  We consider evolutionary games on a population whose underlying topology of
interactions is determined by a binomial random graph $G(n,p)$. Our focus is on
2-player symmetric games with 2 strategies played between the incident members
of such a population. Players update their strategies synchronously. At each
round, each player selects the strategy that is the best response to the
current set of strategies its neighbours play. We show that such a system
reduces to generalised majority and minority dynamics. We show rapid
convergence to unanimity for $p$ in a range that depends on a certain
characteristic of the payoff matrix. In the presence of a bias among the pure
Nash equilibria of the game, we determine a sharp threshold on $p$ above which
the largest connected component reaches unanimity with high probability. For
$p$ below this critical value, where this does not happen, we identify those
substructures inside the largest component that remain discordant throughout
the evolution of the system.
