Abstract
The following analysis demonstrates that network efficiency is a very delicate matter. Neither the restriction to suitable core-subnets, nor intelligent enlargements guarantee efficient results in any case. For instance, the numerical example in Myerson (1977) contains a prisoners’ dilemma situation for some agents in the network. Of course, the outcome is inefficient from the perspective of these players – but not for the unrestricted network. Breass’ paradox shows that the enlargement of a network can lead to an inefficient outcome in the Nash equilibrium even if all players are taken into consideration. Restricting the network can create a Pareto efficient outcome. A third model discusses the strategic problem of a cyber network attack in the form of an inspection game. From the defender’s point of view, the question arises which nodes of the network are essential attack targets and thus need special security attention. In principle two types of nodes are critical: important ones and unimportant ones. Important nodes, as they connect to many other essential nodes and are therefore suitable multipliers for network malware and information capture, and unimportant nodes, from the attacker’s point of view, which are, in general, not in the focus of security attention, such that infiltration via them may be undetected for a long time.
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- 1.
For a power index analysis of networks, i.e., based on cooperative game theory, see Holler and Rupp (2019).
- 2.
Take, for instance, Y({1:2, 1:3, 2:3}) = (5, 5, 2) = (x, y, z). According to 1:2: x − 3 = y − 3. According to 1:3: x − 6 = z − 3. According to 2:3: y − 6 = z − 3. These equations determine (5, 5, 2).
- 3.
In the original story of the prisoners’ dilemma (see, e.g., Luce and Raiffa 1957: 93), if we take the district attorney’s preferences into consideration then the outcome is no longer inefficient – however, then the game is no longer a prisoners’ dilemma. See Holler and Klose-Ullmann (2020: 49ff) for a discussion.
- 4.
Of course. if the interaction of players 1 and 2 is with “unforeseeable end” (and the discounting of future payoffs is moderate) then we have no longer a prisoners’ dilemma situation.
- 5.
The following discussion of the Braess paradox is based on Stefan Napel’s elaboration of the paradox in Holler et al. (2019) – also to simplify the notation and the calculations.
- 6.
This assumption is to simplify the analysis. In Braess (1968) all costs vary with the intensity of traffic.
- 7.
Note for an equilibrium we have to specify the strategies (i.e., the path) that individual drivers will choose. In the minimum case of only four cars there are already six equilibria that are consistent with the (½, ½) on AB and AC, alternatively.
- 8.
This concept was introduced in Selten (1975).
- 9.
Nagurney and Boyce (2005: 443) introduce user optimization and system optimization in this context.
- 10.
A prisoners’ dilemma is characterized by strictly dominant strategies: players choose dominant strategy, irrespective of the choices they observe or they expected of the other players.
- 11.
BloodHound uses graph theory to reveal the hidden and often unintended relationships within an Active Directory environment. Attacks can use BloodHound to easily identify highly complex attack paths that would otherwise be impossible to quickly identify. Defenders can use BloodHound to identify and eliminate those same attack paths. Both blue and red teams can use BloodHound to easily gain a deeper understanding of privilege relationships in an Active Directory environment, see https://github.com/BloodHoundAD/Bloodhound/wiki.
- 12.
Nash’s proof holds for finite games: a game is finite if each player has only a finite number of pure strategies.
- 13.
It follows immediately that a pure strategy may be considered a special form of a mixed strategy to be played with probability 1.
- 14.
For further details, see Holler (1990).
- 15.
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Holler, M.J., Rupp, F. (2020). Some Strategic Decision Problems in Networks. In: Nguyen, N.T., Kowalczyk, R., Mercik, J., Motylska-Kuźma, A. (eds) Transactions on Computational Collective Intelligence XXXV. Lecture Notes in Computer Science(), vol 12330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-62245-2_9
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