We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.

A Random Geometric Graph (RGG) in two dimensions is constructed by distributing n nodes independently and uniformly at random in and creating edges between every pair of nodes having Euclidean distance at most r, for some prescribed r. We analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that with probability 1 – (n−1) this algorithm informs every node in the largest connected component of an RGG within rounds. This holds for any value of r larger than the critical value for the emergence of a connected component with Ω(n) nodes. In order to prove this result, we show that for any two nodes sufficiently distant from each other in , the length of the shortest path between them in the RGG, when such a path exists, is only a constant factor larger than the optimum. This result has independent interest and, in particular, gives that the diameter of the largest connected component of an RGG is , which surprisingly has been an open problem so far.

We study the relationship between the social cost of correlated equilibria and the social cost of Nash equilibria. In contrast to previous work focusing on the possible benefits of a benevolent mediator, we define and bound the Price of Mediation (PoM): the ratio of the cost of the worst correlated equilibrium to the cost of the worst Nash. We observe that in practice, the heuristics used for mediation are frequently non-optimal, and from an economic perspective mediators may be inept or self-interested. Recent results on computation of equilibria also motivate our work.