A Network Game with Attacker and Protector Entities

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. – We derive a characterization of graphs possessing matching Nash equilibria. The characterization enables a linear time algorithm to compute a matching Nash equilibrium on any such graph with a given independent set and vertex cover. – Bipartite graphs are shown to satisfy the characterization. So, using a polynomialtime algorithm to compute a perfect matching in a bipartite graph, we obtain, as our main result, an efficient graph-theoretic algorithm to compute a matching Nash equilibrium on any instance of the game with a bipartite graph. Throughout the paper, some missing proofs can be found in the attached Appendix. It may be read at the discretion of the Program Committee. ∗This work was partially supported by the IST Programs of the European Union under contract numbers IST2001-33116 (FLAGS) and IST-2004-001907 (DELIS). †Department of Computer Science, University of Cyprus, P.O. Box 20537, Nicosia CY-1678, Cyprus. Email: {mavronic,viki,annap}@ucy.ac.cy ‡Department of Computer Engineering and Informatics, University of Patras, Rion, 26500 Patras, Greece. Email: [email protected]