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July  2019, 15(3): 1017-1048. doi: 10.3934/jimo.2018084

## Global and local advertising strategies: A dynamic multi-market optimal control model

 Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Av. Diagonal Las Torres 2640, Peñalolén, 7941169, Santiago, Chile

* Corresponding author: Tel: +56 2 23311491 - email: marcelo.villena@uai.cl

Received  March 2016 Revised  March 2018 Published  July 2018

Differential games have been widely used to model advertising strategies of companies. Nevertheless, most of these studies have concentrated on the dynamics and market structure of the problem, neglecting their multi-market dimension. Since nowadays competition typically operates on multi-product contexts and usually in geographically separated markets, the optimal advertising strategies must take into consideration the different levels of disaggregation, especially, for example, in retail multi-product and multi-store competition contexts. In this paper, we look into the decision-making process of a multi-market company that has to decide where, when and how much money to invest in advertising. For this purpose, we develop a model that keeps the dynamic and oligopolistic nature of the traditional advertising game introducing the multi-market dimension of today's economies, while differentiating global (i.e. national TV) from local advertising strategies (i.e. a price discount promotion in a particular store). It is important to note, however, that even though this problem is real for most multi-market companies, it has not been addressed in the differential games literature. On the more technical side, we steer away from the traditional aggregated dynamics of advertising games in two aspects. Firstly, we can model different markets at once, obtaining a global instead of a local optimum, and secondly, since we are incorporating a variable that is common to markets, the resulting equations systems for every market are now coupled. In other words, one's decision in one market does not only affect one's competition in that particular market; it also affects one's decisions and one's competitors in all markets.

Citation: Marcelo J. Villena, Mauricio Contreras. Global and local advertising strategies: A dynamic multi-market optimal control model. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1017-1048. doi: 10.3934/jimo.2018084
##### References:
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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 [7] P. Davis, Spatial competition in retail markets: Movie theaters, The RAND Journal of Economics, 37 (2006), 964-982.  doi: 10.1111/j.1756-2171.2006.tb00066.x.  Google Scholar [8] E. J. Dockne and S. Jorgensen, New product advertising in dynamic oligopolies, Zeitschrift fur Operations Research, 36 (1992), 459-473.  doi: 10.1007/BF01415762.  Google Scholar [9] E. J. Dockner, S. Jorgensen, N. Van Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, U. K, 2000. doi: 10.1017/CBO9780511805127.  Google Scholar [10] P. B. Ellickson and S. Misra, Supermarket pricing strategies, Marketing Science, 27 (2008), 811-828.  doi: 10.1287/mksc.1080.0398.  Google Scholar [11] G. M. 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Villena, Spatial lanchester models, European Journal of Operational Research, 210 (2011), 706-715.  doi: 10.1016/j.ejor.2010.11.009.  Google Scholar [18] M. L. Huang J. 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 [19] R. S. Jarmin, K. Shawn and J. Miranda, The role of retail chains: National, regional, and industry results, in Producer Dynamics: New Evidence from Micro Data, University of Chicago Press, 2009,237{262. Google Scholar [20] 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 [21] S. Jørgensen and G. Zaccour, Differential Games in Marketing, vol. 15, Springer, 2004. Google Scholar [22] G. Kimball, Some industrial applications of military operations research methods, Operations Research, 5 (1957), 201-204.  doi: 10.1287/opre.5.2.201.  Google Scholar [23] A. Krishnamoorthy, A. Prasad and S. 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 [24] F. W. Lanchester, Aircraft in Warfare: The Dawn of the Fourth Arm, Constable limited, 1916. Google Scholar [25] J. D. Little, Aggregate advertising models: The state of the art, Operations research, 27 (1979), 629-667.   Google Scholar [26] C. Marinelli and S. Savin, Optimal distributed dynamic advertising, Journal of Optimization Theory and Applications, 137 (2008), 569-591.  doi: 10.1007/s10957-007-9350-6.  Google Scholar [27] M. Nerlove and K. Arrow, Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1976), 129-142.  doi: 10.1007/978-3-642-51565-1_54.  Google Scholar [28] A. Prasad and S. P. Sethi, Advertising under uncertainty: A stochastic differential game approach, Journal of Optimization Theory and Applications, 123 (2004), 163-185.  doi: 10.1023/B:JOTA.0000043996.62867.20.  Google Scholar [29] S. P. Sethi, Deterministic and stochastic optimization of a dynamic advertising model, Optimal Control Applications and Methods, 4 (1983), 179-184.  doi: 10.1002/oca.4660040207.  Google Scholar [30] S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, Kluwer, Norwell, MA, 2000.  Google Scholar [31] G. Sorger, Competitive dynamic advertising: A modification of the case game, Journal of Economics Dynamics and Control, 13 (1989), 55-80.  doi: 10.1016/0165-1889(89)90011-0.  Google Scholar [32] M. Vidale and H. 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

show all references

##### References:
 [1] A. Beresteanu, P. Ellickson and S. Misra, The dynamics of retail oligopoly, Working Papers. Google Scholar [2] M. Breton, R. Jarrar and G. Zaccour, A note on feedback sequential equilibria in a lanchester model with empirical application, Management Science, 52 (2006), 804-811.  doi: 10.1287/mnsc.1050.0475.  Google Scholar [3] A. Buratto, L. Grosset and B. Viscolani, Advertising channel selection in a segmented market, Automatica, 42 (2006), 1343-1347.   Google Scholar [4] R. Cellini and L. Lambertini, Advertising in a differential oligopoly game, Journal of Optimization Theory and Applications, 116 (2003), 61-81.  doi: 10.1023/A:1022158102252.  Google Scholar [5] P. K. Chintagunta and D. Jain, Empirical analysis of a dynamic duopoly model of competition, Journal of Economics and Management Strategy, 4 (1995), 109-131.   Google Scholar [6] P. K. Chintagunta and N. 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 [7] P. Davis, Spatial competition in retail markets: Movie theaters, The RAND Journal of Economics, 37 (2006), 964-982.  doi: 10.1111/j.1756-2171.2006.tb00066.x.  Google Scholar [8] E. J. Dockne and S. Jorgensen, New product advertising in dynamic oligopolies, Zeitschrift fur Operations Research, 36 (1992), 459-473.  doi: 10.1007/BF01415762.  Google Scholar [9] E. J. Dockner, S. Jorgensen, N. Van Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, U. K, 2000. doi: 10.1017/CBO9780511805127.  Google Scholar [10] P. B. Ellickson and S. Misra, Supermarket pricing strategies, Marketing Science, 27 (2008), 811-828.  doi: 10.1287/mksc.1080.0398.  Google Scholar [11] G. M. Erickson, Empirical analysis of closed-loop duopoly advertising strategies, Management Science, 38 (1992), 1732-1749.  doi: 10.1287/mnsc.38.12.1732.  Google Scholar [12] G. Erickson, Dynamic Models of Advertising Competition, 2nd edition, Kluwer, Norwell, MA, 2003. Google Scholar [13] G. Erickson, An oligopoly model of dynamic advertising competition, European Journal of Operational Research, 197 (2009), 374-388.  doi: 10.1016/j.ejor.2008.06.023.  Google Scholar [14] G. Feichtinger, R. F. l. Hart and S. P. Sethi, Dynamic optimal control models in advertising: Recent developments, Management Science, 40 (1994), 195-226.  doi: 10.1287/mnsc.40.2.195.  Google Scholar [15] G. Fruchter, The many-player advertising game, Management Science, 45 (1999), 1609-1611.  doi: 10.1287/mnsc.45.11.1609.  Google Scholar [16] G. 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 [17] E. González and M. Villena, Spatial lanchester models, European Journal of Operational Research, 210 (2011), 706-715.  doi: 10.1016/j.ejor.2010.11.009.  Google Scholar [18] M. L. Huang J. 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 [19] R. S. Jarmin, K. Shawn and J. Miranda, The role of retail chains: National, regional, and industry results, in Producer Dynamics: New Evidence from Micro Data, University of Chicago Press, 2009,237{262. Google Scholar [20] 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 [21] S. Jørgensen and G. Zaccour, Differential Games in Marketing, vol. 15, Springer, 2004. Google Scholar [22] G. Kimball, Some industrial applications of military operations research methods, Operations Research, 5 (1957), 201-204.  doi: 10.1287/opre.5.2.201.  Google Scholar [23] A. Krishnamoorthy, A. Prasad and S. 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 [24] F. W. Lanchester, Aircraft in Warfare: The Dawn of the Fourth Arm, Constable limited, 1916. Google Scholar [25] J. D. Little, Aggregate advertising models: The state of the art, Operations research, 27 (1979), 629-667.   Google Scholar [26] C. Marinelli and S. Savin, Optimal distributed dynamic advertising, Journal of Optimization Theory and Applications, 137 (2008), 569-591.  doi: 10.1007/s10957-007-9350-6.  Google Scholar [27] M. Nerlove and K. Arrow, Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1976), 129-142.  doi: 10.1007/978-3-642-51565-1_54.  Google Scholar [28] A. Prasad and S. P. Sethi, Advertising under uncertainty: A stochastic differential game approach, Journal of Optimization Theory and Applications, 123 (2004), 163-185.  doi: 10.1023/B:JOTA.0000043996.62867.20.  Google Scholar [29] S. P. Sethi, Deterministic and stochastic optimization of a dynamic advertising model, Optimal Control Applications and Methods, 4 (1983), 179-184.  doi: 10.1002/oca.4660040207.  Google Scholar [30] S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, Kluwer, Norwell, MA, 2000.  Google Scholar [31] G. Sorger, Competitive dynamic advertising: A modification of the case game, Journal of Economics Dynamics and Control, 13 (1989), 55-80.  doi: 10.1016/0165-1889(89)90011-0.  Google Scholar [32] M. Vidale and H. 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
">Figure 2.  $\mu_{i}(t)$ for the global case in table 2
">Figure 3.  $\lambda^{i}(t)$ for the global case in table 2
">Figure 4.  Phase space diagram ($\lambda^{i}$ versus $x$) for the global case in table 2
">Figure 1.  $x(t)$ for the global case in table 2
Game description
">Figure 6.  Case 1 in table 3
">Figure 7.  Case 2 in table 3
">Figure 8.  Case 3 in table 3
">Figure 9.  Case 4 in table 3
The pure-global case, case 0
The quasi-global case, case 1
The local case, case 2
Notation
 $J_{i}$ Profit function of player $i$ $x_{ik}(t)$ Market share of player $i$ at location $k$ $q_{ik}$ Gross profit rate per unit of market share of player $i$ at location $k$ $Q_{ik}$ Second order gross profit rate per unit of market share of player $i$ at location $k$ $b_{ik}$ Linear local advertising cost of player $i$ at location $k$ $B_{ik}$ Second order local advertising cost of player $i$ at location $k$ $e_{i}$ Linear global advertising cost of player $i$ at location $k$ $E_{i}$ Second order global advertising cost of player $i$ $\sigma_{ik}$ Effectiveness of local advertising of player $i$ at location $k$ $\sigma_{i}$ Effectiveness of global advertising of player $i$ at location $k$ $r_{i}$ Discount rate of player $i$ $\mu_{ik}(t)$ Local advertising effort of player $i$ at location $k$ $\mu_{i}(t)$ Global advertising effort of player $i$
 $J_{i}$ Profit function of player $i$ $x_{ik}(t)$ Market share of player $i$ at location $k$ $q_{ik}$ Gross profit rate per unit of market share of player $i$ at location $k$ $Q_{ik}$ Second order gross profit rate per unit of market share of player $i$ at location $k$ $b_{ik}$ Linear local advertising cost of player $i$ at location $k$ $B_{ik}$ Second order local advertising cost of player $i$ at location $k$ $e_{i}$ Linear global advertising cost of player $i$ at location $k$ $E_{i}$ Second order global advertising cost of player $i$ $\sigma_{ik}$ Effectiveness of local advertising of player $i$ at location $k$ $\sigma_{i}$ Effectiveness of global advertising of player $i$ at location $k$ $r_{i}$ Discount rate of player $i$ $\mu_{ik}(t)$ Local advertising effort of player $i$ at location $k$ $\mu_{i}(t)$ Global advertising effort of player $i$
Data for the pure global game.
 Global parameters $q_{11}$ 0.8 $q_{21}$ 0.3 $Q_{11}$ 0.0 $Q_{21}$ 0.0 $e_{1}$ 0.0 $e_{2}$ 0.0 $E_{1}$ 0.5 $E_{2}$ 0.3 $\sigma_{1}$ 0.95 $\sigma_{2}$ 1.6 $r_{1}$ 0.01 $r_{2}$ 0.05
 Global parameters $q_{11}$ 0.8 $q_{21}$ 0.3 $Q_{11}$ 0.0 $Q_{21}$ 0.0 $e_{1}$ 0.0 $e_{2}$ 0.0 $E_{1}$ 0.5 $E_{2}$ 0.3 $\sigma_{1}$ 0.95 $\sigma_{2}$ 1.6 $r_{1}$ 0.01 $r_{2}$ 0.05
Data of numerical examples
 Case 1 Case 2 Case 3 Case 4 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 1 2 1 2 1 2 1 2 $q_{i1}$ 3 3 3 3 3 3 3 3 $q_{i2}$ 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 $B_{i1}$ 1 1 1 1 1 1 1 1 $B_{i2}$ 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 $E_{i}$ 1 1 1 2 1 1 1 1 $\sigma_{i1}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0.8 $\sigma_{i2}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0 $\sigma_{i}$ 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
 Case 1 Case 2 Case 3 Case 4 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 1 2 1 2 1 2 1 2 $q_{i1}$ 3 3 3 3 3 3 3 3 $q_{i2}$ 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 $B_{i1}$ 1 1 1 1 1 1 1 1 $B_{i2}$ 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 $E_{i}$ 1 1 1 2 1 1 1 1 $\sigma_{i1}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0.8 $\sigma_{i2}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0 $\sigma_{i}$ 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Data for pure-global (case 0), quasi-global (case 1) and local effects (case 2).
 Case 0: pure-global Case 1: quasi-global Case 2: local effects Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm2 \hline $q_{i1}$ 0.8 0.3 0.8 0.3 0.8 0.3 $q_{i2}$ 0.8 0.3 0.8 0.3 0.8 0.3 $Q_{i1}$ 0 0 0 0 0 0 $Q_{i2}$ 0 0 0 0 0 0 $b_{i1}$ 0 0 0 0 0 0 $b_{i2}$ 0 0 0 0 0 0 $B_{i1}$ 0 0 0.001 0.001 5 0.1 $B_{i2}$ 0 0 0.001 0.001 1 2 $e_{i}$ 0 0 0 0 0 0 $E_{i}$ 1 2 1 2 1 2 $\sigma_{i1}$ 0 0 0 0 0.1 0.3 $\sigma_{i2}$ 0 0 0 0 0.6 0.1 $\sigma_{i}$ 1 1.9 1 1.9 1 1.9 $r_{i}$ 0.01 0.05 0.01 0.05 0.01 0.05
 Case 0: pure-global Case 1: quasi-global Case 2: local effects Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm2 \hline $q_{i1}$ 0.8 0.3 0.8 0.3 0.8 0.3 $q_{i2}$ 0.8 0.3 0.8 0.3 0.8 0.3 $Q_{i1}$ 0 0 0 0 0 0 $Q_{i2}$ 0 0 0 0 0 0 $b_{i1}$ 0 0 0 0 0 0 $b_{i2}$ 0 0 0 0 0 0 $B_{i1}$ 0 0 0.001 0.001 5 0.1 $B_{i2}$ 0 0 0.001 0.001 1 2 $e_{i}$ 0 0 0 0 0 0 $E_{i}$ 1 2 1 2 1 2 $\sigma_{i1}$ 0 0 0 0 0.1 0.3 $\sigma_{i2}$ 0 0 0 0 0.6 0.1 $\sigma_{i}$ 1 1.9 1 1.9 1 1.9 $r_{i}$ 0.01 0.05 0.01 0.05 0.01 0.05
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