23506 Chap.11 Repeated Prisoner's Dilemma: Trigger Strategies, Finite vs Infinite Horizon, Folk Theorem
Repeated Prisoner's Dilemma: Trigger Strategies, Finite vs Infinite Horizon, Folk Theorem
Repeated interaction expands strategy from a current action to a rule after every public history. The lecture Prisoner’s Dilemma has payoffs (2,2) under mutual cooperation, temptation 3 against a cooperator and punishment 1 under mutual defection. A history lists past action profiles; the empty history precedes period 1. Unconditional strategies ignore history, while grim trigger chooses Cooperate after an all-cooperation history and Defect forever after any defection.
A known finite horizon unravels backward. In the last period, Defect strictly dominates because no later punishment remains. Once final defection is fixed, the penultimate action cannot improve that continuation, so defection repeats backward to every period. A long finite horizon still has a known endpoint. An uncertain continuation or terminal bonus is a different model and requires a new last-stage analysis.
An infinite horizon values a constant stream k from now at k/(1−δ). Following grim trigger at a clean history gives 2/(1−δ). A one-time defection gives 3 now and 1 forever from next period, worth 3+δ/(1−δ). Cooperation is a best response when δ≥1/2, and mutual defection is credible during punishment. For problem-set payoffs R=10, T=12 and P=3, the condition re-derives to δ≥2/9. Sufficient patience makes cooperative equilibrium possible, not unique: unconditional mutual defection remains an SPE.
What this chapter covers
- 01Public histories and repeated-game strategies
- 02Unconditional and grim-trigger plans
- 03Known finite-horizon unraveling
- 04Infinite geometric continuation values
- 05Grim-trigger cutoff δ≥1/2
- 06Problem-set cutoff δ≥2/9 and folk intuition
Payoff-specific grim threshold
- 2Define grim trigger after clean and punishment histories.
- 2Value cooperation forever from a clean history.
- 2Value one current defection followed by permanent punishment.
- 2Solve the no-deviation inequality exactly.
- 2Verify punishment credibility and interpret equality.
Key terms
- History
- The public list of action profiles before the current period.
- Grim trigger
- Cooperate while no defection has occurred, then defect forever.
- Finite unraveling
- Backward propagation of final-period incentives through a known finite horizon.
- Geometric series
- A discounted constant stream k+kδ+… equal to k/(1−δ).
- No-deviation condition
- An inequality making prescribed continuation at least as valuable as a profitable current deviation.
- Folk-theorem intuition
- The possibility of many feasible individually acceptable outcomes under patient repeated interaction and credible incentives.
Repeated Prisoner's Dilemma: Trigger Strategies, Finite vs Infinite Horizon, Folk Theorem FAQ
What is a repeated-game strategy?
A mapping from every relevant history to a current action.
How does grim trigger begin?
It cooperates after the empty history and after histories containing only mutual cooperation.
Why does finite cooperation unravel?
The last period has no future punishment, so defection is fixed and that continuation propagates backward.
Is 100 periods effectively infinite?
Not in the formal known-horizon model; period 100 is still a known endpoint.
What is a constant infinite stream worth?
k/(1−δ) when δ lies in (0,1).
Why does deviation value have a leading δ on punishment?
Punishment starts next period, while temptation payoff occurs now.
What is the lecture grim cutoff?
δ≥1/2 for cooperation 2, temptation 3 and punishment 1.
What is the problem-set cutoff?
δ≥2/9 for cooperation 10, temptation 12 and punishment 3.
Why check the punishment phase?
A threat deters only if prescribed punishment is optimal after it begins.
Does patient repetition uniquely predict cooperation?
No. It can support cooperation, while mutual unconditional defection and other equilibria may coexist.
What is a complete trigger-strategy proof?
Define the strategy after the empty history, every history containing only mutual cooperation and every history containing a defection. State whether monitoring is public and perfect. At a clean history, value the prescribed cooperation stream from the current period. Value a one-time profitable defection as the current temptation payoff plus the discounted punishment stream beginning next period. Solve the weak no-deviation inequality, noting that 1−δ is positive. Then move to a punishment history and verify that prescribed Defect is a best response to the other player’s Defect; a severe but non-credible punishment cannot support SPE. Treat equality as equilibrium when compliance is still a best response. For a known finite horizon, do not use the infinite formula: start at the last stage and unravel backward. For changed stage payoffs, re-derive the cutoff from R, T and P rather than copying one-half. Conclude that cooperation is sustainable under the stated condition, while acknowledging that unconditional mutual defection can remain another equilibrium.
Exam move
Write grim trigger formally as an action after every history, including the empty history and any history containing defection. For a finite game, start at the known last period and articulate why the same continuation is added to every earlier current action. For an infinite game, write the first three terms of each stream before applying the geometric formula. Use the template R/(1−δ)≥T+δP/(1−δ), verify the sign of T−P and interpret the threshold as temptation gain relative to punishment loss. Re-derive one-half and two-ninths separately, then check punishment actions after dirty histories. End by saying cooperation is sustainable, not inevitable.
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