Coordination and equilibrium selection in games: the role of local effects

Abstract: 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.