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From mean field games to the best reply strategy in a stochastic framework
Evolutionary, mean-field and pressure-resistance game modelling of networks security
1. | Centre for Complexity Science, University of Warwick, Coventry, CV4 7AL, UK |
2. | Department of Statistics, University of Warwick, Associate Member of IPI RAN, Coventry, CV4 7AL, UK |
The recently developed mean-field game models of corruption and bot-net defence in cyber-security, the evolutionary game approach to inspection and corruption, and the pressure-resistance game element, can be combined under an extended model of interaction of large number of indistinguishable small players against a major player, with focus on the study of security and crime prevention. In this paper we introduce such a general framework for complex interaction in network structures of many players, that incorporates individual decision making inside the environment (the mean-field game component), binary interaction (the evolutionary game component), and the interference of a principal player (the pressure-resistance game component). To perform concrete calculations with this overall complicated model, we suggest working, in sequence, in three basic asymptotic regimes; fast execution of personal decisions, small rates of binary interactions, and small payoff discounting in time.
References:
[1] |
R. J. Aumann,
Markets with a continuum of traders, Econometrica: Journal of the Econometric Society, 32 (1964), 39-50.
doi: 10.2307/1913732. |
[2] |
R. Basna, A. Hilbert and V. N. Kolokoltsov,
An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces, Communications on Stochastic Analysis, 8 (2014), 449-468.
doi: 10.31390/cosa.8.4.02. |
[3] |
D. Bauso, H. Tembine and T. Basar,
Robust mean field games, Dynamic Games and Applications, 6 (2016), 277-303.
doi: 10.1007/s13235-015-0160-4. |
[4] |
A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs in Mathematics. Springer, New York, 2013.
doi: 10.1007/978-1-4614-8508-7. |
[5] |
A. Bensoussan, M. H. M. Chau and S. C. P. Yam,
Mean field games with a dominating player, Applied Mathematics & Optimization, 74 (2016), 91-128.
doi: 10.1007/s00245-015-9309-1. |
[6] |
J. Bergin and D. Bernhardt,
Anonymous sequential games with aggregate uncertainty, Journal of Mathematical Economics, 21 (1992), 543-562.
doi: 10.1016/0304-4068(92)90026-4. |
[7] |
P. E. Caines, Mean field games, Encyclopedia of Systems and Control, (2013), 1–6. |
[8] |
M. J. Canty, D. Rothenstein and R. Avenhaus,
Timely inspection and deterrence, European Journal of Operational Research, 131 (2001), 208-223.
doi: 10.1016/S0377-2217(00)00082-5. |
[9] |
P. Cardaliaguet, Notes on mean field games (p. 120), Technical report, 2010. |
[10] |
R. Carmona and F. Delarue,
Probabilistic analysis of mean-field games, SIAM Journal on Control and Optimization, 51 (2013), 2705-2734.
doi: 10.1137/120883499. |
[11] |
R. Carmona and X. Zhu,
A probabilistic approach to mean field games with major and minor players, The Annals of Applied Probability, 26 (2016), 1535-1580.
doi: 10.1214/15-AAP1125. |
[12] |
P. Dubey, A. Mas-Colell and M. Shubik,
Efficiency properties of strategies market games: An axiomatic approach, Journal of Economic Theory, 22 (1980), 339-362.
|
[13] |
D. Friedman,
Evolutionary games in economics, Econometrica: Journal of the Econometric Societ, 59 (1991), 637-666.
doi: 10.2307/2938222. |
[14] |
D. Friedman,
On economic applications of evolutionary game theory, Journal of Evolutionary Economics, 8 (1998), 15-43.
|
[15] |
H. Gintis, Game Theory Evolving: A Problem-centered Introduction to Modeling Strategic
Behavior, Second edition. Princeton University Press, Princeton, NJ, 2009. |
[16] |
D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, Journal de Math'e Matiques Pures et Appliqu'ees, 93 (2010), 308–328.
doi: 10.1016/j.matpur.2009.10.010. |
[17] |
D. A. Gomes, J. Mohr and R. R. Souza,
Continuous time finite state mean field games, Applied Mathematics & Optimization, 68 (2013), 99-143.
doi: 10.1007/s00245-013-9202-8. |
[18] |
D. Gomes, R. M. Velho and M. T. Wolfram, Socio-economic applications of finite state mean field games, Phil. Trans. R. Soc. A, 372 (2014), 20130405, 18pp.
doi: 10.1098/rsta.2013.0405. |
[19] |
D. A. Gomes and J. Saude,
Mean field games models–a brief survey, Dynamic Games and Applications, 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
[20] |
D. Helbing, D. Brockmann, T. Chadefaux, K. Donnay, U. Blanke, O. Woolley-Meza, M. Moussaid, A. Johansson, J. Krause, S. Schutte and M. Perc,
Saving human lives: What complexity science and information systems can contribute, Journal of Statistical Physics, 158 (2015), 735-781.
doi: 10.1007/s10955-014-1024-9. |
[21] |
J. Hofbauer and K. Sigmund,
Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.
doi: 10.1090/S0273-0979-03-00988-1. |
[22] |
M. Huang, R. P. Malham'e and P. E. Caines,
Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information & Systems, 6 (2006), 221-252.
doi: 10.4310/CIS.2006.v6.n3.a5. |
[23] |
M. Huang,
Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM Journal on Control and Optimization, 48 (2010), 3318-3353.
doi: 10.1137/080735370. |
[24] |
B. Jovanovic and R. W. Rosenthal,
Anonymous sequential games, Journal of Mathematical Economics, 17 (1988), 77-87.
doi: 10.1016/0304-4068(88)90029-8. |
[25] |
M. I. Kamien and N. L. Schwartz, Dynamic Optimisation. The Calculus of Variations and Optimal Control in Economics and Management, Second edition. Advanced Textbooks in Economics, 31. North-Holland Publishing Co., Amsterdam, 1991. |
[26] |
S. Katsikas, V. Kolokoltsov and W. Yang, Evolutionary inspection and corruption games, Games, 7 (2016), Paper No. 31, 25 pp.
doi: 10.3390/g7040031. |
[27] |
V. N. Kolokoltsov, Nonlinear Markov Games, Proceedings of the 19th MTNS Symposium, 2010. |
[28] |
V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations (Vol. 182), Cambridge University Press, 2010. |
[29] |
V. Kolokoltsov and W. Yang,
Turnpike theorems for Markov games, Dynamic Games and Applications, 2 (2012), 294-312.
doi: 10.1007/s13235-012-0047-6. |
[30] |
V. N. Kolokoltsov, Nonlinear Markov games on a finite state space (mean-field and binary interactions), International Journal of Statistics and Probability, 1 (2012). |
[31] |
V. N. Kolokoltsov, The evolutionary game of pressure (or interference), resistance and collaboration, Math. Oper. Res., 42 (2017), 915–944, arXiv: 1412.1269, Available online: https://arXiv.org/abs/1412.1269(accessedon3December2014) (toappearinMOR(MathematicsofOperartionResearch))
doi: 10.1287/moor.2016.0838. |
[32] |
V. N. Kolokoltsov and O. A. Malafeyev,
Mean-field-game model of corruption, Dynamic Games and Applications, 7 (2017), 34-47.
doi: 10.1007/s13235-015-0175-x. |
[33] |
V. N. Kolokoltsov and A. Bensoussan,
Mean-field-game model for Botnet defense in Cyber-security, Applied Mathematics & Optimization, 74 (2016), 669-692.
doi: 10.1007/s00245-016-9389-6. |
[34] |
J. M. Lasry and P. L. Lions,
Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[35] |
M. R. D'Orsogna and M. Perc,
Statistical physics of crime: A review, Physics of Life Reviews, 12 (2015), 1-21.
|
[36] |
M. Perc, J. J. Jordan, D. G. Rand, Z. Wang, S. Boccaletti and A. Szolnoki,
Statistical physics of human cooperation, Physics Reports, 687 (2017), 1-51.
doi: 10.1016/j.physrep.2017.05.004. |
[37] |
S. M. Ross, Introduction to Stochastic Dynamic Programming, Probability and Mathematical
Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. |
[38] |
L. Samuelson,
Evolution and game theory, The Journal of Economic Perspectives, 16 (2002), 47-66.
|
[39] |
T. Sandler,
Counterterrorism: A game-theoretic analysis, Journal of Conflict Resolution, 49 (2005), 183-200.
|
[40] |
T. Sandler and D. G. Arce,
Terrorism: A game-theoretic approach, Handbook of Defense Economics, 2 (2007), 775-813.
|
[41] |
T. Sandler and K. Siqueira,
Games and terrorism: Recent developments, Simulation & Gaming, 40 (2009), 164-192.
|
[42] |
G. Szab'o and G. Fath,
Evolutionary games on graphs, Physics Reports, 446 (2007), 97-216.
doi: 10.1016/j.physrep.2007.04.004. |
[43] |
J. M. Smith, Evolution and the theory of games, In Did Darwin Get It Right?, Springer US, (1988), 202–215. |
[44] |
C. Taylor, D. Fudenberg, A. Sasaki and M. A. Nowak,
Evolutionary game dynamics in finite populations, Bulletin of Mathematical Biology, 66 (2004), 1621-1644.
doi: 10.1016/j.bulm.2004.03.004. |
[45] |
H. Tembine, J. Y. Le Boudec, R. El-Azouzi and E. Altman, Mean field asymptotics of Markov decision evolutionary games and teams, In Game Theory for Networks, 2009. GameNets' 09. International Conference on, IEEE, (2009), 140–150. |
[46] |
J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995.
doi: doi. |
[47] |
A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Nonconvex Optimization and its Applications, 80. Springer, New York, 2006. |
show all references
References:
[1] |
R. J. Aumann,
Markets with a continuum of traders, Econometrica: Journal of the Econometric Society, 32 (1964), 39-50.
doi: 10.2307/1913732. |
[2] |
R. Basna, A. Hilbert and V. N. Kolokoltsov,
An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces, Communications on Stochastic Analysis, 8 (2014), 449-468.
doi: 10.31390/cosa.8.4.02. |
[3] |
D. Bauso, H. Tembine and T. Basar,
Robust mean field games, Dynamic Games and Applications, 6 (2016), 277-303.
doi: 10.1007/s13235-015-0160-4. |
[4] |
A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs in Mathematics. Springer, New York, 2013.
doi: 10.1007/978-1-4614-8508-7. |
[5] |
A. Bensoussan, M. H. M. Chau and S. C. P. Yam,
Mean field games with a dominating player, Applied Mathematics & Optimization, 74 (2016), 91-128.
doi: 10.1007/s00245-015-9309-1. |
[6] |
J. Bergin and D. Bernhardt,
Anonymous sequential games with aggregate uncertainty, Journal of Mathematical Economics, 21 (1992), 543-562.
doi: 10.1016/0304-4068(92)90026-4. |
[7] |
P. E. Caines, Mean field games, Encyclopedia of Systems and Control, (2013), 1–6. |
[8] |
M. J. Canty, D. Rothenstein and R. Avenhaus,
Timely inspection and deterrence, European Journal of Operational Research, 131 (2001), 208-223.
doi: 10.1016/S0377-2217(00)00082-5. |
[9] |
P. Cardaliaguet, Notes on mean field games (p. 120), Technical report, 2010. |
[10] |
R. Carmona and F. Delarue,
Probabilistic analysis of mean-field games, SIAM Journal on Control and Optimization, 51 (2013), 2705-2734.
doi: 10.1137/120883499. |
[11] |
R. Carmona and X. Zhu,
A probabilistic approach to mean field games with major and minor players, The Annals of Applied Probability, 26 (2016), 1535-1580.
doi: 10.1214/15-AAP1125. |
[12] |
P. Dubey, A. Mas-Colell and M. Shubik,
Efficiency properties of strategies market games: An axiomatic approach, Journal of Economic Theory, 22 (1980), 339-362.
|
[13] |
D. Friedman,
Evolutionary games in economics, Econometrica: Journal of the Econometric Societ, 59 (1991), 637-666.
doi: 10.2307/2938222. |
[14] |
D. Friedman,
On economic applications of evolutionary game theory, Journal of Evolutionary Economics, 8 (1998), 15-43.
|
[15] |
H. Gintis, Game Theory Evolving: A Problem-centered Introduction to Modeling Strategic
Behavior, Second edition. Princeton University Press, Princeton, NJ, 2009. |
[16] |
D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games, Journal de Math'e Matiques Pures et Appliqu'ees, 93 (2010), 308–328.
doi: 10.1016/j.matpur.2009.10.010. |
[17] |
D. A. Gomes, J. Mohr and R. R. Souza,
Continuous time finite state mean field games, Applied Mathematics & Optimization, 68 (2013), 99-143.
doi: 10.1007/s00245-013-9202-8. |
[18] |
D. Gomes, R. M. Velho and M. T. Wolfram, Socio-economic applications of finite state mean field games, Phil. Trans. R. Soc. A, 372 (2014), 20130405, 18pp.
doi: 10.1098/rsta.2013.0405. |
[19] |
D. A. Gomes and J. Saude,
Mean field games models–a brief survey, Dynamic Games and Applications, 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
[20] |
D. Helbing, D. Brockmann, T. Chadefaux, K. Donnay, U. Blanke, O. Woolley-Meza, M. Moussaid, A. Johansson, J. Krause, S. Schutte and M. Perc,
Saving human lives: What complexity science and information systems can contribute, Journal of Statistical Physics, 158 (2015), 735-781.
doi: 10.1007/s10955-014-1024-9. |
[21] |
J. Hofbauer and K. Sigmund,
Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.
doi: 10.1090/S0273-0979-03-00988-1. |
[22] |
M. Huang, R. P. Malham'e and P. E. Caines,
Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information & Systems, 6 (2006), 221-252.
doi: 10.4310/CIS.2006.v6.n3.a5. |
[23] |
M. Huang,
Large-population LQG games involving a major player: The Nash certainty equivalence principle, SIAM Journal on Control and Optimization, 48 (2010), 3318-3353.
doi: 10.1137/080735370. |
[24] |
B. Jovanovic and R. W. Rosenthal,
Anonymous sequential games, Journal of Mathematical Economics, 17 (1988), 77-87.
doi: 10.1016/0304-4068(88)90029-8. |
[25] |
M. I. Kamien and N. L. Schwartz, Dynamic Optimisation. The Calculus of Variations and Optimal Control in Economics and Management, Second edition. Advanced Textbooks in Economics, 31. North-Holland Publishing Co., Amsterdam, 1991. |
[26] |
S. Katsikas, V. Kolokoltsov and W. Yang, Evolutionary inspection and corruption games, Games, 7 (2016), Paper No. 31, 25 pp.
doi: 10.3390/g7040031. |
[27] |
V. N. Kolokoltsov, Nonlinear Markov Games, Proceedings of the 19th MTNS Symposium, 2010. |
[28] |
V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations (Vol. 182), Cambridge University Press, 2010. |
[29] |
V. Kolokoltsov and W. Yang,
Turnpike theorems for Markov games, Dynamic Games and Applications, 2 (2012), 294-312.
doi: 10.1007/s13235-012-0047-6. |
[30] |
V. N. Kolokoltsov, Nonlinear Markov games on a finite state space (mean-field and binary interactions), International Journal of Statistics and Probability, 1 (2012). |
[31] |
V. N. Kolokoltsov, The evolutionary game of pressure (or interference), resistance and collaboration, Math. Oper. Res., 42 (2017), 915–944, arXiv: 1412.1269, Available online: https://arXiv.org/abs/1412.1269(accessedon3December2014) (toappearinMOR(MathematicsofOperartionResearch))
doi: 10.1287/moor.2016.0838. |
[32] |
V. N. Kolokoltsov and O. A. Malafeyev,
Mean-field-game model of corruption, Dynamic Games and Applications, 7 (2017), 34-47.
doi: 10.1007/s13235-015-0175-x. |
[33] |
V. N. Kolokoltsov and A. Bensoussan,
Mean-field-game model for Botnet defense in Cyber-security, Applied Mathematics & Optimization, 74 (2016), 669-692.
doi: 10.1007/s00245-016-9389-6. |
[34] |
J. M. Lasry and P. L. Lions,
Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[35] |
M. R. D'Orsogna and M. Perc,
Statistical physics of crime: A review, Physics of Life Reviews, 12 (2015), 1-21.
|
[36] |
M. Perc, J. J. Jordan, D. G. Rand, Z. Wang, S. Boccaletti and A. Szolnoki,
Statistical physics of human cooperation, Physics Reports, 687 (2017), 1-51.
doi: 10.1016/j.physrep.2017.05.004. |
[37] |
S. M. Ross, Introduction to Stochastic Dynamic Programming, Probability and Mathematical
Statistics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. |
[38] |
L. Samuelson,
Evolution and game theory, The Journal of Economic Perspectives, 16 (2002), 47-66.
|
[39] |
T. Sandler,
Counterterrorism: A game-theoretic analysis, Journal of Conflict Resolution, 49 (2005), 183-200.
|
[40] |
T. Sandler and D. G. Arce,
Terrorism: A game-theoretic approach, Handbook of Defense Economics, 2 (2007), 775-813.
|
[41] |
T. Sandler and K. Siqueira,
Games and terrorism: Recent developments, Simulation & Gaming, 40 (2009), 164-192.
|
[42] |
G. Szab'o and G. Fath,
Evolutionary games on graphs, Physics Reports, 446 (2007), 97-216.
doi: 10.1016/j.physrep.2007.04.004. |
[43] |
J. M. Smith, Evolution and the theory of games, In Did Darwin Get It Right?, Springer US, (1988), 202–215. |
[44] |
C. Taylor, D. Fudenberg, A. Sasaki and M. A. Nowak,
Evolutionary game dynamics in finite populations, Bulletin of Mathematical Biology, 66 (2004), 1621-1644.
doi: 10.1016/j.bulm.2004.03.004. |
[45] |
H. Tembine, J. Y. Le Boudec, R. El-Azouzi and E. Altman, Mean field asymptotics of Markov decision evolutionary games and teams, In Game Theory for Networks, 2009. GameNets' 09. International Conference on, IEEE, (2009), 140–150. |
[46] |
J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995.
doi: doi. |
[47] |
A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Nonconvex Optimization and its Applications, 80. Springer, New York, 2006. |

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