An efficiency result in a repeated prisoner’s dilemma game under costly observation with nonpublic randomization

We consider an infinitely repeated prisoner’s dilemma game with costly observation, where each player chooses a pairing of an action and an observational decision. If the player observes the opponent, the player pays an observation cost and observes the action just played by the opponent. Otherwise, the player cannot obtain any information about the action chosen by the opponent. We then introduce a correlated signal at the beginning of each stage game (nonpublic randomization) and prove an efficiency result without any implicit communication on the condition that the discount factor is sufficiently close to one and the observation cost is sufficiently small. We find that our results hold under both an arbitrary strongly or weakly correlated signal.