Remarks on APS by Tirole (1988)

Original article (link) posted: 04/08/2005

Tirole (1988) mentions APS in supplementary section. His description about optimal collusions is very clear, so I will quote it here.

Abreu, Pearce and Stacchetti (1986, 1990) show that one can indeed restrict attention to a collusive phase and a punishment phase, characterized by payoffs V+ and V-, where V+ and V- are now the best and worst elements in the set of symmetric perfect equilibrium payoffs. Furthermore, the collusive phase and the punishment phase take simple forms. In the collusive phase, the firms produce output q+. The punishment phase is triggered by a tail test, i.e., it starts if the market price falls under some threshold level p+. Thus, the collusive phase is qualitatively similar to that presumed in Porter (1983) and Green and Porter (1984). The punishment phase, however, does not have a fixed length; rather, it resembles the collusive phase. The two firms produce (presumably high) output q- each. If the market price exceeds a threshold price p-, the game remains in the punishment phase; if it lies below p-, the game goes back to the collusive phase. Thus, the evolution between the two phases follows a Markovian process. The reader may be surprised by the "inverse tail test" in the punishment phase. The idea is that a harsh punishment requires a high output (higher than is even privately desirable); to ensure that the firms produce a high output, it is specified that in the case of a high price (which signals a low output) the game remains in the punishment phase. (Notice that if one restricted punishments to be of the Cournot type, the optimal length of punishment would be T = "infinity", from the APS result on the harshest possible punishment V-.)
(Tirole (1988), p.265)