Original article (link) posted: 03/08/2005
In many economic examples of practical interest, the assumption that players observe one another's past action is inappropriate.
Green and Porter (1984) and Porter (1983) were the first papers to study discounted repeated games in which players receive information related only stochastically to others actions.
For Economists, one of the most attractive features of the model is that it escapes the prediction of dynamically uniform behavior on the most collusive equilibrium path, thereby offering a possible interpretation of observed phenomena such as price wars.
Porter investigated symmetric equilibria that are optimally collusive among a restricted set of "trigger strategy" profiles. (A price realization of less than "p" triggers a T-period phase of Cournot-Nash behavior, after which cooperation resumes until the Cournot phase gets triggered again.)
Porter found that it is often optimal to set T="infinity", that is, revert permanently to the stage game Cournot-Nash equilibrium.
Abreu, Pearce and Stacchetti (1986) dropped the restriction to trigger strategy profiles, and characterized optimal pure strategy summetric equilibria of a class of games that generalize the Green-Porter model. They found that a constrained efficient solution is described by two "acceptance regions" in the signal space and two actions. (In the efficient equilibrium, players choose the strategy1 as long as the value of the signal falls in the region1. Otherwise, they switch to the strategy2, and keep playing that strategy as long as the signal falls in the region2.)
Using a larger region and a less severe punishment will generally result in a loss of efficiency because of the region's poorer ability to discriminate between good and bad behavior. Thus, after one period of the best equilibrium, player will be instructed either to begin the worst equilibrium or to restart the best equilibrium.
Notice that, in every contingency, players are duplicating the behavior of the first period of one of two equilibria (the best or the worst), so only two quantities are ever produced.
(Pearce (1992), p.150-2)
Abreu, Pearce and Stacchetti (1986) "Optimal Cartel Equilibria with Imperfect Monitoring" JET, 39
Green and Porter (1984) "Noncooperative Collusion under Imperfect Price Information" Econometrica, 52
Porter (1983) "Optimal Cartel Trigger Price Strategies" JET, 29