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June 2011 Connectivity and equilibrium in random games
Constantinos Daskalakis, Alexandros G. Dimakis, Elchanan Mossel
Ann. Appl. Probab. 21(3): 987-1016 (June 2011). DOI: 10.1214/10-AAP715


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 are pure Nash equilibria 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 nonexistence certificates for a large family of graphs. Then we show that in any strongly connected graph of n vertices with expansion (1+Ω(1))log2(n) the distribution of the number of equilibria approaches the Poisson distribution with parameter 1, asymptotically as n→+∞.

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 Erdős–Rényi, 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 pure Nash equilibria as a function of the average degree pn, consistent with the nonmonotone behavior of the model. We show that when the average degree satisfies np>(2+Ω(1))loge(n), the number of pure Nash equilibria follows a Poisson distribution with parameter 1, asymptotically as n→∞. When 1/nnp<(0.5−Ω(1))loge(n), pure Nash equilibria fail to exist with high probability. Finally, when np=O(1/n) a pure Nash equilibrium exists with constant probability.


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Constantinos Daskalakis. Alexandros G. Dimakis. Elchanan Mossel. "Connectivity and equilibrium in random games." Ann. Appl. Probab. 21 (3) 987 - 1016, June 2011.


Published: June 2011
First available in Project Euclid: 2 June 2011

zbMATH: 1229.91079
MathSciNet: MR2830610
Digital Object Identifier: 10.1214/10-AAP715

Primary: 91A
Secondary: 60 , 68Q

Keywords: connectivity , Game theory , graphical games , Phase transitions , random constraint satisfaction problems

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.21 • No. 3 • June 2011
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