Connectivity and Equilibrium in Random Games

We study how the structure of the interaction graph of a game affects the existence of pure Nash equilibria. In particular, for a fixed interaction graph, we are interested in whether there exist pure Nash equilibria which arise when random utility tables are assigned to the players. We provide conditions for the structure of the graph under which equilibria are likely to exist and complementary conditions which make the existence of equilibria highly unlikely. Our results have immediate implications for many deterministic graphs and generalize known results for random games on the complete graph. In particular, our results imply that the probability that bounded degree graphs have pure Nash equilibria is exponentially small in the size of the graph and yield a simple algorithm that finds small non-existence certificates for a large family of graphs. We then show that as n → ∞, any graph on n vertices with expansion (1 + Ω(1)) logn will have the number of equilbria distributed as a Poisson random variable with parameter 1. In order to obtain a refined characterization of the degree of connectivity associated with the existence of equilibria, we study the model in the random graph setting. In particular, we look at the case where the interaction graph is drawn from the Erdős-Rényi, G(n, p), where each edge is present independently with probability p. For this model we establish a double phase transition for the existence of pure Nash equilibria as a function of the average degree pn consistent with the non-monotone behavior of the model. We show that when the average degree satisfies np > (2+Ω(1)) log n, the number of pure Nash equilibria follows a Poisson distribution with parameter 1. When 1/n << np < (0.5−Ω(1)) logn pure Nash equilibria fail to exist with high probability. Finally, when np << 1/n a pure Nash equilibrium exists with high probability.