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Abstract
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.
Acknowledgement
I thank two anonymous referees of this journal and Vicki M. Bier for useful comments.
Notes
Notes
1. See http://www.state.gov/s/ct/rls/other/des/123085.htm and http://security.homeoffice.gov.uk/legislation/current-legislation/terrorism-act-2000/proscribed-groups#, retrieved August 6, 2009.
2. In particular, Bier (private communication) observes that simple convexity of the component failure probabilities is not sufficient to yield convexity of their product; one needs log-convexity – which implies, roughly speaking, that the success probability of an attack against any given component decreases faster than exponentially in the level of investment.
3. This superseded earlier infinite recursions of the kind ‘If I think that you think that I think …’
4. An alternative is = v
i
+ ϵ
i
, which is the defender's valuation plus an error term for target i. The error term can reflect attacker lack of information about the defender's valuations, or attacker-specific goals such as prominence of target i, or the cost of attacking target i. I thank Vicky Bier for this suggestion.
5. Bier et al. (Citation2006, p. 316) define the probability of success of an attack on a component as a function of the investment by the defender to strengthen that component, where the probability of attack on the system is exogenously given.
6. The decisiveness m i is a characteristic of the contest. It can be well illustrated by the history of warfare. Low decisiveness occurs for systems that are defendable, predictable, and where the individual components are dispersed, that is, physically distant or separated by barriers of various kinds. Neither the defender nor the attacker can get a significant upper hand. An example is the time prior to the emergence of cannons and modern fortifications in the fifteenth century. Another example is entrenchment combined with the machine gun, in multiply dispersed locations, in World War I. High decisiveness occurs for systems that are less predictable, easier to attack, and where the individual components are concentrated, that is, close to each other or not separated by particular barriers. This may cause ‘winner-take-all’ battles and dictatorship by the strongest. Either the defender or the attacker may get the upper hand. The combination of airplanes, tanks, and mechanised infantry in World War II allowed the offence to concentrate firepower more rapidly than the defence, which intensified the effect of force superiority (Hirshleifer Citation1995, pp. 32–33).
7. In the conflict literature, this is referred to as egalitarian distribution of an asset independent of effort (investment), so that each agent receives 50%. In our context m = 0 gives a certain ‘egalitarianism’ between the defender and the attacker in the sense that the defender obtains half as much reliability as he maximally hopes for. We ignore m < 0 which corresponds in one sense to altruism and in another sense to punishing individual investments and placing a premium on laziness.
8. Hirshleifer (Citation1989, p. 104) argues that ‘in a military context we might expect the ratio form of the Contest Success Function to be applicable when clashes take place under close to ‘idealized’ conditions such as: an undifferentiated battlefield, full information, and unflagging weapons effectiveness. In contrast, the difference form tends to apply where there are sanctuaries and refuges, where information is imperfect, and where the victorious player is subject to fatigue and distraction.’ Hence, applying the difference form, in struggles between nations, one side may surrender rather than resist against an unappeasable opponent, with the expectation of not losing everything, realising the cost to the victor of locating and extracting all the spoils.
9. For the parallel system in Section 7.1, one alternative is to let the attacker equalise the vulnerabilities of the targets as perceived by the defender.
10. Conventional reliability theory distinguishes between independent and dependent systems. Ebeling (Citation1997, 108ff) describes dependent systems as systems where ‘component failures are in some way dependent’. Markov analysis is typically applied. Aside from degraded systems, examples are load-sharing systems and standby systems where the breakdown of one component affects the other components.
11. I thank Vicky Bier for suggesting multi-use systems.