 rn ( = mn λ λ *) n n ( b ( b = n n λ λ *), *), n n ( b ( b n n ns ns *), *), * > * > i i 0, 0, mn mn * * > = 0, 0, M m = 1 mn * + ns * = b n *

24 The Equivalent Network Model The dotted link is the dummy link that absorbs the budget surplus ns.

25 The Variational Inequality Formulation B( *) ( *) 0 S = { 0, n + 1 i= 1 i = C}

26 The Variational Inequality Formulation Let u( λ r ( ) = ( n ( b) = ( Κ = {( λ n, b) mn ( b ( ), n ), m= 1,..., M; n n = 1,..., N) MN+ N = 1,..., N), b) R, + = b } + mn ns n

27 The Variational Inequality Formulation Then, equilibrium online budget b* and its allocation * is a solution o u( *)( *) λ ( b*)( b b*) 0 (, b) Κ

28 This variational inequality can be solved by an iterative scheme where the unction in each o the sub problems is separable and quadratic (see Daermos and Sparrow (1969), Daermos (1980), Zhao and Daermos (1991), Nagurney (1999)). The solution (*,b*) determines the size o the online budget and its allocation.

29 Existence and Uniqueness o the Solution I vector unction (-u(), λ(b)) is strongly monotone on K, then Equilibrium o the competition exists. The equilibrium is unique. The algorithm or inding the equilibrium is convergent.

30 Suicient Conditions or Monotonicity The matrices o second derivatives o -r() and the irst derivatives o λ(b) are positive deinite.

31 Example There are two irms competing over three websites: u 11 = u 21 = u 31 = ;

32 Example Example , 5 90; = = + + = + = + = b b u u u λ λ

33 Solution n n (14.00,12.00,13.00,12.00,20.00,3.00) (10.54,6.21,0.00,4.25,7.60,1.68) (12.30,3.23,0.00,5.93,3.89,2.21) (12.31,3.08,0.00,8.48,0.52,2.93) b n (39,00,35.00) (16.76,13.53) (15.53,12.03) (15.38,11.92)

34 Future Research Modeling the impact o asymmetric inormation on optimal marketing strategies.

35 For more inormation see:

36 Thank you!!!

Key Words: Internet marketing, online advertising, pricing, network optimization, paradoxes.

Key Words: Internet marketing, online advertising, pricing, network optimization, paradoxes. A Network Modeling Approach for the Optimization of Internet-Based Advertising Strategies and Pricing with a Quantitative Explanation of Two Paradoxes Lan Zhao Department of Mathematics and Computer Information