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

Stamatios Katsikas; Vassilli Kolokoltsov

doi:10.3934/jdg.2019021

Article Contents

2019, Volume 6, Issue 4: 315-335. Doi: 10.3934/jdg.2019021

This issue Previous Article From mean field games to the best reply strategy in a stochastic framework Next Article Markovian strategies for piecewise deterministic differential games with continuous and impulse controls

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

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

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 $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.