We consider repeated games with private monitoring that are "close" to repeated games with public/perfect monitoring. A private monitoring information structure is close to a public monitoring information structure when private signals can generate approximately the same distribution of the public signal once they are aggregated into a public signal by some public coordination device. A player's informational size associated with the public coordination device is the key to inducing truth-telling in nearby private monitoring games when communication is possible. A player is informationally small given a public coordination device if she believes that her signal is likely to have a small impact on the public signal generated by the public coordinating device. We show that a uniformly strict equilibrium with public monitoring is robust in a certain sense: it remains an equilibrium in nearby private monitoring repeated games when the associated public coordination device, which makes private monitoring close to public monitoring, keeps every player informationally small at the same time.

We also prove a new folk theorem for repeated games with private monitoring and communication by exploiting the connection between public monitoring games and private monitoring games via public coordination devices.