April  2020, 7(2): 141-162. doi: 10.3934/jdg.2020009

Optimal resource allocation in the difference and differential Stackelberg games on marketing networks

1. 

Saint Petersburg Higher School of Economics, Russian Federation, 3A Khantemirovskaya St., 194100 St. Petersburg, Russia

2. 

Southern Federal University, Rostov-on-Don, Russian Federation, 8A Milchakov St., 344090 Rostov-on-Don, Russia

* Corresponding author: Gennady Ougolnitsky

Received  October 2019 Published  April 2020

Fund Project: The second author is supported by Russian Science Foundation, project 17-19-01038

We consider difference and differential Stackelberg game theoretic models with several followers of opinion control in marketing networks. It is assumed that in the stage of analysis of the network its opinion leaders have already been found and are the only objects of control. The leading player determines the marketing budgets of the followers by resource allocation. In the basic version of the models both the leader and the followers maximize the summary opinions of the network agents. In the second version the leader has a target value of the summary opinion. In all four models we have found the Stackelberg equilibrium and the respective payoffs of the players analytically. It is shown that the hierarchical control system is ideally compatible in all cases.

Citation: Alexei Korolev, Gennady Ougolnitsky. Optimal resource allocation in the difference and differential Stackelberg games on marketing networks. Journal of Dynamics and Games, 2020, 7 (2) : 141-162. doi: 10.3934/jdg.2020009
References:
[1]

D. Acemoglu and A. Ozdaglar, Opinion dynamics and learning in social networks, Dyn. Games Appl., 1 (2011), 3-49.  doi: 10.1007/s13235-010-0004-1.

[2]

M. T. Agieva, A. V. Korolev and G. A. Ougolnitsky, Modeling and simulation of impact and control in social networks, in Modeling and Simulation of Social-Behavioral Phenomena in Creative Societies (eds. N. Agarwal, L. Sakalauskas, G.-W. Weber), Communications in Computer and Information Science, 1079, Springer, Cham, 2019, 29–40.

[3]

S. Z. Alparslan-GökO. DefterliE. Kropat and G.-W. Weber, Modeling, inference and optimization of regulatory networks based on time series data, European J. Oper. Res., 211 (2011), 1-14.  doi: 10.1016/j.ejor.2010.06.038.

[4]

S. Z. Alparslan-GökB. Söyler and G.-W. Weber, A New Mathematical Approach in Environmental and Life Sciences: Gene–Environment Networks and Their Dynamics, Environmental Modeling & Assessment, 14 (2009), 267-288. 

[5]

T. Başar and G. J. Olsder, Dynamic noncooperative game theory, in Mathematics in Science and Engineering, 160, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.

[6]

A. Chkhartishvili, D. Gubanov, and D. Novikov, Social Networks: Models of Information Influence, Control, and Confrontation, Springer Publishers, 2019.

[7]

M. H. De Groot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121. 

[8] E. J. DocknerS. JørgensenN. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511805127.
[9]

M. A. Gorelov and A. F. Kononenko, Dynamic models of conflicts. Ⅲ. Hierarchical games, Autom. Remote Control, 76 (2015), 264-277.  doi: 10.1134/S000511791502006X.

[10] M. O. Jackson, Social and Economic Networks, Princeton University Press, Princeton, NJ, 2008. 
[11]

V. V. Mazalov and A. N. Rettieva, Fish wars with many players, Int. Game Theory Rev., 12 (2010), 385-405.  doi: 10.1142/S0219198910002738.

[12]

C. Papadimitriou, Algorithms, games, and the internet, in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, 749–753, ACM, New York, 2001,749–753. doi: 10.1145/380752.380883.

[13]

F. S. Roberts, Discrete Mathematical Models with Applications to Social, Biological and Environmental Problems, Prentice-Hall, 1976. doi: 10.2307/2322080.

[14]

T. Roughgarden, Algorithmic game theory: Some greatest hits and future directions, in Fifth IFIP International Conference on Theoretical Computer Science–TCS 2008, IFIP Int. Fed. Inf. Process., 273, Springer, New York, 2008, 21–42. doi: 10.1007/978-0-387-09680-3_2.

[15]

A. A. Sedakov and M. Zhen, Opinion dynamics game in a social network with two influence nodes, Vestn. St.-Peterbg. Univ. Prikl. Mat. Inform. Protsessy Upr., 15 (2019), 118-125.  doi: 10.21638/11702/spbu10.2019.109.

[16]

G. A. Ugol'nitskii and A. B. Usov, A study of differential models for hierarchical control systems via their discretization, Autom. Remote Control, 74 (2013), 252-263.  doi: 10.1134/S0005117913020070.

[17]

G. A. Ugol'nitskii and A. B. Usov, Equilibria in models of hierarchically organized dynamical control systems with regard to sustainable development conditions, Autom. Remote Control, 75 (2014), 1055-1068.  doi: 10.1134/S000511791406006X.

[18]

M. Zhen, Stackelberg equilibrium in opinion dynamics game in social network with two influence nodes, Contributions to Game Theory and Management, 12 (2019), 366-386. 

show all references

References:
[1]

D. Acemoglu and A. Ozdaglar, Opinion dynamics and learning in social networks, Dyn. Games Appl., 1 (2011), 3-49.  doi: 10.1007/s13235-010-0004-1.

[2]

M. T. Agieva, A. V. Korolev and G. A. Ougolnitsky, Modeling and simulation of impact and control in social networks, in Modeling and Simulation of Social-Behavioral Phenomena in Creative Societies (eds. N. Agarwal, L. Sakalauskas, G.-W. Weber), Communications in Computer and Information Science, 1079, Springer, Cham, 2019, 29–40.

[3]

S. Z. Alparslan-GökO. DefterliE. Kropat and G.-W. Weber, Modeling, inference and optimization of regulatory networks based on time series data, European J. Oper. Res., 211 (2011), 1-14.  doi: 10.1016/j.ejor.2010.06.038.

[4]

S. Z. Alparslan-GökB. Söyler and G.-W. Weber, A New Mathematical Approach in Environmental and Life Sciences: Gene–Environment Networks and Their Dynamics, Environmental Modeling & Assessment, 14 (2009), 267-288. 

[5]

T. Başar and G. J. Olsder, Dynamic noncooperative game theory, in Mathematics in Science and Engineering, 160, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.

[6]

A. Chkhartishvili, D. Gubanov, and D. Novikov, Social Networks: Models of Information Influence, Control, and Confrontation, Springer Publishers, 2019.

[7]

M. H. De Groot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121. 

[8] E. J. DocknerS. JørgensenN. V. Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511805127.
[9]

M. A. Gorelov and A. F. Kononenko, Dynamic models of conflicts. Ⅲ. Hierarchical games, Autom. Remote Control, 76 (2015), 264-277.  doi: 10.1134/S000511791502006X.

[10] M. O. Jackson, Social and Economic Networks, Princeton University Press, Princeton, NJ, 2008. 
[11]

V. V. Mazalov and A. N. Rettieva, Fish wars with many players, Int. Game Theory Rev., 12 (2010), 385-405.  doi: 10.1142/S0219198910002738.

[12]

C. Papadimitriou, Algorithms, games, and the internet, in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, 749–753, ACM, New York, 2001,749–753. doi: 10.1145/380752.380883.

[13]

F. S. Roberts, Discrete Mathematical Models with Applications to Social, Biological and Environmental Problems, Prentice-Hall, 1976. doi: 10.2307/2322080.

[14]

T. Roughgarden, Algorithmic game theory: Some greatest hits and future directions, in Fifth IFIP International Conference on Theoretical Computer Science–TCS 2008, IFIP Int. Fed. Inf. Process., 273, Springer, New York, 2008, 21–42. doi: 10.1007/978-0-387-09680-3_2.

[15]

A. A. Sedakov and M. Zhen, Opinion dynamics game in a social network with two influence nodes, Vestn. St.-Peterbg. Univ. Prikl. Mat. Inform. Protsessy Upr., 15 (2019), 118-125.  doi: 10.21638/11702/spbu10.2019.109.

[16]

G. A. Ugol'nitskii and A. B. Usov, A study of differential models for hierarchical control systems via their discretization, Autom. Remote Control, 74 (2013), 252-263.  doi: 10.1134/S0005117913020070.

[17]

G. A. Ugol'nitskii and A. B. Usov, Equilibria in models of hierarchically organized dynamical control systems with regard to sustainable development conditions, Autom. Remote Control, 75 (2014), 1055-1068.  doi: 10.1134/S000511791406006X.

[18]

M. Zhen, Stackelberg equilibrium in opinion dynamics game in social network with two influence nodes, Contributions to Game Theory and Management, 12 (2019), 366-386. 

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