Two oligopolists play a prisoner's dilemma and know they will interact exactly 20 times. A consultant recommends implementing a grim trigger strategy to sustain cooperation throughout all 20 rounds. What is the fundamental flaw in this advice?
AThe grim trigger is too lenient — tit-for-tat would be necessary to deter defection over 20 rounds
BIn a finitely repeated game with a known endpoint, backward induction unravels cooperation: both firms defect in round 20 (no future threat exists), then in round 19, and so on, leaving no cooperative equilibrium regardless of the trigger strategy used
CThe grim trigger fails because the discount factors in oligopoly settings are too high to make punishment credible
DTwenty rounds is too short for trigger strategies to establish a cooperation norm
The logic of backward induction destroys cooperation in any finitely repeated prisoner's dilemma with a known endpoint. In the final round, no future interaction exists to threaten, so the dominant strategy is to defect. Both players know this, so in round 19 the future threat is already worthless, and defection dominates there too. This logic propagates all the way back to round 1. The grim trigger (or any trigger strategy) requires a credible future threat, and knowing the game ends in round 20 removes that credibility entirely. Indefinite repetition — not just long repetition — is the essential ingredient.
Question 2 Multiple Choice
A business relationship becomes more long-term, raising a firm's discount factor δ from 0.6 to 0.95. Under a grim trigger strategy in an infinitely repeated prisoner's dilemma, how does this change affect the sustainability of cooperation?
AHigher δ reduces the present value of future payoffs relative to current payoffs, making defection more tempting
BHigher δ makes cooperation more sustainable because the long-run cost of triggering permanent punishment (the lost stream of cooperative surplus) now outweighs the short-run gain from defecting
CThe discount factor is irrelevant in infinitely repeated games because the game has no terminal period
DHigher δ reduces the credibility of the grim trigger by making punishment more costly for both parties
A higher δ means the firm places more weight on future payoffs. Cooperation is sustained when the present value of remaining cooperative forever exceeds the one-time defection gain plus the discounted stream of punishment-phase payoffs. When δ rises, the future cooperative stream (worth 3/(1−δ) in the classic prisoner's dilemma) grows much larger, while the defection gain (worth 5 once) stays constant. Above the critical threshold δ*, the future cooperation value dominates and defection is irrational. Patient players — those who care about the future — cooperate.
Question 3 True / False
Trigger strategies can sustain cooperation in a finitely repeated prisoner's dilemma as effectively as in an infinitely repeated one, provided players choose a sufficiently severe punishment.
TTrue
FFalse
Answer: False
Severity of punishment does not matter when the game has a known endpoint. In the final period, no future threat of any severity exists, so both players defect. Backward induction then unravels cooperation in every preceding period, regardless of the punishment that would apply in a period that never triggers. Only in indefinitely repeated games (where each period has positive probability of continuation) does the threat of future punishment remain credible in every period.
Question 4 True / False
The Folk Theorem implies that infinitely repeated games with sufficiently patient players have many possible equilibrium outcomes — not a single cooperative equilibrium.
TTrue
FFalse
Answer: True
The Folk Theorem shows that any payoff vector giving each player at least their minmax payoff can be sustained as a subgame-perfect equilibrium when δ is close enough to 1. This creates a vast set of sustainable outcomes — from barely better than mutual defection all the way to the cooperative ideal, and many points in between. Which equilibrium actually emerges depends on which strategies players coordinate on. The Folk Theorem explains cooperation but also why predicting the specific cooperative outcome is difficult: repetition opens up a large equilibrium space rather than selecting a unique outcome.
Question 5 Short Answer
Explain why a player's discount factor δ is crucial to whether cooperation can be sustained under a grim trigger strategy in an infinitely repeated prisoner's dilemma.
Think about your answer, then reveal below.
Model answer: The discount factor δ determines how much a player values future payoffs relative to present ones. Under the grim trigger, a player contemplating defection compares a one-time gain (defecting while the opponent cooperates) against the permanent loss of the cooperative surplus (triggering mutual defection forever). If δ is low, future payoffs are heavily discounted and the short-term defection gain dominates — cooperation breaks down. If δ is high (close to 1), future payoffs are nearly as valuable as today's, and the long stream of foregone cooperative surplus makes defection unprofitable. There is a critical threshold δ* above which cooperation is individually rational for every player in every period.
The formal condition is: cooperation payoff stream ≥ defection payoff, or V_coop/(1−δ) ≥ V_defect + δ·V_punish/(1−δ). Rearranging gives δ ≥ (V_defect − V_coop)/(V_defect − V_punish). The intuition is that patience — valuing the future — is what makes threats credible and cooperation self-enforcing.