Achieving sustainable cooperation in generalized Prisoner's dilemma with observation errors

A repeated game is a formal model for analyzing cooperation in long-term relationships, e.g., in the prisoner's dilemma. Although the case where each player observes her opponent's action with some observation errors (imperfect private monitoring) is difficult to analyze, a special type of an equilibrium called belief-free equilibrium is identified to make the analysis in private monitoring tractable. However, existing works using a belief-free equilibrium show that cooperative relations can be sustainable only in ideal situations. We deal with a generic problem that can model both the prisoner's dilemma and the team production problem. We examine a situation with an additional action that is dominated by another action. To our surprise, by adding this seemingly irrelevant action, players can achieve sustainable cooperative relations far beyond the ideal situations. More specifically, we identify a class of strategies called one-shot punishment strategy that can constitute a belief-free equilibrium in a wide range of parameters. Moreover, for a two-player case, the obtained welfare matches a theoretical upper bound.