Mathematics – Probability
Scientific paper
2007-03-29
Annals of Applied Probability, 2011
Mathematics
Probability
Scientific paper
We study how the structure of the interaction graph of a game affects the existence of pure Nash equilibria (PNE). For a fixed interaction graph, we are interested in whether there are PNE arising 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. Then we show that in any strongly connected graph of $n$ vertices with expansion (1+Omega(1)) log_2 n the distribution of the number of equilibria approaches the Poisson distribution with parameter 1, asymptotically as n -> infty. In order to obtain a refined characterization of the degree of connectivity associated with the existence of equilibria, we also study the model in the random graph setting. In particular, we look at the case where the interaction graph is drawn from the Erdos-Renyi, G(n,p), model where each edge is present independently with probability p. For this model we establish a double phase transition for the existence of PNE as a function of the average degree p n, consistent with the non-monotone behavior of the model. We show that when the average degree satisfies n p > (2 + Omega(1)) log_e n, the number of PNE follows a Poisson distribution with parameter 1, asymptotically as n -> infty. When 1/n << n p < (0.5 -Omega(1)) log_e n, PNE fail to exist with high probability. Finally, when n p = O(1/n) a PNE exists with constant probability.
Daskalakis Constantinos
Dimakis Alexandros G.
Mossel Elchanan
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