Trigger Strategy

Trigger Strategy in Repeated Games

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Trigger strategies are central to game theory tutoring and the analysis of repeated games. They explain how cooperation can be sustained even when players have incentives to defect in one‑shot interactions. A trigger strategy rewards cooperation but punishes deviation, making threats credible and long‑run cooperation possible.

Cooperate → until deviation Deviation → trigger punishment

Why do trigger strategies matter? Because they make cooperation self‑enforcing. If the future is valuable enough, the threat of punishment deters deviation today. This logic underlies the Folk Theorem and explains cooperation in oligopoly pricing, trade agreements, and social norms.

1. Define the stage game. Identify payoffs for cooperation and defection.
2. Specify the trigger strategy. “Cooperate unless the opponent defects; if they defect, punish.”
3. Choose the punishment. Grim trigger → punish forever. Tit‑for‑tat → punish for one period. Finite trigger → punish for k periods.
4. Compute incentives. Compare the short‑run gain from deviating with the discounted loss from punishment.
5. Apply the incentive constraint. Cooperation is sustainable if:
   Gain from deviation ≤ Discounted cost of punishment
6. Interpret the result. If players value the future enough (high discount factor), cooperation becomes an equilibrium.

Consider a repeated Prisoner’s Dilemma with discount factor δ.

Payoffs:

• Cooperate–Cooperate: 3 each
• Defect–Cooperate: 5 for defector
• Cooperate–Defect: 0
• Defect–Defect: 1 each

Under grim trigger, cooperation is sustainable if:

5 ≤ 3 + δ·3 + δ²·3 + …

Right side = 3 / (1 − δ). So cooperation requires:

δ ≥ 2/3

If players care enough about the future, cooperation becomes rational.

• Thinking trigger strategies require infinite punishment (they do not).
• Confusing grim trigger with tit‑for‑tat.
• Ignoring the discount factor when checking incentives.
• Assuming cooperation is automatic — it must satisfy incentive constraints.

Trigger strategies explain collusion, cartel stability, trade agreements, social norms, and long‑run cooperation. They are essential for understanding repeated games, the Folk Theorem, and strategic behavior over time.

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