Strategic defense and attack for series and parallel reliability systems

Abstract

A system of independent components is defended by a strategic defender and attacked by a strategic attacker. The reliability of each component depends on how strongly it is defended and attacked, and on the intensity of the contest. In a series system, the attacker benefits from a substitution effect since attacker benefits flow from attacking any of the components, while the defender needs to defend all components. Even for a series system, when the attacker is sufficiently disadvantaged with high attack inefficiencies, and the intensity of the contest is sufficiently high, the defender earns maximum utility and the attacker earns zero utility. The results for the defender (attacker) in a parallel system are equivalent to the results for the attacker (defender) in a series system. Hence, the defender benefits from the substitution effect in parallel systems. With budget constraints the ratio of the investments for each component, and the contest success function for each component, are the same as without budget constraints when replacing the system values for the defender and attacker with their respective budget constraints.

Introduction

A large literature on security and safety applies reliability analysis to determine the most cost-effective risk reduction strategies. Examples are Levitin, 2003a, Levitin, 2003b, Levitin and Lisnianski, 2001, Levitin and Lisnianski, 2003, Levitin et al., 2003. The external threat is usually assumed to be static. Within cyber security, Gordon and Loeb, 2002, Gordon et al., 2003 determine the optimal investment for information protection, and Gal-Or and Ghose (2005) analyze how market characteristics affect security investment. Again the threat is assumed fixed and immutable.

Some research applying game theory considers components in isolation (Major, 2002, Woo, 2002, Woo, 2003, O’Hanlon et al., 2002). For multiple components one strand of literature associates one defender with each component. Hausken (2002) lets each agent dichotomously choose a strategy which for its component causes either reliability zero with no cost of effort or reliability one for a fixed cost of effort. He finds that the series, parallel, and summation systems frequently correspond to the coordination game, the battle of the sexes and the chicken game, and prisoner’s dilemma, respectively.

Kunreuther and Heal (2003) analyze interdependent systems where one target’s defense benefits all targets.¹ Zhuang et al. (forthcoming) explore the effects of heterogeneous discount rates on the optimal defensive strategy in such systems. Hausken (2006) finds that with increasing interdependence, each defending agent free rides by investing less, suffers lower profit, while the attacker enjoys higher profit. Enders and Sandler, 2003, Hausken, 2006 analyze the substitution effect which causes a strategic attacker to substitute into the most optimal attack allocation across multiple targets, and the income effect which eliminates parts of the attacker’s resource base.

Another strand of the literature lets one defender defend entire systems. For series and parallel systems with independent components Bier and Abhichandani, 2002, Bier et al., 2005 assume that the defender minimizes the success probability, and expected damage, respectively, of an attack. The success probability depends on the resources expended by the defender to strengthen each component.² Although the approach implicitly accounts for a strategic attacker (for series systems the defender equalizes the expected damage of attacks against multiple components), a more general approach would assume that the success probability of an attack depends on resource investments by both the defender and the attacker for each component. The attack September 11, 2001 illustrated that major threats today involve strategic attackers. Threats emerge from nature, technology, and humans, but increasing complexity and human involvement suggest that the strategic factor needs to be given increased emphasis in future research.

Azaiez and Bier (2007) discuss various simplifications to this general approach. For example, the level of effort expended by the attacker on each component could be a constant. They choose the simplification that the success probability of an attack on each component is constant, and assume that the defender attempts to deter attacks by making them as costly as possible to the attacker. This enables them to find closed-form results for systems with moderately general structures with both parallel and series subsystems.

The objective of this article is to extend the research where an entire system of independent components is defended by a fully strategic defender and attacked by a fully strategic attacker. The external threat is neither static, fixed, nor immutable. Series and parallel systems are considered. The defender and attacker adapt to each other optimally choosing continuous strategic variables for each component under defense and attack. The reliability of each component depends on the relative investments the defender and attacker direct into defending versus attacking that component. The defender seeks to maximize the reliability of the system, accounting for its assessment of the value of the system, while the attacker seeks to minimize the reliability, accounting for its often different assessment of the system value. One paramount consideration is how each agent substitutes investments across multiple components.

One defense inefficiency and one attack inefficiency are associated with each component, which specify unit costs of defense and attack. Such inefficiencies vary considerably across components. A component such as the US Gold Reserve stored at Ft. Knox has high defense inefficiency and even higher attack inefficiency. It is located for optimal defense, and is very hard to attack. Another component such as the US Statue of Liberty has a more vulnerable location which increases the defense inefficiency and decreases the attack inefficiency. A component such as an underground transport system has high defense inefficiency since it is geographically dispersed, and low attack inefficiency. In contrast, a component buried deep within a mountain has low defense inefficiency and high attack inefficiency.

The article assumes variation in the intensity of the contest between the defender and attacker for each component. The intensity can vary greatly across components. Low intensity occurs for components and systems that are defendable, and where the individual components are dispersed. In such cases neither the defender nor the attacker can easily get a significant upper hand. High intensity occurs for systems that are easier to attack, and where the individual components are concentrated. This may cause “winner-take-all” battles and dictatorship by the strongest agent.

A crucial issue for defense and attack is how the various components, such as in these examples, are interlinked in series and parallel. As is conventional in the literature, components are assumed independent. If they are not, the analysis presumes application of Simon’s (1969, 217) principle of “near decomposability”. Complex or hierarchic systems are frequently nearly decomposable, and intracomponent linkages are generally stronger than intercomponent linkages. Multiple subcomponents that are sufficiently interdependent are joined together to form one larger aggregate component which thus has a more complex internal structure from which the relevant parameters such as the unit costs of defense and attack, and the intensity of the contest, are determined. This process continues until each component is sufficiently independent from the other components so that the analysis can justifiably assume independent components as an approximation.

Section 2 introduces component and system reliabilities, utilities, and contest success functions. Section 3 considers an arbitrarily complex system. Sections 4 Series system, 5 Series system with budget constraints analyze the series system without and with budget constraints. Sections 6 Parallel system, 7 Parallel system with budget constraints analyze the parallel system without and with budget constraints. Section 8 concludes.