Skip to main content

Advertisement

SpringerLink
Log in

Efficient solutions for joint activity based security games: fast algorithms, results and a field experiment on a transit system

  • Published:
Autonomous Agents and Multi-Agent Systems Aims and scope Submit manuscript

Abstract

In recent years, several security agencies have been deploying scheduling systems based on algorithmic advances in Stackelberg security games (SSGs). Unfortunately, none of the existing algorithms can scale up to domains where benefits are accrued from multiple defender resources performing jointly coordinated activities. Yet in many domains, including port patrolling where SSGs are in use, enabling multiple defender resources to perform jointly coordinated activities would significantly enhance the effectiveness of the patrols. To address this challenge, this paper presents four contributions. First, we present Smart (Security games with Multiple coordinated Activities and Resources that are Time-dependent), a novel SSG model that explicitly represents jointly coordinated activities between defender’s resources. Second, we present two branch-and-price algorithms, \(S\textsc {mart}_{\textsc {O}}\,\)—an optimal algorithm, and \(S\textsc {mart}_{\textsc {H}}\,\)—a heuristic approach, to solve Smart instances. The two algorithms present three novel features: (i) a novel approach to generate individual defender strategies by ordering the search space during column generation using insights from the Traveling Salesman Problem(TSP); (ii) exploitation of iterative modification of rewards of multiple defender resources to generate coordinated strategies and (iii) generation of tight upper bounds for pruning using the structure of the problem. Third, we present an extensive empirical and theoretical analysis of both \(S\textsc {mart}_{\textsc {O}}\,\)and \(S\textsc {mart}_{\textsc {H}}\,\). Fourth, we describe a large scale real-world experiment whereby we run the first head-to-head comparison between game-theoretic schedules generated using \(S\textsc {mart}_{\textsc {H}}\,\)against schedules generated by humans on a one-day patrol exercise over one train line of the Los Angeles Metro System. Our results show that game-theoretic schedules were evaluated to be superior to ones generated by humans.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. By convention in security games literature, the defender is referred to as “she” and the adversary as “he”.

  2. We associate effectiveness with activities and not with targets, assuming that each activity is equally effective at all targets.

  3. For the sake of computation, we formulate the LP as a minimization problem (Eq. 6); this will be explained in detail when we describe the slave procedure.

  4. Only considering pure-strategies for the attacker is not a limitation; Stackelberg games always exhibit at least one Strong Stackelberg equilibrium where the attacker’s best response is a pure strategy [30].

  5. \(W \ge \varGamma \) implies that two resources present at the same target at anytime during the patrol are considered to conduct a joint activity.

  6. We are not able to reveal the value of these payoffs due to an agreement with the Los Angeles County Sheriff Department (LASD).

  7. Whereas these estimates of individual and joint effectiveness could potentially be slightly altered, the purpose of our exercise was comparison with human schedulers. Since the comparison was ultimately conducted by security experts who evaluated the game-theoretic schedule to be superior to human-generated one, we may infer that the effectiveness values we obtained for the individual and joint activities in our SMART model for FSE were reasonable.

References

  1. Agmon, N., Kraus, S., & Kaminka, G. A. (2008). Multi-robot perimeter patrol in adversarial settings. In Proceedings of the International Conference on Robotics and Automation (ICRA) (pp. 2339–2345).

  2. Agmon, N., Sadov, V., Kaminka, G. A., & Kraus, S. (2008). The impact of adversarial knowledge on adversarial planning in perimeter patrol. AAMAS, 1, 55–62.

    Google Scholar 

  3. Agmon, N., Kaminka, G. A., & Kraus, S. (2011). Multi-robot adversarial patrolling: Facing a full-knowledge opponent. Journal of Artificial Intelligence Research (JAIR), 42(1), 887–916.

    MathSciNet  Google Scholar 

  4. Barnhart, C., Johnson, E. L., Nemhauser, G. L., Savelsbergh, M. W. P., & Vance, P. H. (1994). Branch and price: Column generation for solving huge integer programs. Operations Research, 46, 316–329.

    Article  MathSciNet  Google Scholar 

  5. Basilico, N., Gatti, N., & Amigoni, F. (2009). Leader-follower strategies for robotic patrolling in environments with arbitrary topologies. In Proceedings of the Eight International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS) (pp. 57–64).

  6. Becker, R., Zilberstein, S., Lesser, V., & Goldman, C. V. (2004). Solving transition independent decentralized Markov decision processes. Journal of Artificial Intelligence Research (JAIR), 22, 423–455.

    MathSciNet  MATH  Google Scholar 

  7. Bertsimas, D., & Tsitsiklis, J. N. (1994). Introduction to Linear Optimization. 1st. Boston: Athena Scientific, 1997.

  8. Brown, A., Camerer, C. F., & Lovallo, D. (2012). To review or not to review? Limited strategic thinking at the movie box office. American Economic Journal: Microeconomics, 4(2), 1–26.

    Google Scholar 

  9. Clarke, R. V. (1996). Preventing mass transit crime—Crime prevention studies (Vol. 6). Monsey: Criminal Justice Press.

    Google Scholar 

  10. Clarke, R. V., & Newman, G. (2007). Police and the prevention of crime. Policing: A Journal of Policy and Practice, 1, 9–20.

    Article  MATH  Google Scholar 

  11. Conitzer, V. (2012). Computing game-theoretic solutions and applications to security. In Proceedings of the AAAI Conference on Artificial Intelligence (AAAI) (pp. 2106–2112).

  12. Conitzer, V., & Sandholm, T. (2006). Computing the optimal strategy to commit to. In Proceedings of the Seventh ACM Conference on Electronic Commerce (EC) (pp. 82–90).

  13. Dickerson, J. P., Simari, G. I., Subrahmanian, V. S., & Kraus, S. (2010). A graph-theoretic approach to protect static and moving targets from adversaries. In Proceedings of the Ninth International Joint Conference on Autonomous Agents and Multiagent Systems (pp. 299–306).

  14. Fang, F., Jiang, A. X., & Tambe, M. (2013). Protecting moving targets with multiple mobile resources. Journal of Artificial Intelligence Research (JAIR), 48, 583–634.

    MathSciNet  Google Scholar 

  15. Gatti, N. (2008). Game theoretical insights in strategic patrolling: Model and algorithm in normal form. In Proceedings of the European Conference on Artificial Intelligence (ECAI) (pp. 403–407).

  16. Gutin, G., Yeo, A., & Zverovich, A. (2002). Traveling salesman should not be greedy: Domination analysis of greedy-type heuristics for the tsp. Discrete Applied Mathematics, 117, 81–86.

    Article  MathSciNet  MATH  Google Scholar 

  17. Jain, M., Kardes, E., Kiekintveld, C., Tambe, M., & Ordonez, F. (2010). Security games with arbitrary schedules: A branch and price approach. In Proceedings of the AAAI Conference on Artificial Intelligence (AAAI) (pp. 792–797).

  18. Jain, M., Pita, J., Tsai, J., Kiekintveld, C., Rathi, S., Ordonez, F., et al. (2010). Software assistants for patrol planning at lax and federal air marshals service. Interfaces, 40(4), 267–290.

    Article  Google Scholar 

  19. Jiang, A. X., Procaccia, A. D., Qian, Y., Shah, N., & Tambe, M. (2013). Defender (mis)coordination in security games. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI) (pp. 199–206).

  20. Jiang, A. X., Yin, Z., Zhang, C., Tambe, M., & Kraus, S. (2013). Game-theoretic randomization for security patrolling with dynamic execution uncertainty. In Proceedings of the Twelfth International Conference on Autonomous Agents and Multiagent Systems (pp. 207–214).

  21. Kiekintveld, C., Jain, M., Tsai, J., Pita, J., Ordóñez, F., & Tambe, M. (2009). Computing optimal randomized resource allocations for massive security games. In Proceedings of The Eighth International Conference on Autonomous Agents and Multiagent Systems (pp. 233–239).

  22. Korzhyk, D., Conitzer, V., & Parr, R. (2011). Security games with multiple attacker resources. In Proceedings of the Twenty-second International Joint Conference on Artificial Intelligence (IJCAI) (pp. 273–279).

  23. Korzhyk, D., Conitzer, V., & Parr, R. (2011). Solving stackelberg games with uncertain observability. In Proceedings of the Tenth International Conference on Agents and Multi-agent Systems (AAMAS) (pp. 1013–1020).

  24. Letchford, J., & Conitzer, V. (2013). Solving security games on graphs via marginal probabilities. In Proceedings of the AAAI Conference on Artificial Intelligence (AAAI) (pp. 591–597).

  25. Letchford, J., & Vorobeychik, Y. (2013). Optimal interdiction of attack plans. In Proceedings of the Twelfth International Conference of Autonomous Agents and Multi-agent Systems (AAMAS) (pp. 199–206).

  26. Letchford, J., MacDermed, L., Conitzer, V., Parr, R., & Isbell, C. L. (2012). Computing optimal strategies to commit to in stochastic games. In Proceedings of the AAAI Conference on Artificial Intelligence (pp. 1380–1386).

  27. Machado, A., Ramalho, G., Zucker, J. D., & Drogoul, A. (2003). Multi-agent patrolling: An empirical analysis of alternative architectures. In Proceedings of the International Conference of Multi-Agent-Based Simulation (pp. 155–170).

  28. Nemhauser, G. L., Wolsey, L. A., & Fisher, M. L. (1978). An analysis of approximations for maximizing submodular set functions-I. Mathematical Programming, 14(1), 265–294.

    Article  MathSciNet  Google Scholar 

  29. Ostling, R., Wang, J., Tao-yi, J., Chou, E. Y., & Camerer, C. F. (2011). Testing game theory in the field: Swedish lupi lottery games. American Economic Journal: Microeconomics, 3(3), 1–33.

    Google Scholar 

  30. Paruchuri, P., Pearce, J. P., Marecki, J., Tambe, M., Ordonez, F., & Kraus, S. (2008). Playing games for security: An efficient exact algorithm for solving bayesian stackelberg games. In Proceedings of the Seventh International Conference on Autonomous Agents and Multiagent Systems (AAMAS) (pp. 539–547).

  31. Pita, J., Jain, M., Western, C., Portway, C., Tambe, M., Ordonez, F., et al. (2008). Deployed armor protection: The application of a game theoretic model for security at the los angeles international airport. In Proceedings of the International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS).

  32. Pita, J., Tambe, M., Kiekintveld, C., Cullen, S., & Steigerwald, E. (2011). GUARDS - Innovative Application of Game Theory for National Airport Security. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI) (pp. 2710–2715).

  33. Shieh, E., An, B., Yang, R., Tambe, M., Baldwin, C., DiRenzo, J., et al. (2012). PROTECT: A Deployed Game Theoretic System to Protect the Ports of the United States. In Proceedings of the Eleventh International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS) (pp. 13–20).

  34. Shieh, E., Jain, M., Jiang, A. X., & Tambe, M. (2013). Efficiently solving joint activity based security games. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI) (pp. 346–352).

  35. Shieh, E., Jiang, A. X., Yadav, A., Varakantham, P., & Tambe, M. (2014). Unleashing dec-mdps in security games: Enabling effective defender teamwork. In Proceedings of the European Conference on Artificial Intelligence (ECAI) (pp. 2340–2345).

  36. Sless, E., Agmon, N., & Kraus, S. (2014). Multi-robot adversarial patrolling: Facing coordinated attacks. In Proceedings of the Thirteenth International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS) (pp. 1093–1100).

  37. Sutton, R. S., & Barto, A. G. (1998). Reinforcement learning: An introduction. Cambridge: MIT Press.

    Google Scholar 

  38. Tambe, M. (2011). Security and game theory: Algorithms, deployed systems, lessons learned. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  39. Tsai, J., Rathi, S., Kiekintveld, C., Ordóñez, F., & Tambe, M. (2009). IRIS—A tool for strategic security allocation in transportation networks. In Proceedings of the Eight International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS) (pp. 831–839).

  40. Vanek, O., Bosansky, B., Jakob, M., & Pechoucek, M. (2010). Transiting areas patrolled by a mobile adversary. In IEEE Symposium on Computational Intelligence and Games (CIG), IEEE (pp. 9–16).

  41. Vanek, O., Jakob, M., Hrstka, O. & Pechoucek, M. (2011). Using multi-agent simulation to improve the security of maritime transit. In MABS 2011.

  42. Varakantham, P., Chuin Lau, H., & Yuan, Z. (2013). Scalable randomized patrolling for securing rapid transit networks. In Proceedings of the Conference for Innovative Applications for Artificial Intelligence (IAAI) (pp. 1563–1568).

  43. Vorobeychik, Y. & Singh, S. (2012). Computing stackelberg equilibria in discounted stochastic games. In Proceedings of the AAAI Conference on Artificial Intelligence (AAAI) (pp. 1478–1484).

  44. Yin, Z., Jiang, A., Johnson, M., Tambe, M., Kiekintveld, C., Leyton-Brown, K., et al. (2012). Trusts: Scheduling randomized patrols for fare inspection in transit systems. In Proceedings of the Conference on Innovative Applications for Artificial Intelligence (IAAI) (pp. 59–72).

Download references

Acknowledgments

The authors would like to acknowledge their appreciation for the collaboration of the Los Angeles Sheriff’s Department (LASD), the Booz-Allen Hamilton Company and the Transportation Security Administration’s (TSA) Intermodal Security Training and Exercise Program (I-STEP). The LASD provided us with exceptional support and preparation which allowed us to organize our experiments with great detail and accuracy. Booz-Allen managed TSA’s I-STEP Los Angeles Mass Transit Full-Scale Exercise, conducted May 16, 2014, thus allowing us to run experiments and collect data in very realistic and practical conditions. This research was supported by the United States Department of Homeland Security (DHS) through the National Center for Risk and Economic Analysis of Terrorism Events (CREATE) at the University of Southern California (USC) under Basic Ordering Agreement HSHQDC-10-A-BOA19, Task Order No. HST02-12-J-MLS151. However, any opinions, findings, and conclusions or recommendations in this document are those of the authors and do not necessarily reflect views of the United States Department of Homeland Security, or the University of Southern California, or CREATE.

Author information

Authors and Affiliations

  1. University of Southern California, Los Angeles, CA , 90089, USA

    Francesco Maria Delle Fave, Eric Shieh, Manish Jain, Albert Xin Jiang, Heather Rosoff & Milind Tambe

  2. Los Angeles County Sheriff’s Department, Los Angeles, CA, USA

    John P. Sullivan

Authors
  1. Francesco Maria Delle Fave
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric Shieh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Manish Jain
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Albert Xin Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Heather Rosoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Milind Tambe
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. John P. Sullivan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francesco Maria Delle Fave.

Additional information

This article has a double first authorship. Both Francesco Maria Delle Fave and Eric Shieh should be considered first authors of this work.

An initial version of the work presented in this article has previously appeared in [34]. In this work, we extend this initial version with the following contributions: (i) we present a large scale real-world experiment, describing in detail how a real-world security deployment problem could be modeled using our Smart framework; (ii) within this experiment, we provide the first head-to-head comparison between game-theoretic schedules (generated by our algorithm \(S\textsc {mart}_{\textsc {H}}\,\)) and human-generated schedules, presenting results showing some of the benefits that game-theoretic scheduling can provide; (iii) we present new simulations where we analyze the performance of TSP ordering heuristic developed to scale up \(S\textsc {mart}_{\textsc {H}}\,\)and (iv) where we evaluate the effectiveness of using OrigamiP to prune nodes of the branch-and-price tree; (v) finally, we provide additional detailed examples and discuss significant new related work and future work.

Appendix

Appendix

This appendix presents two tables:

  • Table 10 depicts the security allocation resulting from the manual allocation process.

  • Table 11 depicts the security allocation resulting from the game-theoretic allocation process based on the \(S\textsc {mart}_{\textsc {H}}\,\)algorithm.

Table 10 The human-generated security allocation
Full size table
Table 11 Security allocation generated by \(S\textsc {mart}_{\textsc {H}}\,\): \(s\) represents the street level of a station, \(m\) the mezzanine level and \(p\) the platform level
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Delle Fave, F.M., Shieh, E., Jain, M. et al. Efficient solutions for joint activity based security games: fast algorithms, results and a field experiment on a transit system. Auton Agent Multi-Agent Syst 29, 787–820 (2015). https://doi.org/10.1007/s10458-014-9270-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10458-014-9270-4

Keywords

Search

Navigation