American Institute of Mathematical Sciences

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October  2019, 6(4): 315-335. doi: 10.3934/jdg.2019021

Evolutionary, mean-field and pressure-resistance game modelling of networks security

Stamatios Katsikas 1, and  Vassilli Kolokoltsov 2,

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

Received  August 2019 Revised  August 2019 Published  October 2019

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.

Keywords: Mean-field game, evolutionary game, pressure-resistance game, counter-terrorism, bot-net defense, cyber-security, inspection, corruption, crime prevention.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Stamatios Katsikas, Vassilli Kolokoltsov. Evolutionary, mean-field and pressure-resistance game modelling of networks security. Journal of Dynamics and Games, 2019, 6 (4) : 315-335. doi: 10.3934/jdg.2019021
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.

Figure 1.  The simplified version of our network: only transitions between neighbours are allowed in $ H $, all transitions are allowed in $ B $, binary interaction occurs only within a common level in $ H $
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