## Statistics Papers

#### Document Type

Journal Article

#### Date of this Version

2011

#### Publication Source

The Annals of Applied Probability

#### Volume

21

#### Issue

3

#### Start Page

987

#### Last Page

1016

#### DOI

10.1214/10-AAP715

#### Abstract

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))log_{2}(*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))log_{e}(*n*), the number of pure Nash equilibria follows a Poisson distribution with parameter 1, asymptotically as *n*→∞. When 1/*n*≪*np*<(0.5−Ω(1))log_{e}(*n*), pure Nash equilibria fail to exist with high probability. Finally, when *np*=*O*(1/*n*) a pure Nash equilibrium exists with constant probability.

#### Keywords

game theory, graphical games, connectivity, phase transitions, random constraint satisfaction problems

#### Recommended Citation

Daskalakis, C.,
Dimakis, A. G.,
&
Mossel, E.
(2011).
Connectivity and Equilibrium in Random Games.
*The Annals of Applied Probability,*
*21*
(3),
987-1016.
http://dx.doi.org/10.1214/10-AAP715

**Date Posted:** 27 November 2017

This document has been peer reviewed.