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Annals of Operations Research

, Volume 236, Issue 1, pp 271–289 | Cite as

Modeling costly learning and counter-learning in a defender-attacker game with private defender information

  • Jie Xu
  • Jun ZhuangEmail author
Article

Abstract

In asymmetric war scenarios (e.g., counter-terrorism), the adversary usually invests a significant time to learn the system structure and identify vulnerable components, before launching attacks. Traditional game-theoretic defender-attacker models either ignore such learning periods or the entailed costs. This paper fills the gap by analyzing the strategic interactions of the terrorist’s costly learning and defender’s counter-learning and defense strategies in a game with private defender information. Our model allows six possible attacker strategies: (a) attack immediately; (b) learn and attack; (c) learn and not attack; (d) learn and attack when appearing vulnerable and not attack when appearing invulnerable; (e) learn and not attack when appearing vulnerable and attack when appearing invulnerable; and (f) not attack. Our results show that four of the six strategies (a, d, e, f) are possible at equilibrium and the other two (b, c) are strictly dominated. Interestingly, we find that the counterintuitive strategy (e) could be at equilibrium, especially when the probability that the target appears vulnerable given it is invulnerable is sufficiently high. Our results also show that the attacker’s learning cost has a significant impact on both the attacker’s best responses and the defender’s equilibrium deception and defense strategies. Finally, we study the attacker’s values of perfect information and imperfect information, which provide additional insights for defense and counter-learning strategies.

Keywords

Defender-attacker games Costly learning Counter-learning Game theory Value of perfect information  Value of imperfect information 

References

  1. Alpern, S., Morton, A., & Papadaki, K. (2011). Patrolling games. Operations Research, 59(5), 1246–1257.CrossRefGoogle Scholar
  2. Bier, V. M., & Haphuriwat, N. (2011). Analytical method to identify the number of containers to inspect at US ports to deter terrorist attacks. Annals of Operations Research, 187(1), 137–158.CrossRefGoogle Scholar
  3. Bier, V. M., Nagaraj, A., & Abhichandani, V. (2005). Protection of simple series and parallel systems with components of different values. Reliability Engineering and System Safety, 87(3), 315–323.CrossRefGoogle Scholar
  4. Bohme, R., & Moore, T. (2009). The iterated weakest link—a model of adaptive security investment. In Workshop on the economics of information security (WEIS), University College, London, UK. Available at http://weis09.infosecon.net/files/152/paper152.pdf. Accessed in August, 2014.
  5. Brown, G., Carlyle, M., Diehl, D., Kline, J., & Wood, K. (2005). A two-sided optimization for theater ballistic missile defense. Operations Research, 53(5), 745–763.CrossRefGoogle Scholar
  6. CNN. (2010). Dutch arrest two men after flight from US Available at http://news.blogs.cnn.com/2010/08/30/two-men-arrested-at-amsterdam-airport/. Accessed in August, 2014.
  7. Cobb, B. R., & Basuchoudhary, A. (2009). A decision analysis approach to solving the signaling game. Decision Analysis, 6(4), 239–255.CrossRefGoogle Scholar
  8. DePaulo, B. M., Wetzel, C., Sternglanz, R. W., & Wilson, M. J. W. (2003). Verbal and nonverbal dynamics of privacy, secrecy, and deceit. Journal of Social Issues, 59(2), 391–410.CrossRefGoogle Scholar
  9. Dutta, P. K. (1999). Strategies and games: Theory and practice. Cambridge, Massachusetts: MIT Press.Google Scholar
  10. Global Terrorism Database. (2013). Available at http://www.start.umd.edu/gtd/search/IncidentSummary.aspx?gtdid=200911140007. Accessed in August, 2014.
  11. Hausken, K., & Levitin, G. (2009). Protection vs. false targets in series systems. Reliability Engineering and System Safety, 94(5), 973–981.CrossRefGoogle Scholar
  12. Hausken, K., & Zhuang, J. (2011). Governments’ and terrorists’ defense and attack in a T-period game. Decision Analysis, 8(1), 46–70.CrossRefGoogle Scholar
  13. Hespanha, J. P., Ateskan, Y. S., & Kizilocak, H. H. (2000). Deception in non-cooperative games with partial information. In Proceedings of the 2nd DARPA-JFACC symposium on advances in enterprise control. Citeseer. Available at http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.158.4664&rep=rep1&type=pdf. Accessed in August, 2014.
  14. Insua, D. R., Rios, J., & Banks, D. (2009). Adversarial risk analysis. Journal of the American Statistical Association, 104(486), 841–854.CrossRefGoogle Scholar
  15. Joint Chiefs of Staff. (1996). Joint doctrine for military deception. Joint Publication 3–13.4 Available at http://www.dtic.mil/doctrine/jel/new_pubs/jp3_13_4.pdf. Accessed in August, 2014.
  16. Mail Online. (2010). Ink bomb defused ‘with 17 minutes to spare’: Device at UK airport was ready to explode. Available at http://www.dailymail.co.uk/news/article-1326552/Yemen-ink-bomb-defused-17-minutes-spare-Device-ready-explode.html. Accessed in August, 2014.
  17. Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic theory. New York, NY: Oxford University Press.Google Scholar
  18. National Commission on Terrorist Attacks Upon the United States. (2004). The 9/11 commission report: Final report of the national commission on terrorist attacks upon the United States. W. W. Norton and Company, New York, NY.Google Scholar
  19. Powell, R. (2007). Allocating defensive resources with private information about vulnerability. The American Political Science Review, 101(4), 799–809.CrossRefGoogle Scholar
  20. Powell, R. (2009). Sequential, nonzero-sum “Blotto”: Allocating defensive resources prior to attack. Games and Economic Behavior, 67(2), 611–615.CrossRefGoogle Scholar
  21. Roberson, B. (2006). The colonel blotto game. Economic Theory, 29(1), 1–24.CrossRefGoogle Scholar
  22. Sandler, T., & Siqueira, K. (2006). Global terrorism: Deterrence versus pre-emption. Canadian Journal of Economics, 39(4), 1370–1387.CrossRefGoogle Scholar
  23. Schelling, T. C. (Ed.). (1966). Arms and influence. Yale University Press, New Haven, CT.Google Scholar
  24. Shan, X., & Zhuang, J. (2013a). Cost of equity in homeland security resource allocation in the face of a strategic attacker. Risk Analysis, 33(6), 1083–1099.CrossRefGoogle Scholar
  25. Shan, X., & Zhuang, J. (2013b). Hybrid defensive resource allocations in the face of partially strategic attackers in a sequential defender-attacker game. European Journal of Operational Research, 228(1), 262–272.CrossRefGoogle Scholar
  26. Swire, P. P. (2001). What should be hidden and open in computer security: Lessons from deception, the art of war, law, and economic theory. ArXiv Computer Science e-prints cs/0109089.
  27. US Department of Homeland Security. (2011). Risk management fundamentals—homeland security risk management doctrine. Available at http://www.dhs.gov/xlibrary/assets/rma-risk-management-fundamentals.pdf. Accessed in August, 2014.
  28. Wang, C., & Bier, V. M. (2013). Expert elicitation of adversary preferences using ordinal judgments. Operations Research, 61(2), 372–385.CrossRefGoogle Scholar
  29. Zangwill, W. I., & Kantor, P. B. (1998). Toward a theory of continuous improvement and the learning curve. Management Science, 44(7), 910–920.CrossRefGoogle Scholar
  30. Zhuang, J. (2010). Impacts of subsidized security on stability and total social costs of equilibrium solutions in an n-player game with errors. The Engineering Economist, 55(2), 131–149.CrossRefGoogle Scholar
  31. Zhuang, J., & Bier, V. M. (2007). Balancing terrorism and natural disasters-defensive strategy with endogenous attacker effort. Operations Research, 55(5), 976–991.CrossRefGoogle Scholar
  32. Zhuang, J., & Bier, V. M. (2010). Reasons for secrecy and deception in homeland-security resource allocation. Risk Analysis, 30(12), 1737–1743.CrossRefGoogle Scholar
  33. Zhuang, J., & Bier, V. M. (2011). Secrecy and deception at equilibrium, with applications to anti-terrorism resource allocation. Defence and Peace Economics, 22(1), 43–61.CrossRefGoogle Scholar
  34. Zhuang, J., Bier, V. M., & Alagoz, O. (2010). Modeling secrecy and deception in a multiple-period attacker-defender signaling game. European Journal of Operational Research, 203(2), 409–418.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Industrial and System EngineeringSUNY at BuffaloBuffaloUSA

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