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A sequential partial information bomber‐defender shooting problem

Author

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  • Krishna Kalyanam
  • David Casbeer
  • Meir Pachter

Abstract

A bomber carrying homogenous weapons sequentially engages ground targets capable of retaliation. Upon reaching a target, the bomber may fire a weapon at it. If the target survives the direct fire, it can either return fire or choose to hold fire (play dead). If the former occurs, the bomber is immediately made aware that the target is alive. If no return fire is seen, the true status of the target is unknown to the bomber. After the current engagement, the bomber, if still alive, can either re‐engage the same target or move on to the next target in the sequence. The bomber seeks to maximize the expected cumulative damage it can inflict on the targets. We solve the perfect and partial information problems, where a target always fires back and sometimes fires back respectively using stochastic dynamic programming. The perfect information scenario yields an appealing threshold based bombing policy. Indeed, the marginal future reward is the threshold at which the control policy switches and furthermore, the threshold is monotonic decreasing with the number of weapons left with the bomber and monotonic nondecreasing with the number of targets left in the mission. For the partial information scenario, we show via a counterexample that the marginal future reward is not the threshold at which the control switches. In light of the negative result, we provide an appealing threshold based heuristic instead. Finally, we address the partial information game, where the target can choose to fire back and establish the Nash equilibrium strategies for a representative two target scenario.

Suggested Citation

  • Krishna Kalyanam & David Casbeer & Meir Pachter, 2020. "A sequential partial information bomber‐defender shooting problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(3), pages 223-235, April.
  • Handle: RePEc:wly:navres:v:67:y:2020:i:3:p:223-235
    DOI: 10.1002/nav.21892
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    References listed on IDEAS

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    1. Richard Weber, 2013. "ABCs of the bomber problem and its relatives," Annals of Operations Research, Springer, vol. 208(1), pages 187-208, September.
    2. Gao, Xing & Zhong, Weijun & Mei, Shue, 2013. "A game-theory approach to configuration of detection software with decision errors," Reliability Engineering and System Safety, Elsevier, vol. 119(C), pages 35-43.
    3. David V. Mastran & Clayton J. Thomas, 1973. "Decision rules for attacking targets of opportunity," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 20(4), pages 661-672, December.
    4. Takasi Kisi, 1976. "Suboptimal decision rule for attacking targets of opportunity," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 23(3), pages 525-533, September.
    5. Hu, Xiaoxiao & Xu, Maochao & Xu, Shouhuai & Zhao, Peng, 2017. "Multiple cyber attacks against a target with observation errors and dependent outcomes: Characterization and optimization," Reliability Engineering and System Safety, Elsevier, vol. 159(C), pages 119-133.
    6. Takashi Kamihigashi, 2017. "41 Counterexamples to property (B) of the discrete time bomber problem," Annals of Operations Research, Springer, vol. 248(1), pages 579-588, January.
    7. Gregory Levitin & Kjell Hausken, 2012. "Resource Distribution in Multiple Attacks with Imperfect Detection of the Attack Outcome," Risk Analysis, John Wiley & Sons, vol. 32(2), pages 304-318, February.
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    Cited by:

    1. Hunt, Kyle & Zhuang, Jun, 2024. "A review of attacker-defender games: Current state and paths forward," European Journal of Operational Research, Elsevier, vol. 313(2), pages 401-417.

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