Skip to main content
SpringerLink
Log in

Differential Terror Queue Games

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

We present models of differential terror queue games, wherein terrorists seek to determine optimal attack rates over time, while simultaneously the government develops optimal counterterror staffing levels. The number of successful and interdicted terror attacks is determined via an underlying dynamic terror queue model. Different information structures and commitment abilities derive from different assumptions regarding what the players in the game can and cannot deduce about the underlying model. We compare and explain the impact of different information structures, i.e., open loop, closed loop, and asymmetric. We characterize the optimal controls for both the terrorists and the government in terms of the associated state and costate variables and deduce the costate equations that must be solved numerically to yield solutions to the game for the different cases. Using recently assembled data describing both terror attack and staffing levels, we compare the differential game models to each other as well as to the optimal control model of Seidl et al. (Eur J Oper Res 248:246–256, 2016). The paper concludes with a discussion of the lessons learned from the entire modeling exercise.

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

Similar content being viewed by others

Notes

  1. In the subsequent time, argument t is omitted for notational convenience unless necessary.

  2. For a more detailed discussion on the different information structures, we refer to [3, 5] and Long (2000).

  3. Thus, players do not deduce how their opponent actually determines its decision rule despite having full information about the game. This is probably the main reason why in the literature closed-loop or feedback solutions are seen as more reasonable than open-loop strategies.

  4. In general, nonzero-sum differential games which admit a unique open-loop Nash equilibrium solution have uncountably many Nash equilibrium solutions if the information structure is extended (see, e.g., [2]). This informationally nonuniqueness is based on the dynamic nature of the information. However, using the above method implies a canonical system of ordinary differential equations which has a unique steady state.

  5. Note, however, that this is also strongly related to the used parameters. If the costs of terror plots are lower or the utility is higher, results are the opposite.

  6. The reason for the sign of \(\eta _y\) is that for the terrorists a high Y is beneficial since then interdiction prevents agents from detecting new plots.

References

  1. Appelbaum B (2011) As US agencies put more value on a life Businesses fret. New York Times, Feb 16

  2. Basar T (1977) Informationally nonunique equilibrium solutions in differential games. SIAM J Control Opt 15:636–660

    Article  MathSciNet  MATH  Google Scholar 

  3. Basar T, Olsder GJ (1982) Dynamic noncooperative game theory. Academic Press, London

    MATH  Google Scholar 

  4. Cellini R, Lambertini L (2005) R&D incentives and market structure: dynamic analysis. J Optim Theory Appl 126:85–96

    Article  MathSciNet  MATH  Google Scholar 

  5. Dockner E, Jorgensen S, Long NV, Soger G (2000) Differential games in economics and management science. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  6. Enders W, Sandler T (2012) The political economy of terrorism. Cambridge University Press, New York

    Google Scholar 

  7. Feinstein JS, Kaplan EH (2010) Analysis of a strategic terror organization. J Confl Resolut 54:281–302

    Article  Google Scholar 

  8. Grass D, Caulkins JP, Feichtinger G, Tragler G, Behrens D (2008) Optimal control of nonlinear processes: with applications in drugs, corruption and terror. Springer, Berlin

    Book  MATH  Google Scholar 

  9. Jacobson D, Kaplan EH (2007) Suicide bombings and targeted killings in (counter-) terror games. J Confl Resol 51:772–792

    Article  Google Scholar 

  10. Kaplan EH, Kress M, Szechtman R (2010) Confronting entrenched insurgents. Oper Res 58:329–341

    Article  MathSciNet  MATH  Google Scholar 

  11. Kaplan EH (2010) Terror queues. Oper Res 58:773–784

    Article  MathSciNet  MATH  Google Scholar 

  12. Kaplan EH (2012) Estimating the duration of Jihadi terror plots in the United States. Stud Confl Terr 35:880–894

    Article  Google Scholar 

  13. Kaplan EH (2013) Staffing models for covert counterterrorism agencies. Soc Econ Plan Sci 47:2–8

    Article  Google Scholar 

  14. Kaplan EH (2015) Socially efficient detection of terror plots. Oxf Econ Pap 67:104–115

    Article  Google Scholar 

  15. Long NV (2010) A survey of dynamic games in economics. Surveys of theories in economics and business administration, 1st edn. World Scientific, Singapore

    Book  Google Scholar 

  16. Seidl A, Kaplan EH, Caulkins JP, Wrzaczek S, Feichtinger G (2016) Optimal control of a terror queue. Eur J Oper Res 248:246–256

    Article  MathSciNet  MATH  Google Scholar 

  17. Strom K, Pope M, Hollywood J, Weintraub G, Daye C, Gemeinhardt D (2010) Building on clues: examining successes and failures in detecting US terrorist plots 1999–2009. Institute for Homeland Security Solutions, Research Triangle Park, North Carolina

    Google Scholar 

  18. Viscusi WK, Aldy JE (2003) The value of a statistical life: a critical review of market estimates throughout the world. J Risk Uncertain 27:5–76

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This research was supported by the Austrian Science Fund (FWF) under Grants P25979-N25 and P25275-G11.

Author information

Authors and Affiliations

  1. Wittgenstein Centre for Demography and Global Human Capital (IIASA, VID/ÖAW, WU), Vienna Institute of Demography, Austrian Academy of Sciences, Vienna, Austria

    Stefan Wrzaczek & Gustav Feichtinger

  2. Yale School of Management, Yale School of Public Health, Yale School of Engineering and Applied Science, New Haven, CT, USA

    Edward H. Kaplan

  3. H. John Heinz III College, Carnegie Mellon University, Pittsburgh, PA, USA

    Jonathan P. Caulkins

  4. Department of Business Administration, University of Vienna, Vienna, Austria

    Andrea Seidl

  5. Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria

    Gustav Feichtinger

Authors
  1. Stefan Wrzaczek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edward H. Kaplan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jonathan P. Caulkins
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Andrea Seidl
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Gustav Feichtinger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Wrzaczek.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wrzaczek, S., Kaplan, E.H., Caulkins, J.P. et al. Differential Terror Queue Games. Dyn Games Appl 7, 578–593 (2017). https://doi.org/10.1007/s13235-016-0195-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-016-0195-1

Keywords

Search

Navigation