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A Differential Games Solution to a Model of Competition Between a Thief and the Police

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  • Gustav Feichtinger

    (University of Technology, Vienna, Austria)

Abstract

This paper contains an extension of an optimal control model considered by Sethi (Sethi, S. P. 1979. Optimal Pilfenng policies for dynamic continuous thieves. Management Sci. 25 (6, June) 535--542.) to a differential game situation. It is assumed that Sethi's concave thief, i.e., a risk-averter, plays against the police, whose objective function incorporates convex costs of law enforcement, a one-shot utility at the time of arrest and a continuous utility or cost rate after the thief is arrested. The probability that the thief is caught by time t is influenced not only by the pilfering rate (as is the case in Sethi's model) but also by the instrumental variable of the police, namely its rate of law enforcement. Our aim is to analyze noncooperative Nash solutions of the differential game sketched above. Due to the special structure of the Hamiltonians, a system of two nonlinear differential equations for the control variables of both players can be derived. The system is solved explicitly under special assumptions. It is shown that the optimal rate of law enforcement of police increases monotonically whereas for the Nash-optimal pilfering rate three cases may be characterized in which increase, decrease or constancy occur. Thus the Nash-optimal solutions show in each case monotonic behavior, i.e., changes in trend never occur. Moreover, the dependency of the terminal values of control can be described. Finally, an interpretation of the model as market entrance game is sketched.

Suggested Citation

  • Gustav Feichtinger, 1983. "A Differential Games Solution to a Model of Competition Between a Thief and the Police," Management Science, INFORMS, vol. 29(6), pages 686-699, June.
  • Handle: RePEc:inm:ormnsc:v:29:y:1983:i:6:p:686-699
    DOI: 10.1287/mnsc.29.6.686
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    Cited by:

    1. Motchenkova, Evgenia, 2008. "Determination of optimal penalties for antitrust violations in a dynamic setting," European Journal of Operational Research, Elsevier, vol. 189(1), pages 269-291, August.
    2. Alain Bensoussan & Suresh P. Sethi, 2007. "The Machine Maintenance and Sale Age Model of Kamien and Schwartz Revisited," Management Science, INFORMS, vol. 53(12), pages 1964-1976, December.
    3. E. Motchenkova & P. M. Kort, 2006. "Analysis of Current Penalty Schemes for Violations of Antitrust Laws," Journal of Optimization Theory and Applications, Springer, vol. 128(2), pages 431-451, February.
    4. A. Bensoussan & Q. Feng & S. P. Sethi, 2022. "Integrating equipment investment strategy with maintenance operations under uncertain failures," Annals of Operations Research, Springer, vol. 317(2), pages 353-386, October.
    5. Huseyin Cavusoglu & Byungwan Koh & Srinivasan Raghunathan, 2010. "An Analysis of the Impact of Passenger Profiling for Transportation Security," Operations Research, INFORMS, vol. 58(5), pages 1287-1302, October.
    6. Avinadav, Tal & Chernonog, Tatyana & Fruchter, Gila E. & Prasad, Ashutosh, 2020. "Contract design when quality is co-created in a supply chain," European Journal of Operational Research, Elsevier, vol. 286(3), pages 908-918.
    7. Motchenkova, E. & Kort, P.M., 2004. "Analysis of the Properties of Current Penalty Schemes for Violations of Antitrust Law," Discussion Paper 2004-97, Tilburg University, Center for Economic Research.
    8. Luca Colombo & Paola Labrecciosa & Agnieszka Rusinowska, 2022. "A Dynamic Analysis of Criminal Networks," Documents de travail du Centre d'Economie de la Sorbonne 22006r, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Jun 2023.
    9. Luiz de Mello, 2009. "Avoiding the Value Added Tax," Public Finance Review, , vol. 37(1), pages 27-46, January.
    10. Steve Alpern & Alec Morton & Katerina Papadaki, 2011. "Patrolling Games," Operations Research, INFORMS, vol. 59(5), pages 1246-1257, October.
    11. Guzmán, Cristóbal & Riffo, Javiera & Telha, Claudio & Van Vyve, Mathieu, 2022. "A sequential Stackelberg game for dynamic inspection problems," European Journal of Operational Research, Elsevier, vol. 302(2), pages 727-739.
    12. Huseyin Cavusoglu & Young Kwark & Bin Mai & Srinivasan Raghunathan, 2013. "Passenger Profiling and Screening for Aviation Security in the Presence of Strategic Attackers," Decision Analysis, INFORMS, vol. 10(1), pages 63-81, March.
    13. Guzman, Cristobal & Riffo, Javiera & Telha, Claudio & Van Vyve, Mathieu, 2021. "A Sequential Stackelberg Game for Dynamic Inspection Problems," LIDAM Discussion Papers CORE 2021036, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Iván Marinovic & Martin Szydlowski, 2022. "Monitoring with career concerns," RAND Journal of Economics, RAND Corporation, vol. 53(2), pages 404-428, June.
    15. Herbert Dawid & Engelbert Dockner & Richard Hartl & Josef Haunschmied & Ulrike Leopold-Wildburger & Mikulas Luptacik & Alexander Mehlmann & Alexia Prskawetz & Marion Rauner & Gerhard Sorger & Gernot T, 2010. "Gustav Feichtinger celebrates his 70th birthday," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 18(4), pages 437-451, December.
    16. Suresh P. Sethi & Sushil Gupta & Vipin K. Agrawal & Vijay K. Agrawal, 2022. "Nobel laureates’ contributions to and impacts on operations management," Production and Operations Management, Production and Operations Management Society, vol. 31(12), pages 4283-4303, December.
    17. Huseyin Cavusoglu & Birendra Mishra & Srinivasan Raghunathan, 2005. "The Value of Intrusion Detection Systems in Information Technology Security Architecture," Information Systems Research, INFORMS, vol. 16(1), pages 28-46, March.

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