Abstract: Consider a network vulnerable to viral infection. The system security software can guarantee
safety only to a limited part of the network. Such limitations result from economy costs or processing
costs. The problem raised is to which part of the network the security software should
be installed, so that to secure as much as possible the network. We model this practical network
scenario as a non-cooperative multi-player game on a graph, with two kinds of players, a set
of attackers and a protector player, representing the viruses and the system security software,
respectively. Each attacker player chooses a node of the graph (or a set of them, via a probability
distribution) to infect. The protector player chooses independently, in a basic case of the
problem, a simple path or an edge of the graph (or a set of them, via a probability distribution)
and cleans this part of the network from attackers. Each attacker wishes to maximize the probability
of escaping its cleaning by the protector. In contrast, the protector aims at maximizing
the expected number of cleaned attackers. We call the two games obtained from the two basic
cases considered, as the Path and the Edge model, respectively. For these two games, we are
interested in the associated Nash equilibria, where no network entity can unilaterally improve
its local objective. We obtain the following results:
• The problem of existence of a pure Nash equilibrium is NP-complete for the Path model.
This opposed to that, no instance of the Edge model possesses a pure Nash equilibrium,
proved in [7].
• In [7] a characterization of mixed Nash equilibria for the Edge model is provided. However,
that characterization only implies an exponential time algorithm for the general case.
Here, combining it with clever exploration of properties of various practical families of
graphs, we compute, in polynomial time, mixed Nash equilibria on corresponding graph
instances. These graph families include, regular graphs, graphs that can be decomposed, in
polynomially time, into vertex disjoint r-regular subgraphs, graphs with perfect matchings
and trees.
• We utilize the notion of social cost [6] for measuring system performance on such scenario;
here is defined to be the utility of the protector. We prove that the corresponding Price of
Anarchy in any mixed Nash equilibria of the game is upper and lower bounded by a linear
function of the number of vertices of the graph.

Abstract: Pervasive games are a new type of digital games that combines game and physical reality within the gameplay. This novel game type raises unprecedented research and design challenges for developers and urges the exploration of new technologies and methods to create high quality game experiences and design novel and compelling forms of content for the players. This chapter follows a systematic approach to explore the landscape of pervasive gaming. First, the authors approach pervasive games from a theoretical point of view, defining the four axes of pervasive games design, introducing the concept of game world persistency, and describing aspects of spatially/temporally/socially expanded games. Then, they present ten pervasive game projects, classified in five genres based on their playing environment and features. Following that, the authors present a comparative view of those projects with respect to several design aspects: communication and localization, context and personal awareness aspects, information model, player equipment, and game space visualization. Last, the authors highlight current trends, design principles, and future directions for pervasive games development.