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Individual versus overarching protection and attack of assets

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Abstract

Two agents protect and attack a collection of assets overarchingly versus individually. Examples of overarching protection are border security, counter intelligence, and public health measures. Both layers of protection have to be breached for an attack to be successful. We consider a simultaneous game, and a two period game with overarching contest in period 1 and individual contests in period 2 if the attacker wins period 1. With reasonable assumptions, such as contest intensities not exceeding one, the defender prefers two protection layers, while the attacker prefers one protection layer. When the unit effort costs of overarching protection and attack are equal, and the agents’ valuations for each asset are equal, in the simultaneous game defender and attacker efforts are equal in the overarching contest. In contrast, for the two period game, the defender invests more than the attacker in the overarching contest to prevent the occurrence of period 2. If the attacker nevertheless wins period 1, both agents exert larger efforts in period 2 compared with the individual contests in the simultaneous game. Framed within the Colonel Blotto literature, the attacker must win the first battlefield (overarching contest) in order to engage in the contests over the n other battlefields (individual contests).

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Notes

  1. Both simultaneous and sequential games are common in the literature. One example where a sequential game is often argued to be realistic is when a defender has built a substantial and virtually impenetrable and insurmountable defense wall in period 1 which the attacker can observe and take as given when choosing his effort in period 2. Even for this example, a defender building such a wall usually or always does so anticipating and sometimes assuming a certain attack capacity by the attacker. No defender designs a defense in a vacuum not assuming anything about possible attacks. For example, the solidity and thickness of the wall is usually designed to withstand certain modes of attack, and not to withstand other modes of attack. Assuming that a defender knows nothing about an attack, while the attacker knows everything about a defense, is questionable. For defenses that are more fluid and movable than defense walls, for example security personnel, detectors, surveillance equipment, such defenses have the capacity to adapt themselves to expected modes of behavior by the attacker. In this paper we assume that the defender and attacker are on an equal footing with respect to observing versus not observing characteristics about each other ahead of time. The attacker cannot know the actual defense effort before testing it, and the defender cannot know the attack effort before experiencing it. Future research may analyze sequential games where either the defender or attacker moves first. Solving sequential games with backward induction requires setting the contest intensities to \(\text{ m}=\text{ m}_\mathrm{i}=1\).

  2. In this paper we do not consider responsive overarching protection which requires analyzing a sequential game where the attacker moves first and the defender moves second as a response to the attack.

Abbreviations

\(n\) :

Number of system assets

\(t_{i}\) :

Defender’s individual protection effort for asset i

\(T_{i}\) :

Attacker’s individual attack effort for asset i

\(t\) :

Defender’s overarching protection effort

\(T\) :

Attacker’s overarching attack effort

\(s_{i }\) :

Defender’s valuation of asset i, \(s_{i}\,{\ge }\,0\)

\(S_{i }\) :

Attacker’s valuation of asset i, \(S_{i}\,{\ge }\,0\)

\(c_{i }\) :

Defender’s unit cost of protecting asset i

\(C_{i }\) :

Attacker’s unit cost of attacking asset i

\(m_{i }\) :

Contest intensity for asset i

\(Q_i =\frac{C_i /S_i }{c_i /s_i }\) :

Unit cost over asset valuation ratio for asset i

\(c\) :

Defender’s unit cost for overarching protection

\(C\) :

Attacker’s unit cost for overarching attack

\(Q=\frac{C/{\sum _{i=1}^n {\frac{S_i }{1+Q_i ^{m_i }}}} }{c/{\sum _{i=1}^n {\frac{s_i }{1+Q_i ^{m_i }}}}}\) :

Adjusted unit cost ratio for overarching attack and protection

\(m\) :

Contest intensity for overarching protection and attack

\(V_{i }\) :

Attack success probability for asset i due to individual protection and attack

\(d_{i }\) :

Defender’s expected damage for asset i

\(D_{i }\) :

Attacker’s expected damage for asset i

\(V\) :

Attack success probability for the collection of assets due to overarching protection and attack

\(d\) :

Defender’s expected damage for the collection of assets

\(D\) :

Attacker’s expected damage the collection of assets

\(r\) :

Defender’s expenditure ratio

\(R\) :

Attacker’s expenditure ratio

\(u\) :

Defender’s utility

\(U\) :

Attacker’s utility

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  1. Faculty of Social Sciences, University of Stavanger, 4036 , Stavanger, Norway

    Kjell Hausken

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Hausken, K. Individual versus overarching protection and attack of assets. Cent Eur J Oper Res 22, 89–112 (2014). https://doi.org/10.1007/s10100-012-0271-6

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