Infinite-Horizon Repeated Prisoner's Dilemma

Content map: SMU H3 Game Theory Map

Setup

Definition:

Infinite-Horizon Repeated Prisoner’s Dilemma

Players: Two players, Player 1 and Player 2.

Strategies: Tit for Tat: cooperate in period 1, then copy the opponent’s previous-period action.

Rules

Start with an infinitely repeated Prisoner’s Dilemma stage game; Players choose $C$ or $D$ each period under Tit for Tat.
The player who rewards cooperation and punishes deviation supports reciprocal cooperation.
The stage game is repeated infinitely many times.
Future payoffs are discounted by $\delta \in (0,1)$ ; The relevant states are the previous-period outcomes $(C,C)$ , $(D,D)$ , $(C,D)$ , and $(D,C)$ .

Payoff Matrix

	C	D
C	4, 4	-2, 6
D	6, -2	0, 0

Derivation (Best Response Analysis)

Under Tit for Tat:
- $(C,C)$ is absorbing, so

V(C,C)=\frac{4}{1-\delta}.

$(D,D)$ is absorbing, so

V(D,D)=0.

Starting from $(C,D)$ , future play alternates between $(D,C)$ and $(C,D)$ , so for Player 1:

V(C,D)=6-2\delta+6\delta^2-2\delta^3+\cdots =\frac{6-2\delta}{1-\delta^2}.

Starting from $(D,C)$ , future play alternates in the opposite order, so:

V(D,C)=-2+6\delta-2\delta^2+6\delta^3+\cdots =\frac{-2+6\delta}{1-\delta^2}.

At state $(C,C)$ , Tit for Tat prescribes $C$ . This is optimal if

V(C,C)\geq V(C,D),

which is equivalent to $\delta \geq \tfrac{1}{3}$ .

At state $(D,D)$ , Tit for Tat prescribes $D$ . This is optimal if

V(D,D)\geq V(D,C),

which is equivalent to $\delta \leq \tfrac{1}{3}$ .

At state $(C,D)$ , Tit for Tat prescribes $D$ . This is optimal if

V(C,D)\geq V(C,C),

which is equivalent to $\delta \leq \tfrac{1}{3}$ .

At state $(D,C)$ , Tit for Tat prescribes $C$ . This is optimal if

V(D,C)\geq V(D,D),

which is equivalent to $\delta \geq \tfrac{1}{3}$ .

Derivation (Nash Equilibrium)

All four one-shot deviation conditions must hold simultaneously.
Combining the inequalities gives

\delta = \frac{1}{3}.

Nash Equilibrium

Result:

Tit for Tat is a subgame perfect Nash equilibrium in this repeated Prisoner’s Dilemma only at the knife-edge discount factor

\delta=\frac{1}{3}.

The efficient path remains perpetual cooperation.
Tit for Tat can mimic cooperative behaviour, but it does not robustly support it as an equilibrium.

Insights

Insight:

Tit for Tat is forgiving and easy to understand, but equilibrium credibility is weaker than under Grim Trigger.

Behavioural success in tournaments is not the same as subgame perfection.