Communication in repeated network games with imperfect monitoring

•I consider repeated games on a network with private monitoring and local interaction.•A player's payoff depends on his own and his neighbors' actions only.•Monitoring is private: each player observes his stage payoff only.•Players can communicate costlessly at each stage, publicly or privately.•A folk theorem holds if and only if no two players have the same neighbors.

I consider repeated games with private monitoring played on a network. Each player has a set of neighbors with whom he interacts: a player's payoff depends on his own and his neighbors' actions only. Monitoring is private and imperfect: each player observes his stage payoff but not the actions of his neighbors. Players can communicate costlessly at each stage: communication can be public, private or a mixture of both. Payoffs are assumed to be sensitive to unilateral deviations. First, for any network, a folk theorem holds if some Joint Pairwise Identifiability condition regarding payoff functions is satisfied. Second, a necessary and sufficient condition on the network topology for a folk theorem to hold for all payoff functions is that no two players have the same set of neighbors not counting each other.