Kjell Hausken

Strategic defense and attack for series and parallel reliability systems

Kovenock and Roberson’s (2010) paper has the potential to advance the research frontier, but has deficiencies. This paper suggests how Kovenock and Roberson’s (2010) paper can be developed into a more substantial paper. Kovenock and Roberson’s (2010) paper consists of three sections. The first section is an introduction which is OK but has no results. The second section, titled “Model and Main Result”, provides no contribution beyond Hausken (2008a). It consists of equations (1)-(10) which are equivalent to equations developed by Hausken (2008a), and equation (11) which is equivalent to the requirement u≥0 and U≥0 provided after equation (17) in Hausken (2008a). The third section quotes Hausken (2008a) once in one sentence which means that section 3 does not belong as a comment on the paper written by Hausken (2008a). The authors are encouraged to develop a new paper based on many interesting ideas in this note. The new paper should develop further the idea of mixed strategies presented in section 3. The new paper may be titled: “Strategic Defense and Attack for Series and Parallel Reliability Systems when Allowing Mixed Strategies”.

Returns to information security investment: The effect of alternative information security breach functions on optimal investment and sensitivity to vulnerability

Four kinds of marginal returns to security investment to protect an information set are decrease, first increase and then decrease (logistic function), increase, and constancy. Gordon, L. A. and Loeb, M. (ACM Trans. Inf. Syst. Secur., 5:438–457, 2002). find for decreasing marginal returns that a firm invests maximum 37% (1 / e) of the expected loss from a security breach, and that protecting moderately rather than extremely vulnerable information sets may be optimal. This article presents classes of all four kinds where the optimal investment is no longer capped at 1 / e. First, investment in information security activities for the logistic function is zero for low vulnerabilities, jumps in a limited “bang-bang” manner to a positive level for intermediate vulnerabilities, and thereafter increases concavely in absolute terms. Second, we present an alternative class with decreasing marginal returns where the investment increases convexly in the vulnerability until a bound is reached, investing most heavily to protect the extremely vulnerable information sets. For the third and fourth kinds the optimal investment is of an all-out “bang-bang” nature, that is, zero for low vulnerabilities, and jumping to maximum investment for intermediate vulnerabilities.

Minmax defense strategy for complex multi-state systems ☆

Abstract This paper presents a general optimization methodology that merges game theory and multi-state system survivability theory. The defender has multiple alternatives of defense strategy that presumes separation and protection of system elements. The attacker also has multiple alternatives of its attack strategy based on a combination of different possible attack actions against different groups of system elements. The defender minimizes, and the attacker maximizes, the expected damage caused by the attack (taking into account the unreliability of system elements and the multi-state nature of complex series–parallel systems). The problem is defined as a two-period minmax non-cooperative game between the defender who moves first and the attacker who moves second. An exhaustive minmax optimization algorithm is presented based on a double-loop genetic algorithm for determining the solution. A universal generating function technique is applied for evaluating the losses caused by system performance reduction. Illustrative examples with solutions are presented.

Protection vs. redundancy in homogeneous parallel systems

The article considers defense resource allocation in a system exposed to external intentional attack. The defender distributes its resource between deploying redundant elements and their protection from attacks. The attacker distributes its effort evenly among all of the elements or among elements from a chosen subset. The vulnerability of each element is determined by an attacker–defender contest success function. The expected damage caused by the attack is evaluated as system unsupplied demand. The article considers both the cases without and with performance redundancy.

False targets efficiency in defense strategy

The paper analyzes the efficiency of deploying false targets as part of a defense strategy. It is assumed that the defender has a single object that can be destroyed by the attacker. The defender distributes its resource between deploying false targets and protecting the object from outside attacks. The attacker cannot distinguish the false targets from the defended object (genuine target). Therefore the attacker has no preferences for attacking one target rather than another target. The defender decides how many false targets to deploy whereas the attacker decides how many targets to attack. The article assumes that both the defender and attacker have complete information and full rationality. The optimal number of false targets and the attacked targets are obtained for the case of fixed and variable resources of the defender and the attacker as solutions of a non-cooperative game between the two agents.

Governments' and Terrorists' Defense and Attack in a T-Period Game

We analyze how a government allocates its resources between attacking to downgrade a terrorists resources and defending against a terrorist attack. Analogously, the terrorist allocates its resources between attacking a governments asset and defending its own resources. A two-stage game is considered where the government moves first and the terrorist moves second. We show that (a) when the terrorists resources are low, the government attacks the terrorists resources sufficiently to deter the terrorist from attacking and does not defend; (b) when the terrorists resources are high, both the government and terrorist defend and attack. We analyze T periods of the two-stage game between two myopic players. First we assume no linkages between periods. We show that after an attack the government may enjoy incoming resources, which deter the terrorist for some periods. Between periods the terrorists resources may change because of arithmetically and geometrically changing incoming funds. We allow the governments and the terrorists resources to be determined randomly in each time period. We also allow the governments resources in one period to depend on the terrorists attacks in earlier time periods for three dynamics, where the terrorists resources are drawn from a normal distribution or change arithmetically or geometrically.

Defense and attack of complex and dependent systems

A framework is constructed for how to analyze the strategic defense of an infrastructure subject to attack by a strategic attacker. Merging operations research, reliability theory, and game theory for optimal analytical impact, the optimization program for the defender and attacker is specified. Targets can be in parallel, series, combined series-parallel, complex, k-out-of-n redundancy, independent, interdependent, and dependent. The defender and attacker determine how much to invest in defending versus attacking each of multiple targets. A target can have economic, human, and symbolic values, subjectively assessed by the defender and attacker. A contest success function determines the probability of a successful attack on each target, dependent on the investments by the defender and attacker into each target, and on characteristics of the contest. The defender minimizes the expected damage plus the defense costs. The attacker maximizes the expected damage minus the attack costs. Each agent is concerned about how his investments vary across the targets, and the impact on his utilities. Interdependent systems are analyzed where the defense and attack on one target impacts all targets. Dependent systems are analyzed applying Markov analysis and repeated games where a successful attack on one target in the first period impacts the unit costs of defense and attack, and the contest intensity, for the other target in the second period.

Protection vs. false targets in series systems

The paper analyses the optimal distribution of the defense resources between protecting the genuine system elements and deploying false elements (targets) in a series system, which is destroyed when any genuine element is destroyed. False and genuine elements cannot be distinguished by the attacker. We analyze a two-period game where the defender builds the defense in the first period, whereas the attacker attacks in the second period. Three cases are considered: the attacker attacks only one element, the attacker attacks all system elements, the attacker chooses the number of elements to attack that maximizes the overall system vulnerability. The probability of element destruction in the case of attack is defined as a contest function depending on the ratio of the defenders and attackers effort and on a contest intensity parameter. The dependence of the minmax defense strategy (number of false elements) and the most harmful attack strategy (number of attacked elements) on the amount of resources available to the counterparts, on the number of genuine system elements and on the contest intensity is analyzed. Illustrative examples are presented.

Strategic defense and attack for reliability systems

This article illustrates a method by which arbitrarily complex series/parallel reliability systems can be analyzed. The method is illustrated with the series–parallel and parallel–series systems. Analytical expressions are determined for the investments and utilities of the defender and the attacker, depend on their unit costs of investment for each component, the contest intensity for each component, and their evaluations of the value of system functionality. For a series–parallel system, infinitely many components in parallel benefit the defender maximally regardless of the finite number of parallel subsystems in series. Conversely, infinitely many components in series benefit the attacker maximally regardless of the finite number of components in parallel in each subsystem. For a parallel–series system, the results are opposite. With equivalent components, equal unit costs for defender and attacker, equal intensity for all components, and equally many components in series and parallel, the defender always prefers the series–parallel system rather than the parallel–series system, and converse holds for the attacker. Hence from the defenders perspective, ceteris paribus, the series–parallel system is more reliable, and has fewer “cut sets†or failure modes.

Protecting complex infrastructures against multiple strategic attackers

Infrastructures are analysed subject to defence by a strategic defender and attack by multiple strategic attackers. A framework is developed where each agent determines how much to invest in defending versus attacking each of multiple targets. A target can have economic, human and symbolic values, which generally vary across agents. Investment expenditure functions for each agent can be linear in the investment effort, concave, convex, logistic, can increase incrementally, or can be subject to budget constraints. Contest success functions (e.g., ratio and difference forms) determine the probability of a successful attack on each target, dependent on the relative investments of the defender and attackers on each target, and on characteristics of the contest. Targets can be in parallel, in series, interlinked, interdependent or independent. The defender minimises the expected damage plus the defence expenditures. Each attacker maximises the expected damage minus the attack expenditures. The number of free choice variables equals the number of agents times the number of targets, or lower if there are budget constraints. Each agent is interested in how his investments vary across the targets, and the impact on his utilities. Alternative optimisation programmes are discussed, together with repeated games, dynamic games and incomplete information. An example is provided for illustration.