A folk theorem in infinitely repeated prisoner's dilemma with small observation cost

We consider an infinitely repeated prisoner's dilemma under costly observation. If a player observes his opponent, then he pays an observation cost and knows the action chosen by his opponent. If a player does not observe his opponent, he cannot obtain any information about his opponent's action. Furthermore, no player can statistically identify the observational decision of his opponent. We prove an efficiency without any signals. Next, we consider a kind of delayed observations. Players decide their actions and observation decisions in the same period, but they choose observation decisions after they choose their actions. We introduce an interim public randomization instead of public randomization just before observation decision. We present a folk theorem with an interim public randomization device for a sufficiently small observation cost when players are sufficiently patient.