Finitely Repeated Games with Automatic and Optional Monitoring

We study a new class of finitely repeated games with optional monitoring, where each player automatically receives complete information about the other players’ actions with some exogenously given probability. Only when the automatic information did not arrive, the player privately decides whether to exercise a costless monitoring option or not. We show that a weak decrease in the vector of the players’ probabilities of automatic monitoring is a necessary and sufficient condition for any repeated game with automatic and optional monitoring to have a weakly greater sequential equilibrium payoff vector set. This result considerably strengthens our earlier result, which only compares purely optional monitoring and the standard model of purely automatic monitoring. We also provide examples where existing folk theorems hold under any automatic and optional monitoring structure but not under the standard model.