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Modeling costly learning and counter-learning in a defender-attacker game with private defender information

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Abstract

In asymmetric war scenarios (e.g., counter-terrorism), the adversary usually invests a significant time to learn the system structure and identify vulnerable components, before launching attacks. Traditional game-theoretic defender-attacker models either ignore such learning periods or the entailed costs. This paper fills the gap by analyzing the strategic interactions of the terrorist’s costly learning and defender’s counter-learning and defense strategies in a game with private defender information. Our model allows six possible attacker strategies: (a) attack immediately; (b) learn and attack; (c) learn and not attack; (d) learn and attack when appearing vulnerable and not attack when appearing invulnerable; (e) learn and not attack when appearing vulnerable and attack when appearing invulnerable; and (f) not attack. Our results show that four of the six strategies (a, d, e, f) are possible at equilibrium and the other two (b, c) are strictly dominated. Interestingly, we find that the counterintuitive strategy (e) could be at equilibrium, especially when the probability that the target appears vulnerable given it is invulnerable is sufficiently high. Our results also show that the attacker’s learning cost has a significant impact on both the attacker’s best responses and the defender’s equilibrium deception and defense strategies. Finally, we study the attacker’s values of perfect information and imperfect information, which provide additional insights for defense and counter-learning strategies.

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Notes

  1. This assumption is reasonable in many security scenarios (especially those involving new and less-than-fully tested technology), where even the defender could be uncertain about the system vulnerability (US Department of Homeland Security 2011).

  2. For simplicity, we focus on binary attacker effort (i.e., attack or not attack). This might be relevant in some high-level strategic decision-making situations, concerning which targets are likely to be attacked (rather than the level of attack effort on each targets). However, we acknowledge that the attack effort may be different among attacked targets and future work could consider continuous-level attack.

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Authors and Affiliations

  1. Department of Industrial and System Engineering, SUNY at Buffalo, Buffalo, NY, 14260, USA

    Jie Xu & Jun Zhuang

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  1. Jie Xu
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  2. Jun Zhuang
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Correspondence to Jun Zhuang.

Additional information

This research was partially supported by the United States Department of Homeland Security (DHS) through the National Center for Risk and Economic Analysis of Terrorism Events (CREATE) under award number 2010-ST-061-RE0001. This research was also patricianly supported by the United States National Science Foundation under award numbers 1200899 and 1334930. However, any opinions, findings, and conclusions or recommendations in this document are those of the authors and do not necessarily reflect views of the DHS, CREATE, or NSF.

Appendix: Definitions for the \(C_i(d,l),~i=1,\ldots ,12\)

Appendix: Definitions for the \(C_i(d,l),~i=1,\ldots ,12\)

$$\begin{aligned} C_1(d, l)&\equiv \left\{ [P_0(l)V_A+W]P_V-W>[(P_0(l)V_A+W)P_{V|V'}(d)-W]P_{V'}(d)-C(d);\right. \\&\left. [P_0(l)V_A+W]P_V-W>[(P_0(l)V_A+W)P_{V|NV'}(d)-W][1-P_{V'}(d)]-C(d);\right. \\&\left. [P_0(l)V_A+W]P_V-W>0 \right\} ;\\ C_2(d, l)&\equiv \left\{ [(P_0(l)V_A+W)P_{V|V'}(d)-W]P_{V'}(d)-C(d)\ge W (1-P_{V}) (1-P_{V'|NV});\right. \\&\left. [(P_0(l)V_A+W)P_{V|V'}(d)-W]P_{V'}(d)\ge [(P_0(l)V_A+W)P_{V|NV'}(d)-W][1-P_{V'}(d)];\right. \\&\left. [(P_0(l)V_A+W)P_{V|V'}(d)-W]P_{V'}(d)-C(d)>0\right\} ;\\ C_3(d, l)&\equiv \left\{ [(P_0(l)V_A+W)P_{V|NV'}(d)-W][1-P_{V'}(d)]-C(d)\ge W (1-P_{V}) P_{V'|NV}; \right. \\&\left. [(P_0(l)V_A+W)P_{V|NV'}(d)-W][1-P_{V'}(d)]>[(P_0(l)V_A+W)P_{V|V'}(d)-W]P_{V'}(d);\right. \\&\left. [(P_0(l)V_A+W)P_{V|NV'}(d)-W][1-P_{V'}(d)]-C(d)>0\right\} ;\\ C_4(d, l)&\equiv \left\{ [P_0(l)V_A+W]P_V-W\ge 0;[(P_0(l)V_A+W)P_{V|V'}(d)-W]P_{V'}(d)-C(d)\ge 0;\right. \\&\left. [(P_0(l)V_A+W)P_{V|NV'}(d)-W][1-P_{V'}(d)]-C(d)\ge 0\right\} ;\\ C_5(d, l)&\equiv \left\{ [P_0(l)V_A+W]P_V-W>[(P_0(l)V_A+W) P_{V|V'}(d)-W] P_{V'}(d);\right. \\&\left. [P_0(l)V_A+W]P_V-W>[(P_0(l)V_A+W) P_{V|NV'}(d)-W] (1-P_{V'}(d));\right. \\&\left. [P_0(l)V_A+W]P_V-W>0\right\} ;\\ C_6(d, l)&\equiv \left\{ [P_0(l)V_A+W]P_V-W\le 0; [(P_0(l)V_A+W) P_{V|V'}(d)-W] P_{V'}(d)\le 0;\right. \\&\left. [(P_0(l)V_A+W) P_{V|NV'}(d)-W] (1-P_{V'}(d))\le 0\right\} ;\\ C_7(d, l)&\equiv \left\{ [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)-C(d)\ge [P_0(l)V_A+W]P_V-W;\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)\ge [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)];\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)-C(d)>0\right\} ;\\ C_8(d, l)&\equiv \left\{ [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)]-C(d)\ge [P_0(l)V_A+W]P_V-W;\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)<[(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)];\right. \\&\left. [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)]-C(d)>0\right\} ;\\ C_9(d, l)&\equiv \left\{ [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)\ge [P_0(l)V_A+W]P_V-W;\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)\ge [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)];\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)>0;\right. \\&\left. [P_0(l)V_A+W]P_V-W>[(P_0(l)V_A+W)P_{V|V'}(d)-W]P_{V'}(d)-C(d);\right. \\&\left. [P_0(l)V_A+W]P_V-W>[(P_0(l)V_A+W)P_{V|NV'}(d)-W][1-P_{V'}(d)]-C(d);\right. \\&\left. [P_0(l)V_A+W]P_V-W>0 \right\} ; \end{aligned}$$
$$\begin{aligned} C_{10}(d, l)&\equiv \left\{ [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)]-C(d)\ge [P_0(l)V_A+W]P_V-W;\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)<[(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)];\right. \\&\left. [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)]-C(d)>0;\right. \\&\left. [P_0(l)V_A+W]P_V-W>[(P_0(l)V_A+W)P_{V|V'}(d)-W]P_{V'}(d)-C(d);\right. \\&\left. [P_0(l)V_A+W]P_V-W>[(P_0(l)V_A+W)P_{V|NV'}(d)-W][1-P_{V'}(d)]-C(d);\right. \\&\left. [P_0(l)V_A+W]P_V-W>0 \right\} ;\\ C_{11}(d, l)&\equiv \left\{ [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)\ge [P_0(l)V_A+W]P_V-W;\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)\ge [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)];\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)>0;[(P_0(l)V_A+W) P_{V|V'}(d)-W] P_{V'}(d)\le 0;\right. \\&\left. [P_0(l)V_A+W]P_V-W\le 0;[(P_0(l)V_A+W) P_{V|NV'}(d)-W] (1-P_{V'}(d))\le 0\right\} ;\\ C_{12}(d, l)&\equiv \left\{ [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)]-C(d)\ge [P_0(l)V_A+W]P_V-W;\right. \\&\left. [(P_0(l)V_A+W) P_{V|V'}(d)-W]P_{V'}(d)<[(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)];\right. \\&\left. [(P_0(l)V_A+W) P_{V|NV'}(d)-W] [1-P_{V'}(d)]-C(d)>0;\right. \\&\left. [P_0(l)V_A+W]P_V-W\le 0; [(P_0(l)V_A+W) P_{V|V'}(d)-W] P_{V'}(d)\le 0;\right. \\&\left. [(P_0(l)V_A+W) P_{V|NV'}(d)-W] (1-P_{V'}(d))\le 0\right\} .\\ \end{aligned}$$

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Xu, J., Zhuang, J. Modeling costly learning and counter-learning in a defender-attacker game with private defender information. Ann Oper Res 236, 271–289 (2016). https://doi.org/10.1007/s10479-014-1722-3

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