Dynamic Cooperative Game Theoretic Models of Opinion Control
Abstract
In this paper we consider dynamic cooperative game theoretic models of control on networks. We suppose that all strong subgroups are determined in the stage of analysis of the influence digraph, and the control impact is exerted only to the members of those subgroups because they determine all stable final opinions. An agent’s opinion is interpreted as his expenses for buying goods (services) of a firm. We show that due to the model assumptions the characteristic functions by Neumann-Morgenstern, Petrosyan-Zaccour, and Gromova-Petrosyan coincide. We find Shapley value for this common characteristic function, prove its time inconsistency, and built an imputation distribution procedure. We compared the components of Shapley value with players’ payoffs for different forms of non-cooperative behavior.
References
J. R. French, “A formal theory of social power,” The Psychological Review, vol. 63, pp. 181–194, 1956; doi: 10.1037/h0046123
F. Harary, “A Criterion for Unanimity in French’s Theory of Social Power,” in Studies in Social Power, Ann Arbor, MI, USA: Institute of Sociological Research, pp. 168–182, 1959.
M. H. DeGroot, “Reaching a Consensus,” J. of American Statistical Association, vol. 69, pp. 118–121, 1974; doi:10.1080/01621459.1974.10480137
M. Jackson, “Social and Economic Networks,” Princeton, NJ, USA: Princeton University Press, 2008.
M. O. Jackson and A. Wolinsky, “A Strategic Model of Social and Economic Networks,” Journal of Economic Theory, vol. 71, no. 1, pp. 44–74, 1996; doi:10.1006/jeth.1996.0108
D. A. Gubanov, D. A. Novikov, and A. G. Chkhartishvili, “Models of influence in social networks (a survey),” Large-Scale Systems Control, vol. 27, pp. 205–281, 2009.
A. Chkhartishvili, D. Gubanov, and D. Novikov, Social Networks: Models of Information Influence, Control, and Confrontation, Springer Publishers, 2019; doi:10.1007/978-3-030-05429-8
D. A. Novikov, “Games and networks,” Automation and Remote Control, vol. 75, pp. 1145–1154, 2014; doi:10.1134/s0005117914060149
M. Jackson and Y. Zenou, “Games on networks,” in Handbook of game theory, P. Young and S. Zamir, eds., Amsterdam: Elsevier Science, pp. 95–163, 2015; doi:10.1016/b978-0-444-53766-9.00003-3
S. Jorgensen and G. Zaccour, Differential Games in Marketing, Totnes, Devon, UK: Kluwer Academic Publishers, 2004.
S. Jorgensen and G. Zaccour, “A Survey of Game-Theoretic Models of Cooperative Advertising,” European J. of Operations Research, vol. 237(1), pp. 1–14, 2014; doi:10.1016/j.ejor.2013.12.017
A. Sedakov and M. Zhen, “Opinion dynamics game in a social network with two influence nodes,” Vestnik SPb Gos. Univ. Ser. Appl. Math. Inform. Sci. Contr. Proc., vol. 15, no. 1, pp. 118–125, 2019; doi:10.21638/11701/spbu10.2019.109
M. Zhen, “Stackelberg equilibrium in opinion dynamics game in social network with two influence nodes,” in Contributions to game theory and management, L. A. Petrosyan and N. A. Zenkevich, eds., vol. 12, pp. 366–386, 2019.
M. T. Agieva, A. V. Korolev, and G. A. Ougolnitsky, “Modeling and Simulation of Impact and Control in Social Networks” Modeling and Simulation of Social-Behavioral Phenomena in Creative Societies, pp. 29–40, 2019; doi:10.1007/978-3-030-29862-3_3
A. Korolev and G. Ougolnitsky, “Optimal resource allocation in the difference and differential Stackelberg games on marketing networks,” Journal of Dynamics and Games, vol. 7, no. 2, pp. 141–162, 2020; doi:10.3934/jdg.2020009
M. T. Agieva, A. V. Korolev, and G. A. Ougolnitsky, “Modeling and Simulation of Impact and Control in Social Networks with Application to Marketing,” Mathematics, vol. 8, no. 9, p. 1529, 2020; doi:10.3390/math8091529
M. T. Agieva, A. V. Korolev, and G. A. Ougolnitsky, “Game Theoretic Models of Sustainable Management in Marketing Networks,” Contributions to Game Theory and Management, vol. 13, pp. 24–56, 2020.
M. T. Agieva, O. I. Gorbaneva, and G. A. Ougolnitsky, “Dynamic SPICE-Model of Resource Allocation in Marketing Networks with Co-Directed Interests,” in Proc. of 2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA), pp. 161–165, 2020; doi:10.1109/summa50634.2020.9280699
M. Agieva, A. Korolev, and G. Ougolnitsky, “Static and Dynamic Game Theoretic Models of Opinion Control in Marketing Network,” Contributions to Game Theory and Management, vol. 14, pp. 8–19, 2021.
N. Nisan et al., eds, Algorithmic Game Theory, Cambridge, UK: Cambridge University Press, 2007.
T. Roughgarden, Selfish Routing and the Price of Anarchy, Cambridge, UK: MIT Press, 2005.
C. H. Papadimitriou, “Algorithms, games, and the Internet,” in Proc. 33rd Symp. Theory of Computing, pp. 749–753, 2001.
T. Basar and Q. Zhu, “Prices of Anarchy, Information, and Cooperation in Differential Games,” Dynamic Games and Applications, vol. 1(1), pp. 50–73, 2011.
J. von Neumann and O. Morgenstern, “Theory of Games and Economic Behavior,” Cambridge, UK: Princeton University Press, 1953.
L. Petrosjan and G. Zaccour, “Time-consistent Shapley value allocation of pollution cost reduction,” J. of Economic Dynamics and Control, vol. 27(3), pp. 381–398, 2003.
E. V. Gromova and L. A. Petrosyan, “On an approach to constructing a characteristic function in cooperative differential games,” Automation and Remote Control, vol. 78, 1680–1692, 2017.
L. S. Shapley, “17. A Value for n-Person Games,” Contributions to the Theory of Games (AM-28), vol. 2, pp. 307–318, 1953; doi:10.1515/9781400881970-018
A. Tur and L. Petrosyan, “Cooperative optimality principals in differential games on networks,” Mathematical Game Theory and Applications, vol. 12, no. 4, pp. 93–111, 2020; doi
17076/mgta_2020_4_27
L. A. Petrosyan and D. W. K. Yeung, “Shapley value for differential network games: theory and application,” J. of Dynamics and Games, vol. 8, no. 2, pp. 151–166, 2021; doi:10.3934/jdg.2020021
A. Tur and L. Petrosyan, “Strong Time-Consistent Solution for Cooperative Differential Games with Network Structure,” Mathematics, vol. 9, no. 7, p. 755, 2021; doi:10.3390/math9070755
N. M. Galieva, A. V. Korolev, and G. A. Ougolnitsky, “Dynamic Resource Allocation Networks in Marketing: Comparing the Effectiveness of Control Methods,” Dynamic Games and Applications, vol. 14, no. 2, pp. 362–395, 2023; doi:10.1007/s13235-023-00494-y
This work is licensed under a Creative Commons Attribution 4.0 International License.
