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. BassA. KrishnamoorthyA. 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. BassA. KrishnamoorthyA. 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. DragoneL. 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. HuangM. 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. JarrarG. 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. JørgensenG. 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. KrishnamoorthyA. 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. LiuS. 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. PrasadS. 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

show all references

References:
[1]

F. M. BassA. KrishnamoorthyA. 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. BassA. KrishnamoorthyA. 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. DragoneL. 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. HuangM. 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. JarrarG. 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. JørgensenG. 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. KrishnamoorthyA. 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. LiuS. 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. PrasadS. 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

Figure 1.  Solutions to (11)–(12) and closed-loop strategies (13)
Figure 2.  Solutions to (47)–(49) and closed-loop strategies (50)
Figure 3.  Values of $u_k^2$ from (82) that Maximize $\dot{s}_k$ for varying $\rho_k/\sigma_k$
Figure 4.  Dependence of $s_1$ on $c_1$ and $c_2$ in (85)
Figure 5.  Contour plot of $u_1^2$ that maximizes $s_1$ for $c_1$ and $c_2$ in (85)
Figure 6.  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 $s_1$
Figure 7.  Value of $u_1$ that minimizes $\varepsilon$ in (92)
Table 1.  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
[1]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091

[2]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51

[3]

Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics & Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1

[4]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153

[5]

Shunfu Jin, Haixing Wu, Wuyi Yue, Yutaka Takahashi. Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-18. doi: 10.3934/jimo.2019060

[6]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123

[7]

Xiaolin Xu, Xiaoqiang Cai. Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium. Journal of Industrial & Management Optimization, 2008, 4 (4) : 843-859. doi: 10.3934/jimo.2008.4.843

[8]

Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010

[9]

Mei Ju Luo, Yi Zeng Chen. Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 1-15. doi: 10.3934/jimo.2016.12.1

[10]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1

[11]

Janos Kollar. The Nash conjecture for threefolds. Electronic Research Announcements, 1998, 4: 63-73.

[12]

Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537

[13]

William Geller, Bruce Kitchens, Michał Misiurewicz. Microdynamics for Nash maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1007-1024. doi: 10.3934/dcds.2010.27.1007

[14]

F. M. Bass, A. Krishnamoorthy, A. Prasad, Suresh P. Sethi. Advertising competition with market expansion for finite horizon firms. Journal of Industrial & Management Optimization, 2005, 1 (1) : 1-19. doi: 10.3934/jimo.2005.1.1

[15]

Filipe Martins, Alberto A. Pinto, Jorge Passamani Zubelli. Nash and social welfare impact in an international trade model. Journal of Dynamics & Games, 2017, 4 (2) : 149-173. doi: 10.3934/jdg.2017009

[16]

Qinglei Zhang, Wenying Feng. Detecting coalition attacks in online advertising: A hybrid data mining approach. Big Data & Information Analytics, 2016, 1 (2&3) : 227-245. doi: 10.3934/bdia.2016006

[17]

Kemal Kilic, Menekse G. Saygi, Semih O. Sezer. Exact and heuristic methods for personalized display advertising in virtual reality platforms. Journal of Industrial & Management Optimization, 2019, 15 (2) : 833-854. doi: 10.3934/jimo.2018073

[18]

Simon Hoof. Cooperative dynamic advertising via state-dependent payoff weights. Journal of Dynamics & Games, 2019, 0 (0) : 1-15. doi: 10.3934/jdg.2019014

[19]

Cristina Brändle, Arturo De Pablo. Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1161-1178. doi: 10.3934/cpaa.2018056

[20]

Margarida Carvalho, João Pedro Pedroso, João Saraiva. Electricity day-ahead markets: Computation of Nash equilibria. Journal of Industrial & Management Optimization, 2015, 11 (3) : 985-998. doi: 10.3934/jimo.2015.11.985

 Impact Factor: 

Metrics

  • PDF downloads (9)
  • HTML views (3)
  • Cited by (0)

Other articles
by authors

[Back to Top]