Bounding payoffs in repeated games with private monitoring: n-player games

We provide a simple upper bound on the Nash equilibrium payoff set at a fixed discount factor in repeated games with imperfect private monitoring. The bound admits a tractable recursive characterization and can thus be applied “off-the-shelf” to any repeated game. The bound is not tight in general, but it is tight if the stage game is “concave” or if a certain form of observable mixed actions is allowed. We illustrate our results with applications to the repeated prisoners' dilemma and to canonical public goods and oligopoly games.