Network Game with Attacker and Protector Entities

Abstract

Consider an information network with harmful procedures called attackers (e.g., viruses); each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is the system protector scanning and cleaning from attackers some part of the network (e.g., an edge or a path), which it chooses independently using another probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the system protector; towards a conflicting objective, the system protector aims at maximizing the expected number of cleaned attackers. We model this network scenario as a non-cooperative strategic game on graphs. We focus on the special case where the protector chooses a single edge. We are interested in the associated Nash equilibria, where no network entity can unilaterally improve its local objective. We obtain the following results: • No instance of the game possesses a pure Nash equilibrium. • Every mixed Nash equilibrium enjoys a graph-theoretic structure, which enables a (typically exponential) algorithm to compute it. • We coin a natural subclass of mixed Nash equilibria, which we call matching Nash equilibria, for this game on graphs. Matching Nash equilibria are defined using structural parameters of graphs, such as independent sets and matchings.

Nonfunctional Requirements Validation Using Nash Equilibria

“…Thus, in order to evaluate network security we evaluate the Nash equilibria of the game of [MPPS05c,MPPS05b]. Indeed they showed a result which is interpreted in our terms as follows: The result combined with equation (1) above implies that the network of Figure 1 has security level equal to 2/n100=2/8100=25, since n=8.…”

“…Recent work by [KO04,ACY05] and [MPPS05b,MPPS05c], initiated the introduction of strategic games on graphs (and the study of their associated Nash equilibria) as a means of studying security problems in networks with selfish entities. By selfish we mean that each entity in the game aims to maximize its utility.…”

“…The ultimate safety of each player may depend in a complex way on the actions of the entire population. [MPPS05b,MPPS05c] considers a security problem on a distributed network modeled as a multi-player non-cooperative game with attackers (e.g., viruses) and a defender (e.g., a security software) entities. More specifically, there are two classes of confronting randomized players on a graph:  attackers, each choosing vertices and wishing to minimize the probability of being caught, and a single defender, who chooses edges and gains the expected number of attackers it kills.…”

“…We model both network and security specifications presented in section 3.1.1. using two graph-theoretic games introduced and investigated in [MPPS05c,MPPS05b,MMPPS06]. The game is played on a graph G representing the network N. The players of the game are of two kinds: the attackers players and the defender players, representing the attacks and the security software of the network.…”

Algorithmic Game Theory and Applications

“…An extensive collection of trade-offs between price of defense and the computational complexity of Nash equilibria is provided in the work of Mavronicolas et al [64]. Most interestingly, the work of Mavronicolas et al [64,[66][67][68] introduce certain natural classes of Nash equilibria for their network security game on graphs, including matching Nash equilibria [67,68] and perfect matching Nash equilibria [64]; they prove that deciding the existence of equilibria from such classes is precisely equivalent to the recognition problem for König-Egervary graphs [25,54]. So, this establishes a very interesting (and perhaps unexpected) link to some classical pearls in graph theory.…”