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Members of a cartel cannot see each other’s output. They see only the market price, and the price also moves with demand nobody controls. A low price might mean someone cheated, or it might just mean a slow quarter. How do you police a deal you cannot observe?

Repeated games with imperfect monitoring

Grim trigger assumes you can tell when someone cheated. Often you cannot: you see a noisy signal that reflects both the other player’s action and pure chance. Then punishment can no longer be reserved for real defections, and keeping cooperation alive means accepting that innocent players sometimes get punished too.

From earlier lessons

Two quick ones before we start.

Grim trigger punishes a defection by...

Cooperation in the repeated PD needs players to be...

Predict, then play

Both players intend to cooperate every round. But the only thing observed is a public signal, and noise can make it look bad even when nobody cheated. Set the noise and the punishment length, then run it.

Why did that happen?

Take a guess, then reveal the reason

With noise, a bad signal does not prove anyone cheated. But if you never punish a bad signal, cheating becomes free, so some punishment must follow bad signals regardless of the cause. That means wars will sometimes strike honest players. Grim trigger takes this to the extreme: the first unlucky signal ends everything. The fix is to make punishment finite. You accept that occasional wars are the price of policing a deal you cannot see, and you forgive so cooperation can resume.

The math

Suppose the public signal reads “bad” with probability $q$ after cooperation and probability $d>q$ after a deviation. The gap $d-q$ measures how informative the signal is about cheating. Under grim trigger (punish forever after one bad signal), the relationship survives only until the first false alarm. Since a false alarm arrives with probability $q$ each round, it is essentially certain eventually, so cooperation collapses to zero.

A trigger-and-forgive strategy punishes for $T$ rounds after a bad signal, then resets. The long-run fraction of time spent cooperating is

\text{coop fraction} = \frac{1}{1 + qT},

which is positive for finite $T$ and falls as noise $q$ or punishment length $T$ rise.

Definition, imperfect public monitoring (Green-Porter)

Players observe only a public signal correlated with the profile of actions, not the actions themselves. On the equilibrium path, punishments must be triggered by signals, so they sometimes fire when no one deviated. Sustaining cooperation requires accepting these on-path punishment phases; the optimal ones are finite and forgiving, not permanent.

For a one-round deviation, the immediate gain is $T-R$ . A bad signal adds probability $d-q$ of entering a punishment phase, whose discounted loss over $L$ rounds is $(R-P)\sum_{k=1}^{L}\delta^k$ . The displayed strategy deters that deviation only if

T-R \le (d-q)(R-P)\sum_{k=1}^{L}\delta^k.

So finite punishment is not automatically optimal or sufficient. It must be long enough and the signal informative enough, while false alarms make longer punishment more costly on the equilibrium path.

Work it out

Why grim fails, and forgiveness saves value

Let the false-alarm rate be $q = 0.1$ . Under grim trigger, cooperation ends the first time the signal reads bad. The chance of surviving $n$ rounds is $0.9^n$ , which decays toward zero, so cooperation almost surely collapses and stays collapsed. All the value is lost to a punishment that punishes no one in particular.

Now use trigger-and-forgive with $T = 4$ . The cooperative fraction is $1/(1 + 0.1\times 4) = 1/1.4 \approx 0.71$ . Wars still happen, about once every ten cooperative rounds, and last four rounds each, but the relationship recovers every time. A little forgiveness converts a doomed relationship into a mostly cooperative one.

Key idea

When you can only see a noisy signal, punishment must sometimes fall on the innocent. Permanent punishment destroys the relationship; finite, forgiving punishment sustains cooperation with occasional wars no one caused.

Check yourself

Check yourselfapply

Under noisy monitoring, why does grim trigger destroy a cooperative relationship?

Check yourselfanalyse

You observe a price war between two firms. What can you conclude about whether someone cheated?

Check yourselfbrings back: repeated games

Compared with the perfect-monitoring case, why must punishments happen on the equilibrium path here?

Sources

Green, E. J., and Porter, R. H. (1984). Noncooperative collusion under imperfect price information. Econometrica, 52(1), 87-100.
Abreu, D., Pearce, D., and Stacchetti, E. (1990). Toward a theory of discounted repeated games with imperfect monitoring. Econometrica, 58(5), 1041-1063.