  We study mechanisms of synchronisation, coordination, and equilibrium
selection in two-player coordination games on multilayer networks. We apply the
approach from evolutionary game theory with three possible update rules: the
replicator dynamics (RD), the best response (BR), and the unconditional
imitation (UI). Players interact on a two-layer random regular network. The
population on each layer plays a different game, with layer I preferring the
opposite strategy to layer II. We measure the difference between the two games
played on the layers by a difference in payoffs $\Delta S$ while the
inter-connectedness is measured by a node overlap parameter $q$. We discover a
critical value $q_c(\Delta S)$ below which layers do not synchronise. For
$q>q_c$ in general both layers coordinate on the same strategy. Surprisingly,
there is a symmetry breaking in the selection of equilibrium -- for RD and UI
there is a phase where only the payoff-dominant equilibrium is selected. Our
work is an example of previously observed differences between the update rules
on a single network. However, we took a novel approach with the game being
played on two inter-connected layers. As we show, the multilayer structure
enhances the abundance of the Pareto-optimal equilibrium in coordination games
with imitative update rules.
