Before revision:  Moreover, we establish a new link between percolation models and game theory to shed light on various aspects of Nash equilibria. Through this connection , we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows.
After revision:  Moreover, we establish a new link between percolation models and game theory , we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows.