  We study the role of local effects and finite size effects in reaching
coordination and in equilibrium selection in different types of two-player
coordination games. We investigate three update rules -- the replicator
dynamics (RD), the best response (BR), and the unconditional imitation (UI) --
for coordination games on random graphs. Local effects turn out to me
significantly more important for the UI update rule. For the pure coordination
game with two equivalent strategies we find a transition from a disordered
state to a state of full coordination for a critical value of the network
connectivity. The transition is system-size-independent for the BR and RD
update rules. For the IU update rule it is system size dependent, but
coordination can always be reached below the connectivity of a complete graph.
We also consider the general coordination game which covers a range of games,
such as the stag hunt. For these games there is a payoff-dominant strategy and
a risk-dominant strategy with associated states of equilibrium coordination. We
analyse equilibrium selection analytically and numerically. For the RD and BR
update rules mean-field predictions agree with simulations and the
risk-dominant strategy is evolutionary favoured independently of local effects.
When players use the unconditional imitation, however, we observe coordination
in the payoff-dominant strategy. Surprisingly, the selection of pay-off
dominant equilibrium only occurs below a critical value of the network
connectivity and it disappears in complete graphs. As we show, it is a
combination of local effects and update rule that allows for coordination on
the payoff-dominant strategy.
