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Choosing what to protect when attacker resources and asset valuations are uncertain

Hausken K.

Operations Research and Decisions

2014

Vol. 24, No. 3

23--44

The situation has been modelled where the attacker’s resources are unknown to the defender. Protecting assets presupposes that the defender has some information on the attacker’s resource capabilities. An attacker targets one of two assets. The attacker’s resources and valuations of these assets are drawn probabilistically. We specify when the isoutility curves are upward sloping (the defender prefers to invest less in defense, thus leading to higher probabilities of success for attacks on both assets) or downward sloping (e.g. when one asset has a low value or high unit defense cost). This stands in contrast to earlier research and results from the uncertainty regarding the level of the attacker’s resources. We determine which asset the attacker targets depending on his type, unit attack costs, the contest intensity, and investment in defense. A two stage game is considered, where the defender moves first and the attacker moves second. When both assets are equivalent and are treated equivalently by both players, an interior equilibrium exists when the contest intensity is low, and a corner equilibrium with no defense exists when the contest intensity is large and the attacker holds large resources. Defense efforts are inverse U shaped in the attacker’s resources.

Distributive atomic efect algebras

Riečanová Z.

Demonstratio Mathematica

2003

Vol. 36, nr 2

247--259

Motivated by the use of fuzzy or unsharp quantum logics as carriers of probability measures there have been recently introduced effect algebras (D-posets). We extend a result by Greechie, Foulis and Pulmannova of finite distributive effect algebras to all Archimedean atomic distributive effect algebras. We show that every such an effect algebra is join and meet dense in a complete effect algebra being a direct product of finite chains and distributive diamonds. This proves that every such effect algebra has a MacNeille completion being again a distributive effect algebra and both these effect algebras are continuous lattices. Moreover, we show that every faithful or (o)-continuous state (probability) on such an effect algebra is a valuation, hence a subadditive state. Its existence is also proved. Finally, we prove that every complete atomic distributive effect algebra E is a homomorphic image of a complete modular atomic ortholattice regarded as effect algebra and E is an MV-effect algebra (MV-algebra) if and only if it is a homomorphic image of a Boolean algebra regarded as effect algebra.