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Repeated Games

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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 $T$ or infinite)
A discount factor $\delta \in [0, 1)$ for each player
Payoffs: sum of discounted stage-game payoffs For infinitely repeated games:

U_i = \sum_{t=1}^{\infty} \delta^{t-1} u_i(s^t)

Average Payoff

Sometimes we use average payoff:

(1-\delta) \sum \delta^{t-1} u_i

→ as

\delta \to 1

, this approaches the per-period average.

6.3 Finitely Repeated Games

The Unravelling Problem

Consider Prisoner's Dilemma repeated

T

times. Last period $T$ : No future to worry about → both defect (unique NE of stage game). Period $T-1$ : Both know they'll defect in

T

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:

\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 + 3\delta + 3\delta^2 + \cdots = \frac{3}{1-\delta}

One-period deviation (defect):

5 + 1\delta + 1\delta^2 + \cdots = 5 + \frac{\delta}{1-\delta}

Cooperation is sustainable if:

\frac{3}{1-\delta} \geq 5 + \frac{\delta}{1-\delta}

\frac{3}{1-\delta} - \frac{\delta}{1-\delta} \geq 5

\frac{3 - \delta}{1-\delta} \geq 5

3 - \delta \geq 5 - 5\delta

4\delta \geq 2

\delta \geq \frac{1}{2}

\delta \geq 0.5

, Grim Trigger sustains cooperation.

6.5 Trigger Strategies Comparison

Strategy	Description	Pros	Cons
Grim Trigger	Cooperate until defection, then punish forever	Simple, strong deterrent	No forgiveness — one mistake ends cooperation forever
Tit-for-Tat	Mirror opponent's last move	Forgiving, reciprocating	Can get stuck in alternating defection (C,D,C,D...)
Forgiving Tit-for-Tat	Occasionally forgive defection	More robust to noise	More complex
Trigger (n-period)	Punish for $n$ periods, then forgive	Proportionate response	Need to choose $n$

Tit-for-Tat in Repeated Prisoner's Dilemma

Tit-for-Tat: "Cooperate in period 1. In period

t

, do whatever your opponent did in period

t-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 game	Collude	Cheat
Collude	(100, 100)	(30, 120)
Cheat	(120, 30)	(50, 50)

Sustaining collusion with Grim Trigger: Collude forever:

\frac{100}{1-\delta}

Cheat then compete:

120 + \frac{50\delta}{1-\delta}

Collusion sustainable when:

\frac{100}{1-\delta} \geq 120 + \frac{50\delta}{1-\delta}

100 \geq 120(1-\delta) + 50\delta

100 \geq 120 - 120\delta + 50\delta

100 \geq 120 - 70\delta

70\delta \geq 20

\delta \geq \frac{2}{7} \approx 0.286

📊 Formula Summary

Concept	Formula
Discounted payoff	$U_i = \sum_{t=1}^{\infty} \delta^{t-1} u_i$
Grim Trigger cooperation condition	$\delta \geq 1 - \frac{1}{\text{gain from defection}}$
Cartel stability condition	$\delta \geq \frac{\pi_{\text{cheat}} - \pi_{\text{collude}}}{\pi_{\text{cheat}} - \pi_{\text{competition}}}$

✅ Practice Questions

Q1: In the Prisoner's Dilemma with payoffs (C: 3,3; 0,5 / D: 5,0; 1,1), find the minimum

\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: $5 + 1\delta + 3\delta^2 + 3\delta^3 + \cdots = 5 + \delta + \frac{3\delta^2}{1-\delta}$

Wait — after TFT punishes one period, if both return to cooperation: $5 + \delta(1) + \delta^2(3) + \delta^3(3) + \cdots = 5 + \delta + \frac{3\delta^2}{1-\delta}$

Cooperation: $\frac{3}{1-\delta}$

$\frac{3}{1-\delta} \geq 5 + \delta + \frac{3\delta^2}{1-\delta}$ $3 \geq 5(1-\delta) + \delta(1-\delta) + 3\delta^2$ $3 \geq 5 - 5\delta + \delta - \delta^2 + 3\delta^2$ $3 \geq 5 - 4\delta + 2\delta^2$ $2\delta^2 - 4\delta + 2 \geq 0$ $\delta^2 - 2\delta + 1 \geq 0$ $(\delta - 1)^2 \geq 0$ (always true)

So TFT sustains cooperation for any $\delta > 0$ in this game! But TFT+TFT equilibrium is not subgame perfect — need to verify. Q2: For the cartel example, find the minimum $\delta$ if the competitive profit is 30 instead of 50. Solution

$\frac{100}{1-\delta} \geq 120 + \frac{30\delta}{1-\delta}$ $100 \geq 120(1-\delta) + 30\delta$ $100 \geq 120 - 120\delta + 30\delta$ $100 \geq 120 - 90\delta$ $90\delta \geq 20$ $\delta \geq \frac{2}{9} \approx 0.222$

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