*There has been no systematic attempt to study the maximal degree of collusion sustainable by credible threats for arbitrary values of the discount factor. In view of the motivation for moving from static to repeated models, this has some claims to being the essential question at issue.*

As argued by

A particular class of paths called two-phase punishments plays a central role. A two-phase punishment is symmetric; in addition it is stationary after the first period, i.e., in the second phase. I show that the optimal two-phase punishment is:

(1) Globally optimal for a certain range of parameter values.

(2) An optimal symmetric punishment.

(3) More severe than Cournot-Nash reversion.

(4) Easily calculated. (It is completely characterized by a pair of simultaneous equations.)

(5) The second phase of the optimal two-phase punishment is the most collusive symmetric output level which can be sustained by the optimal two-phase punishment itself.

...

(5) says that the optimal two-phase punishment consists of a

Optimal asymmetric punishments have a rather complicated structure and thus far elude description as complete as that provided for optimal symmetric punishments in the earlier section.

As argued by

**Abreu (1983, Ph.D thesis)**, the fundamental determinant of the limits of collusion is the severity of punishments with which potential deviants from cooperative behavior can credibly be threatened. Accordingly, this paper concentrates principally on characterizing strategy profiles which yield optimal (in the sense of most severe) punishments.A particular class of paths called two-phase punishments plays a central role. A two-phase punishment is symmetric; in addition it is stationary after the first period, i.e., in the second phase. I show that the optimal two-phase punishment is:

(1) Globally optimal for a certain range of parameter values.

(2) An optimal symmetric punishment.

(3) More severe than Cournot-Nash reversion.

(4) Easily calculated. (It is completely characterized by a pair of simultaneous equations.)

(5) The second phase of the optimal two-phase punishment is the most collusive symmetric output level which can be sustained by the optimal two-phase punishment itself.

...

(5) says that the optimal two-phase punishment consists of a

**stick and carrot**; furthermore, the carrot phase is the most attractive collusive regime which can credibly be offered when optimal two-phase punishments are used to deter defections. Thus when we solve for optimal two-phase punishments, we simultaneously determine the maximal degree of collusion sustainable by optimal symmetric punishments. It is worth remarking that the stick-and-carrot property is not a curiosum, but arises naturally from the structure of the problem.Optimal asymmetric punishments have a rather complicated structure and thus far elude description as complete as that provided for optimal symmetric punishments in the earlier section.

(All quoted from

**Abreu (1986)**)

I would like to say something about asymmetric cases below.

For delta large enough, optimal symmetric punishments yield 0 payoff hence they are the most severe punishment (notice that a firm's minmax payoff in the component game is zero). And in this case, all firms simultaneously minmax one another in the first phase of the punishment.

However, if an optimal symmetric punishment yields firms positive payoffs (more than their minmax payoffs), then it is not globally optimal.

Abreu gives characterizations of optimal asymmetric punishments. But they are complicated and much less sharper than those with symmetric cases.

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