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: Each player chooses $C$ or $D$ in every period; Grim Trigger: start with $C$; after any defection by any player, choose $D$ forever.

Rules

• Start with an infinitely repeated Prisoner’s Dilemma stage game; Players choose $C$ or $D$ each period under grim-trigger strategies.
• The player who sustains cooperation by making deviation unprofitable supports the cooperative outcome.
• The stage game is repeated infinitely many times.
• Future payoffs are discounted by $\delta \in (0,1)$; Total utility is the present discounted value of the payoff stream.

Payoff Matrix

Derivation (Best Response Analysis)

• There are two relevant states under Grim Trigger:
  • cooperative state: no deviation has occurred,
  • punishment state: some deviation has occurred.
• In the cooperative state, obeying Grim Trigger yields

• Deviating once yields

because all future play switches to $(D,D)$.

• Cooperation is optimal in the cooperative state if and only if

• In the punishment state, sticking with $D$ gives $0$ forever.
• Deviating to $C$ gives $-2$ immediately and does not restore cooperation, so it is worse than staying with $D$.

Derivation (Nash Equilibrium)

• By the one-shot deviation principle, Grim Trigger is subgame perfect exactly when no profitable one-period deviation exists in either state.
• The punishment state is always incentive compatible.
• The cooperative state is incentive compatible if and only if $\delta \geq \tfrac{1}{3}$.

Nash Equilibrium

Result:

If $\delta \geq \tfrac{1}{3}$, the strategy profile in which both players use Grim Trigger is a subgame perfect Nash equilibrium and sustains $(C,C)$ forever on the equilibrium path.

Social Optimum

• The socially optimal path is perpetual cooperation, giving each player $\frac{4}{1-\delta}$.
• Permanent defection yields each player $0$.
• Grim Trigger can decentralise the social optimum when players are sufficiently patient.

Insights

Insight:

• A severe and credible punishment can overturn the one-shot dominance of defection.
• The condition $\delta \geq \tfrac{1}{3}$ measures how much players value the future relative to the current gain from cheating.