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We treat PNE-GG, the problem of deciding the existence of a Pure Nash Equilibrium in a graphical game, and the role of treewidth in this problem. PNE-GG is known to be N P-complete in general, but polynomially solvable for graphical games of bounded treewidth. We prove that PNE-GG is W [1]-Hard when parameterized by treewidth. On the other hand, we give a dynamic programming approach that solves the problem in O * (α w ) time, where α is the cardinality of the largest strategy set and w is the treewidth of the input graph (and O * hides polynomial factors). This proves that PNE-GG is in F PT for the combined parameter (α, w). Moreover, we prove that there is no algorithm that solves PNE-GG in O * ((α − ) w ) time for any > 0, unless the Strong Exponential Time Hypothesis fails. Our lower bounds implicitly assume that α ≥ 3; we show that for α = 2 the problem can be solved in polynomial time. Finally, we discuss the implication for computing pure Nash equilibria in graphical games (PNE-GG) of O(log n) treewidth, the existence of polynomial kernels for PNE-GG parameterized by treewidth, and the construction of a sample and maximum-payoff pure Nash equilibrium.

The concept of social networks in conjunction with concepts from economics has attracted considerable attention in recent years. In this paper we propose the Stochastic Diffusion Market Search (SDMS), a novel contextual advertising method for mutual advertisement hosting among participating entities, where each owns a web site. In the scenario considered each participating agent/web-site buys or sells advertising links. In the proposed method the advertising market and network that formed into the system emerge from agents preferences and their social behavior into the network. SDMS consists of a variation of Stochastic Diffusion Search, a swarm intelligence metaheuristic, and an algorithm for market equilibria. We present an evaluation of the model and the experimental results show that the network potentially converges to a stable stage and the distribution of market prices adheres to power law properties. -1-4503-0403-0…$5.00.J to a divisible good. The quantity of J is the number of impression f t J that is able to sell in period t. We define thatare the visitors at J's web site in period t−1. In order to initialize the quantities f t=1 J , we apply an period t = 0 where the Center computes the number of visitors ν t=0 J

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