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Repeated Games

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Repeated Games

🎯 Learning Objectives

  • Distinguish finitely vs. infinitely repeated games
  • Apply the backward-unravelling argument for finite games
  • Understand how discount factors affect cooperation
  • Analyse trigger strategies (Grim Trigger, Tit-for-Tat)
  • Identify conditions for collusion sustainability

6.1 Intuition: The Shadow of the Future

When the same game is played repeatedly, the possibility of future punishment can sustain cooperation today. If defecting today means losing cooperation tomorrow, even short-term temptations can be resisted.
🔑 Key Insight: Repeated interaction can solve the Prisoner's Dilemma — but only if the future matters enough. Real-world examples:
  • OPEC oil cartel: members must resist cheating on production quotas
  • Repeated business transactions: trust is valuable because of future dealings
  • International trade agreements: retaliation threats sustain cooperation

6.2 Formal Definitions

Stage Game

The one-shot game played each period (e.g., Prisoner's Dilemma).

Repeated Game

A repeated game consists of:
  • A stage game (normal form)
  • A time horizon (finite TT or infinite)
  • A discount factor δ∈[0,1)\delta \in [0, 1) for each player
  • Payoffs: sum of discounted stage-game payoffs For infinitely repeated games:
Ui=∑t=1∞δt−1ui(st)U_i = \sum_{t=1}^{\infty} \delta^{t-1} u_i(s^t)

Average Payoff

Sometimes we use average payoff: (1−δ)∑δt−1ui(1-\delta) \sum \delta^{t-1} u_i → as δ→1\delta \to 1, this approaches the per-period average.

6.3 Finitely Repeated Games

The Unravelling Problem

Consider Prisoner's Dilemma repeated TT times. Last period TT: No future to worry about → both defect (unique NE of stage game). Period T−1T-1: Both know they'll defect in TT regardless of what happens now → current period is effectively a one-shot game → both defect. By backward induction: Defection in every period.
⚠️ Result: Any finitely repeated game with a unique stage-game NE has a unique SPNE where the stage-game NE is played every period.

Exceptions

If the stage game has multiple Nash equilibria, cooperation can be sustained in finite games (through "good" and "bad" equilibrium phases).

6.4 Infinitely Repeated Games

In infinitely repeated games, there's no last period — so backward unravelling doesn't apply. Cooperation becomes possible.

Grim Trigger Strategy

"Start by cooperating. If anyone ever defects, defect forever." Prisoner's Dilemma:
CDC(3,3)(0,5)D(5,0)(1,1)\begin{array}{c|cc} & C & D \\ \hline C & (3, 3) & (0, 5) \\ D & (5, 0) & (1, 1) \end{array}
Check if Grim Trigger is SPNE: Cooperation payoff: 3+3δ+3δ2+⋯=31−δ3 + 3\delta + 3\delta^2 + \cdots = \frac{3}{1-\delta} One-period deviation (defect): 5+1δ+1δ2+⋯=5+δ1−δ5 + 1\delta + 1\delta^2 + \cdots = 5 + \frac{\delta}{1-\delta} Cooperation is sustainable if:
31−δ≥5+δ1−δ\frac{3}{1-\delta} \geq 5 + \frac{\delta}{1-\delta} 31−δ−δ1−δ≥5\frac{3}{1-\delta} - \frac{\delta}{1-\delta} \geq 5 3−δ1−δ≥5\frac{3 - \delta}{1-\delta} \geq 5 3−δ≥5−5δ3 - \delta \geq 5 - 5\delta 4δ≥24\delta \geq 2 δ≥12\delta \geq \frac{1}{2}
If δ≥0.5\delta \geq 0.5, Grim Trigger sustains cooperation.

6.5 Trigger Strategies Comparison

StrategyDescriptionProsCons
Grim TriggerCooperate until defection, then punish foreverSimple, strong deterrentNo forgiveness — one mistake ends cooperation forever
Tit-for-TatMirror opponent's last moveForgiving, reciprocatingCan get stuck in alternating defection (C,D,C,D...)
Forgiving Tit-for-TatOccasionally forgive defectionMore robust to noiseMore complex
Trigger (n-period)Punish for nn periods, then forgiveProportionate responseNeed to choose nn

Tit-for-Tat in Repeated Prisoner's Dilemma

Tit-for-Tat: "Cooperate in period 1. In period tt, do whatever your opponent did in period t−1t-1." Properties:
  • Nice: Never defects first
  • Retaliatory: Punishes defection immediately
  • Forgiving: Returns to cooperation if opponent does
  • Clear: Easy for opponent to understand Axelrod's tournaments showed Tit-for-Tat is remarkably successful.

6.6 Cartel Stability

Two firms collude to charge high price. Each can cheat by lowering price.
Stage gameColludeCheat
Collude(100, 100)(30, 120)
Cheat(120, 30)(50, 50)
Sustaining collusion with Grim Trigger: Collude forever: 1001−δ\frac{100}{1-\delta} Cheat then compete: 120+50δ1−δ120 + \frac{50\delta}{1-\delta} Collusion sustainable when:
1001−δ≥120+50δ1−δ\frac{100}{1-\delta} \geq 120 + \frac{50\delta}{1-\delta} 100≥120(1−δ)+50δ100 \geq 120(1-\delta) + 50\delta 100≥120−120δ+50δ100 \geq 120 - 120\delta + 50\delta 100≥120−70δ100 \geq 120 - 70\delta 70δ≥2070\delta \geq 20 δ≥27≈0.286\delta \geq \frac{2}{7} \approx 0.286

📊 Formula Summary

ConceptFormula
Discounted payoffUi=∑t=1∞δt−1uiU_i = \sum_{t=1}^{\infty} \delta^{t-1} u_i
Grim Trigger cooperation conditionδ≥1−1gain from defection\delta \geq 1 - \frac{1}{\text{gain from defection}}
Cartel stability conditionδ≥πcheat−πcolludeπcheat−πcompetition\delta \geq \frac{\pi_{\text{cheat}} - \pi_{\text{collude}}}{\pi_{\text{cheat}} - \pi_{\text{competition}}}

✅ Practice Questions

Q1: In the Prisoner's Dilemma with payoffs (C: 3,3; 0,5 / D: 5,0; 1,1), find the minimum δ\delta for Tit-for-Tat to sustain cooperation.
Solution
For Tit-for-Tat, a defection triggers defection in the next period. If both play TFT, a deviation gives 5 today but 1 tomorrow (because opponent punishes). Then it returns to 3.
Deviation payoff: 5+1δ+3δ2+3δ3+⋯=5+δ+3δ21−δ5 + 1\delta + 3\delta^2 + 3\delta^3 + \cdots = 5 + \delta + \frac{3\delta^2}{1-\delta}
Wait — after TFT punishes one period, if both return to cooperation: 5+δ(1)+δ2(3)+δ3(3)+⋯=5+δ+3δ21−δ5 + \delta(1) + \delta^2(3) + \delta^3(3) + \cdots = 5 + \delta + \frac{3\delta^2}{1-\delta}
Cooperation: 31−δ\frac{3}{1-\delta}
31−δ≥5+δ+3δ21−δ\frac{3}{1-\delta} \geq 5 + \delta + \frac{3\delta^2}{1-\delta} 3≥5(1−δ)+δ(1−δ)+3δ23 \geq 5(1-\delta) + \delta(1-\delta) + 3\delta^2 3≥5−5δ+δ−δ2+3δ23 \geq 5 - 5\delta + \delta - \delta^2 + 3\delta^2 3≥5−4δ+2δ23 \geq 5 - 4\delta + 2\delta^2 2δ2−4δ+2≥02\delta^2 - 4\delta + 2 \geq 0 δ2−2δ+1≥0\delta^2 - 2\delta + 1 \geq 0 (δ−1)2≥0(\delta - 1)^2 \geq 0 (always true)
So TFT sustains cooperation for any δ>0\delta > 0 in this game! But TFT+TFT equilibrium is not subgame perfect — need to verify. Q2: For the cartel example, find the minimum δ\delta if the competitive profit is 30 instead of 50. Solution
1001−δ≥120+30δ1−δ\frac{100}{1-\delta} \geq 120 + \frac{30\delta}{1-\delta} 100≥120(1−δ)+30δ100 \geq 120(1-\delta) + 30\delta 100≥120−120δ+30δ100 \geq 120 - 120\delta + 30\delta 100≥120−90δ100 \geq 120 - 90\delta 90δ≥2090\delta \geq 20 δ≥29≈0.222\delta \geq \frac{2}{9} \approx 0.222
Lower competitive profit makes cheating less attractive → easier to sustain collusion. Q3: Why can't cooperation be sustained in finitely repeated Prisoner's Dilemma? Solution
By backward induction. In the last period, there's no future to reward/punish → both defect. In the second-last period, knowing last period's outcome is already determined (defect, defect) regardless of current actions → current period is effectively one-shot → both defect. This unravelling continues to the first period, giving all-period defection. Cooperation requires the possibility of future punishment, which disappears when the end is known. Join Discord PreviousSubgame Perfect EquilibriumNextRepeated Games Extensions
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