# American Institute of Mathematical Sciences

April  2017, 4(2): 97-124. doi: 10.3934/jdg.2017007

## Dynamic modeling of nontargeted and targeted advertising strategies in an oligopoly

 Department of Computer Science, College of Charleston, Charleston, SC 29424, USA

* Corresponding author: J. S. Howell

Received  May 2016 Revised  December 2016 Published  March 2017

With the growing collection of sales and marketing data and depth of detailed knowledge of consumer habits and trends, firms are gaining the capability to discern customers of other firms from the potential market of uncommitted consumers. Firms with this capability will be able to implement a strategy where the advertising effort towards customers of competing firms may differ from that towards uncommitted consumers. In this work, dynamic models for advertising in an oligopoly setting with fixed total market size and sales decay are presented. Two models are described in detail: a nontargeted model in which the advertising effort is the same for both categories of prospective customers, and a targeted model that gives firms the capability to allocate effort across the two categories differently. In the differential game setting, open-loop and closed-loop Nash equilibrium strategies are derived for both models. Several strategic questions that a firm may face when practicing targeted advertising on a fixed budget are discussed and addressed.

Citation: Chloe A. Fletcher, Jason S. Howell. Dynamic modeling of nontargeted and targeted advertising strategies in an oligopoly. Journal of Dynamics & Games, 2017, 4 (2) : 97-124. doi: 10.3934/jdg.2017007
##### References:
 [1] F. M. Bass, A. Krishnamoorthy, A. Prasad and S.P. Sethi, Advertising competition with market expansion for finite horizon firms, Journal of Industrial and Management Optimization, 1 (2005), 1-19.  doi: 10.3934/jimo.2005.1.1.  Google Scholar [2] F. M. Bass, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Generic and brand advertising strategies in a dynamic duopoly, Marketing Science, 24 (2005), 556-568.  doi: 10.1287/mksc.1050.0119.  Google Scholar [3] P. K. Chintagunta and N. J. Vilcassim, An empirical investigation of advertising strategies in a dynamic duopoly, Management Science, 38 (1992), 1230-1244.  doi: 10.1287/mnsc.38.9.1230.  Google Scholar [4] E. J. Dockner, S. Jørgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, 2000. doi: 10.1017/CBO9780511805127.  Google Scholar [5] D. Dragone, L. Lambertini and A. Palestini, The Leitmann-Schmitendorf advertising game with $n$ players and time discounting, Applied Mathematics and Computation, 217 (2010), 1010-1016.  doi: 10.1016/j.amc.2010.02.031.  Google Scholar [6] G. M. Erickson, Differential game models of advertising competition, European Journal of Operational Research, 83 (1995), 431-438.  doi: 10.1016/0377-2217(94)00232-2.  Google Scholar [7] G. Feichtinger, The Nash solution of an advertising differential game: Generalization of a model by Leitmann and Schmitendorf, Automatic Control, IEEE Transactions on, 28 (1983), 1044-1048.  doi: 10.1109/TAC.1983.1103174.  Google Scholar [8] G. E. Fruchter, The many-player advertising game, Management Science, 45 (1999), 1609-1611.  doi: 10.1287/mnsc.45.11.1609.  Google Scholar [9] G. E. Fruchter, Oligopoly advertising strategies with market expansion, Optimal Control Applications and Methods, 20 (1999), 199-211.  doi: 10.1002/(SICI)1099-1514(199907/08)20:4<199::AID-OCA653>3.0.CO;2-M.  Google Scholar [10] G. E. Fruchter, Advertising in a competitive product line, Int. Game Theory Rev., 3 (2001), 301-314.  doi: 10.1142/S0219198901000439.  Google Scholar [11] G. E. Fruchter and S. Kalish, Closed-loop advertising strategies in a duopoly, Management Science, 43 (1997), 54-63.  doi: 10.1287/mnsc.43.1.54.  Google Scholar [12] G. E. Fruchter and Z. J. Zhang, Dynamic targeted promotions a customer retention and acquisition perspective, Journal of Service Research, 7 (2004), 3-19.   Google Scholar [13] R. F. Hartl and P. M. Kort, Advertising directed towards existing and new customers, in Optimal Control and Dynamic Games (eds. C. Deissenberg and R. 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Jørgensen and S.-P. Sigué, Defensive, offensive, and generic advertising in a lanchester model with market growth, Dynamic Games and Applications, 1–17, URL http://dx.doi.org/10.1007/s13235-015-0147-1. doi: 10.1007/s13235-015-0147-1.  Google Scholar [19] S. Jørgensen and G. Zaccour, Differential Games in Marketing, Springer, 2004. Google Scholar [20] G. E. Kimball, Some industrial applications of military operations research methods, Operations Res., 5 (1957), 201-204.  doi: 10.1287/opre.5.2.201.  Google Scholar [21] A. Krishnamoorthy, A. Prasad and S. P. Sethi, Optimal pricing and advertising in a durable-good duopoly, European Journal of Operational Research, 200 (2010), 486-497.  doi: 10.1016/j.ejor.2009.01.003.  Google Scholar [22] F. W. Lanchester, Aircraft in Warfare: The Dawn of the Fourth Arm, Appleton, New York, 1916. Google Scholar [23] G. Leitmann and W. Schmitendorf, Profit maximization through advertising: a nonzero sum differential game approach, Automatic Control, IEEE Transactions on, 23 (1978), 645-650.  doi: 10.1109/TAC.1978.1101794.  Google Scholar [24] D. Liu, S. Kumar and V. S. Mookerjee, Advertising strategies in electronic retailing: A differential games approach, Information Systems Research, 23 (2012), 903-917.   Google Scholar [25] H. I. Mesak and A. F. Darrat, A competitive advertising model: Some theoretical and empirical results, The Journal of the Operational Research Society, 44 (1993), 491-502.   Google Scholar [26] K. S. Moorthy, Competitive marketing strategies: Game-theoretic models, Handbooks in operations research and management science, 5 (1993), 143-190.   Google Scholar [27] M. Nerlove and K. J. Arrow, Optimal advertising policy under dynamic conditions, Economica, 29 (1962), 129-142.  doi: 10.1007/978-3-642-51565-1_54.  Google Scholar [28] D. Nguyen and L. Shi, Competitive advertising strategies and market-size dynamics: A research note on theory and evidence, Management Science, 52 (2006), 965-973.  doi: 10.1287/mnsc.1060.0509.  Google Scholar [29] A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets, Automatica, 48 (2012), 2882-2887.  doi: 10.1016/j.automatica.2012.08.031.  Google Scholar [30] J. Qi and D.-w. Wang, Optimal control strategies for an advertising competing model, Systems Engineering -Theory & Practice, 27 (2007), 39-44.  doi: 10.1016/S1874-8651(08)60001-0.  Google Scholar [31] S. P. Sethi, Dynamic optimal control models in advertising: A survey, SIAM review, 19 (1977), 685-725.  doi: 10.1137/1019106.  Google Scholar [32] M. L. Vidale and H. B. Wolfe, An operations-research study of sales response to advertising, Operations Research, 5 (1957), 370-381.  doi: 10.1287/opre.5.3.370.  Google Scholar [33] Q. Wang and Z. Wu, A duopolistic model of dynamic competitive advertising, European Journal of Operational Research, 128 (2001), 213-226.  doi: 10.1016/S0377-2217(99)00346-X.  Google Scholar

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##### References:
 [1] F. M. Bass, A. Krishnamoorthy, A. Prasad and S.P. Sethi, Advertising competition with market expansion for finite horizon firms, Journal of Industrial and Management Optimization, 1 (2005), 1-19.  doi: 10.3934/jimo.2005.1.1.  Google Scholar [2] F. M. Bass, A. Krishnamoorthy, A. Prasad and S. P. Sethi, Generic and brand advertising strategies in a dynamic duopoly, Marketing Science, 24 (2005), 556-568.  doi: 10.1287/mksc.1050.0119.  Google Scholar [3] P. K. Chintagunta and N. J. Vilcassim, An empirical investigation of advertising strategies in a dynamic duopoly, Management Science, 38 (1992), 1230-1244.  doi: 10.1287/mnsc.38.9.1230.  Google Scholar [4] E. J. Dockner, S. Jørgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, 2000. doi: 10.1017/CBO9780511805127.  Google Scholar [5] D. Dragone, L. Lambertini and A. Palestini, The Leitmann-Schmitendorf advertising game with $n$ players and time discounting, Applied Mathematics and Computation, 217 (2010), 1010-1016.  doi: 10.1016/j.amc.2010.02.031.  Google Scholar [6] G. M. Erickson, Differential game models of advertising competition, European Journal of Operational Research, 83 (1995), 431-438.  doi: 10.1016/0377-2217(94)00232-2.  Google Scholar [7] G. Feichtinger, The Nash solution of an advertising differential game: Generalization of a model by Leitmann and Schmitendorf, Automatic Control, IEEE Transactions on, 28 (1983), 1044-1048.  doi: 10.1109/TAC.1983.1103174.  Google Scholar [8] G. E. Fruchter, The many-player advertising game, Management Science, 45 (1999), 1609-1611.  doi: 10.1287/mnsc.45.11.1609.  Google Scholar [9] G. E. Fruchter, Oligopoly advertising strategies with market expansion, Optimal Control Applications and Methods, 20 (1999), 199-211.  doi: 10.1002/(SICI)1099-1514(199907/08)20:4<199::AID-OCA653>3.0.CO;2-M.  Google Scholar [10] G. E. Fruchter, Advertising in a competitive product line, Int. Game Theory Rev., 3 (2001), 301-314.  doi: 10.1142/S0219198901000439.  Google Scholar [11] G. E. Fruchter and S. Kalish, Closed-loop advertising strategies in a duopoly, Management Science, 43 (1997), 54-63.  doi: 10.1287/mnsc.43.1.54.  Google Scholar [12] G. E. Fruchter and Z. J. Zhang, Dynamic targeted promotions a customer retention and acquisition perspective, Journal of Service Research, 7 (2004), 3-19.   Google Scholar [13] R. F. Hartl and P. M. Kort, Advertising directed towards existing and new customers, in Optimal Control and Dynamic Games (eds. C. Deissenberg and R. Hartl), vol. 7 of Advances in Computational Management Science, Springer US, 2005, 3–18, URL http://dx.doi.org/10.1007/0-387-25805-1_1. Google Scholar [14] J. Huang, M. Leng and L. Liang, Recent developments in dynamic advertising research, European Journal of Operational Research, 220 (2012), 591-609.  doi: 10.1016/j.ejor.2012.02.031.  Google Scholar [15] R. Jarrar, G. Martín-Herrán and G. Zaccour, Markov perfect equilibrium advertising strategies of lanchester duopoly model: A technical note, Management Science, 50 (2004), 995-1000.  doi: 10.1287/mnsc.1040.0249.  Google Scholar [16] S. Jørgensen, A survey of some differential games in advertising, Journal of Economic Dynamics and Control, 4 (1982), 341-369.  doi: 10.1016/0165-1889(82)90024-0.  Google Scholar [17] S. Jørgensen, G. Martín-Herrán and G. Zaccour, The Leitmann-Schmitendorf advertising differential game, Applied Mathematics and Computation, 217 (2010), 1110-1116.  doi: 10.1016/j.amc.2010.01.047.  Google Scholar [18] S. Jørgensen and S.-P. Sigué, Defensive, offensive, and generic advertising in a lanchester model with market growth, Dynamic Games and Applications, 1–17, URL http://dx.doi.org/10.1007/s13235-015-0147-1. doi: 10.1007/s13235-015-0147-1.  Google Scholar [19] S. Jørgensen and G. Zaccour, Differential Games in Marketing, Springer, 2004. Google Scholar [20] G. E. Kimball, Some industrial applications of military operations research methods, Operations Res., 5 (1957), 201-204.  doi: 10.1287/opre.5.2.201.  Google Scholar [21] A. Krishnamoorthy, A. Prasad and S. P. Sethi, Optimal pricing and advertising in a durable-good duopoly, European Journal of Operational Research, 200 (2010), 486-497.  doi: 10.1016/j.ejor.2009.01.003.  Google Scholar [22] F. W. Lanchester, Aircraft in Warfare: The Dawn of the Fourth Arm, Appleton, New York, 1916. Google Scholar [23] G. Leitmann and W. Schmitendorf, Profit maximization through advertising: a nonzero sum differential game approach, Automatic Control, IEEE Transactions on, 23 (1978), 645-650.  doi: 10.1109/TAC.1978.1101794.  Google Scholar [24] D. Liu, S. Kumar and V. S. Mookerjee, Advertising strategies in electronic retailing: A differential games approach, Information Systems Research, 23 (2012), 903-917.   Google Scholar [25] H. I. Mesak and A. F. Darrat, A competitive advertising model: Some theoretical and empirical results, The Journal of the Operational Research Society, 44 (1993), 491-502.   Google Scholar [26] K. S. Moorthy, Competitive marketing strategies: Game-theoretic models, Handbooks in operations research and management science, 5 (1993), 143-190.   Google Scholar [27] M. Nerlove and K. J. Arrow, Optimal advertising policy under dynamic conditions, Economica, 29 (1962), 129-142.  doi: 10.1007/978-3-642-51565-1_54.  Google Scholar [28] D. Nguyen and L. Shi, Competitive advertising strategies and market-size dynamics: A research note on theory and evidence, Management Science, 52 (2006), 965-973.  doi: 10.1287/mnsc.1060.0509.  Google Scholar [29] A. Prasad, S. P. Sethi and P. A. Naik, Understanding the impact of churn in dynamic oligopoly markets, Automatica, 48 (2012), 2882-2887.  doi: 10.1016/j.automatica.2012.08.031.  Google Scholar [30] J. Qi and D.-w. Wang, Optimal control strategies for an advertising competing model, Systems Engineering -Theory & Practice, 27 (2007), 39-44.  doi: 10.1016/S1874-8651(08)60001-0.  Google Scholar [31] S. P. Sethi, Dynamic optimal control models in advertising: A survey, SIAM review, 19 (1977), 685-725.  doi: 10.1137/1019106.  Google Scholar [32] M. L. Vidale and H. B. Wolfe, An operations-research study of sales response to advertising, Operations Research, 5 (1957), 370-381.  doi: 10.1287/opre.5.3.370.  Google Scholar [33] Q. Wang and Z. Wu, A duopolistic model of dynamic competitive advertising, European Journal of Operational Research, 128 (2001), 213-226.  doi: 10.1016/S0377-2217(99)00346-X.  Google Scholar
Solutions to (11)–(12) and closed-loop strategies (13)
Solutions to (47)–(49) and closed-loop strategies (50)
Values of $u_k^2$ from (82) that Maximize $\dot{s}_k$ for varying $\rho_k/\sigma_k$
Dependence of $s_1$ on $c_1$ and $c_2$ in (85)
Contour plot of $u_1^2$ that maximizes $s_1$ for $c_1$ and $c_2$ in (85)
The solid curve indicates the value of $u_1^2$ that maximizes $s_1-s_2$, while the dotted curve represents the value of $u_1^2$ that maximizes $s_1$
Value of $u_1$ that minimizes $\varepsilon$ in (92)
Summary of Related Works
 [11] [8] [9] [33] [12] [18] This work Model Type Duo. Oligo. Oligo. Duo. Duo. Duo. Oligo. Sales Decay No No No Yes No No Yes Effort To Market Potential Yes Yes Yes Yes No Yes Yes Effort To Competitors' Customers Yes Yes Yes Yes Yes Yes Yes Targeting No No No No Yes Yes Yes Time Horizon Infinite Infinite Infinite Finite Infinite Finite Finite Open-loop NE Yes Yes Yes Yes Yes No Yes Closed-loop NE Yes Yes Yes Yes Yes Yes Yes
 [11] [8] [9] [33] [12] [18] This work Model Type Duo. Oligo. Oligo. Duo. Duo. Duo. Oligo. Sales Decay No No No Yes No No Yes Effort To Market Potential Yes Yes Yes Yes No Yes Yes Effort To Competitors' Customers Yes Yes Yes Yes Yes Yes Yes Targeting No No No No Yes Yes Yes Time Horizon Infinite Infinite Infinite Finite Infinite Finite Finite Open-loop NE Yes Yes Yes Yes Yes No Yes Closed-loop NE Yes Yes Yes Yes Yes Yes Yes
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