Supermodular network games

We study supermodular games on graphs as a benchmark model of cooperation in networked systems. In our model, each agent's payoff is a function of the aggregate action of its neighbors and it exhibits strategic complementarity. We study the largest Nash equilibrium which, in turn, is the Pareto optimal equilibrium in the presence of positive externalities. We show that the action of a node in the largest NE depends on its centrality in the network. In particular, the action of nodes that are in the k-core of the graph is lower bounded by a threshold that is nondecreasing in k. The main insight of the result is that the degree of a node may not be the right indicator of the strength and influence of a node in the equilibrium. We also consider Bayesian supermodular games on networks, in which each node knows only its own degree. In this setting, we show that the largest symmetric Bayesian equilibrium is monotone in the edge perspective degree distribution.