Modeling a Multitarget Attacker–Defender Game with Budget Constraints

Abstract

Though the choices of terrorists' attack targets are vast, their resources are limited. In this paper, a game-theoretical model is proposed to study both the defender's (government) and the attacker's (terrorist) expenditures among multiple targets under budget constraints to guide investment in defense. We study how the defender's and the attacker's equilibrium allocations depend on the budget constraints, target valuations, cost effectiveness of their investments, and inherent defense levels of targets in both sequential-move and simultaneous-move games. The equilibrium solutions are provided using the Karush-Kuhn-Tucker conditions. At the subgame-perfect Nash equilibrium, the defender's total marginal effects are the same among targets. Moreover, the defender's total marginal effects can be decomposed into direct and indirect effects. We also use the multiple-infrastructure and multiple-urban-area data sets to demonstrate the model results. The regression analysis shows that both the attacker's and the defender's equilibrium investments increase with their own target valuations, because a higher valuation for themselves means a higher attractiveness. Interestingly, the attacker's equilibrium investment is negatively correlated with the defender's target valuations, since a higher defender's valuation would make it more difficult for the attacker to successfully attack the target. By contrast, the defender's equilibrium investment is positively correlated with the attacker's target valuations, as a higher attacker's valuations would increase the urgency for the defender to protect the target. To show the utility of the new model, we compare the results of this model and the model in which the defender assumes that only a single target will be attacked when there could actually be multiple targets attacked. Our results show that the defender will suffer higher expected losses if she assumes that the attacker will attack only one target. The analysis of the attacker's and the defender's budget constraints show that (a) the higher the budget the defender has, the less likely it is that her most valuable target will be attacked; (b) a higher proportion of defense resources should be allocated to the most valuable target if the defender's budget is low; and (c) the attacker is less concentrated on attacking the most valuable target and spreads the resources to attack more targets as his budget increases.

A game‐theoretic model for resource allocation with deception and defense efforts

“…Therefore, appropriate resource allocations can greatly protect the defender from possible attacks, whereas improper decisions could lead to a waste of resources or even unprecedented damages to the defender's interest. Thus, knowing how to make the best use of resources is highly important, especially under circumstances of limited resources …”

“…Thus, knowing how to make the best use of resources is highly important, especially under circumstances of limited resources. 2,3 To date, many researchers have addressed the attacker-defender game, where each player seeks to maximize a single objective function under certain constraints. Game theory and the concept of Nash equilibrium have been recognized as suitable tools for studying such strategy-interaction resource allocation problems for a long time.…”

“…(1) consisting of multiple elements interacting with each other, (2) following an established or organized procedure, and (3) aiming to accomplish certain objectives. In fact, from a system perspective, the resource allocation game between two players matches these specific characteristics.…”

Allocating Limited Resources to Protect a Massive Number of Targets Using a Game Theoretic Model

“…No matter whether an action is successful, the actor will incur some cost by taking the action. Recent studies about the effort of deterrence and risk preferences in the security games [11], [12] have analyzed the impact of risk preference on the defense effort and deterrence level, and the impact of defender's cost on the investment strategy. Meanwhile, statistics show that a large data center costs between $10 million and $25 million per year and the corresponding maintenance costs account for nearly 80% of its total cost [13].…”

When Hackers Err: The Impacts of False Positives on Information Security Games

“…Extensive studies have been conducted to investigate how false positives affect information security investment, security systems design, and security systems performance (e.g. Cavusoglu et al 2005, 2009, Ulvila and Gaffney 2004, Guan et al 2017, Salman et al 2014, Duan et al 2012.…”