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Recursive inspection games

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  • von Stengel, Bernhard

Abstract

We consider a sequential inspection game where an inspector uses a limited number of inspections over a larger number of time periods to detect a violation (an illegal act) of an inspectee. Compared with earlier models, we allow varying rewards to the inspectee for successful violations. As one possible example, the most valuable reward may be the completion of a sequence of thefts of nuclear material needed to build a nuclear bomb. The inspectee can observe the inspector, but the inspector can only determine if a violation happens during a stage where he inspects, which terminates the game; otherwise the game continues. Under reasonable assumptions for the payoffs, the inspector’s strategy is independent of the number of successful violations. This allows to apply a recursive description of the game, even though this normally assumes fully informed players after each stage. The resulting recursive equation in three variables for the equilibrium payoff of the game, which generalizes several other known equations of this kind, is solved explicitly in terms of sums of binomial coefficients. We also extend this approach to nonzero-sum games and “inspector leadership” where the inspector commits to (the same) randomized inspection schedule, but the inspectee acts legally (rather than mixes as in the simultaneous game) as long as inspections remain.

Suggested Citation

  • von Stengel, Bernhard, 2016. "Recursive inspection games," LSE Research Online Documents on Economics 68299, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:68299
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    File URL: http://eprints.lse.ac.uk/68299/
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    References listed on IDEAS

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    1. Bernhard von Stengel, 2016. "Recursive Inspection Games," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 935-952, August.
    2. von Stengel, Bernhard & Zamir, Shmuel, 2010. "Leadership games with convex strategy sets," Games and Economic Behavior, Elsevier, vol. 69(2), pages 446-457, July.
    3. Avenhaus, Rudolf & Canty, Morton John, 2005. "Playing for time: A sequential inspection game," European Journal of Operational Research, Elsevier, vol. 167(2), pages 475-492, December.
    4. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, April.
    5. Avenhaus, Rudolf & Von Stengel, Bernhard & Zamir, Shmuel, 2002. "Inspection games," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 51, pages 1947-1987, Elsevier.
    6. Thomas S. Ferguson & Costis Melolidakis, 2000. "Games with finite resources," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(2), pages 289-303.
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    Cited by:

    1. Bernhard von Stengel, 2016. "Recursive Inspection Games," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 935-952, August.
    2. Deutsch, Yael & Goldberg, Noam & Perlman, Yael, 2019. "Incorporating monitoring technology and on-site inspections into an n-person inspection game," European Journal of Operational Research, Elsevier, vol. 274(2), pages 627-637.
    3. Ederlina Ganatuin‐Nocon & Tyrone Ang, 2020. "Revisiting inspection game and inspector leadership through reaction networks," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(6), pages 438-452, September.
    4. Stefanos Leonardos & Costis Melolidakis, 2018. "On the Commitment Value and Commitment Optimal Strategies in Bimatrix Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-28, September.
    5. Ben Hermans & Herbert Hamers & Roel Leus & Roy Lindelauf, 2019. "Timely exposure of a secret project: Which activities to monitor?," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(6), pages 451-468, September.
    6. Deutsch, Yael, 2021. "A polynomial-time method to compute all Nash equilibria solutions of a general two-person inspection game," European Journal of Operational Research, Elsevier, vol. 288(3), pages 1036-1052.

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    More about this item

    Keywords

    inspection game; multistage game; recursive game; Stackelberg leadership; binominal coefficients;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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