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.

翻译：我们研究了多层网络上两人协调博弈中的同步、协调与均衡选择机制。采用演化博弈理论的方法，考虑三种可能的更新规则：复制子动力学（RD）、最优反应（BR）和无条件模仿（UI）。参与者在一个两层随机正则网络上进行博弈。每层上的种群参与不同的博弈，第一层倾向于与第二层相反的策略。我们通过收益差异 $\Delta S$ 来度量两层所进行博弈的差异，而层间互联程度则由节点重叠参数 $q$ 衡量。我们发现存在一个临界值 $q_c(\Delta S)$，当低于该值时，各层无法实现同步。当 $q>q_c$ 时，通常两层会在同一策略上达成协调。令人惊讶的是，均衡选择中存在对称性破缺——对于RD和UI，存在一个阶段，仅选择收益占优均衡。我们的工作印证了先前在单一网络上观察到的更新规则之间的差异。然而，我们采用了新颖的方法，使博弈在两层互联网络上进行。正如我们所展示的，多层结构增强了模仿更新规则下协调博弈中帕累托最优均衡的丰富性。