Finite Repeated Prisoner's Dilemma

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Finite Repeated Prisoner’s Dilemma

• Players: Two players, Player 1 and Player 2.
• Strategies: In each round, each player chooses $C$ or $D$; A full strategy specifies one round-1 action and one round-2 action after each possible round-1 history; Since there are four possible round-1 histories, each player has $2 \cdot 2^4 = 32$ pure strategies.

Rules

• Start with a Prisoner’s Dilemma stage game repeated twice; Players choose $C$ or $D$ in each round after observing previous actions.
• The player who best responds in every subgame reaches the rollback outcome.
• The Prisoner’s Dilemma stage game is played twice.
• Round-1 actions are observed before round 2; Total payoff is the sum of the two stage payoffs.

Payoff Matrix

Derivation (Best Response Analysis)

• Consider round 2 first.
• After any history $(C,C)$, $(C,D)$, $(D,C)$, or $(D,D)$, the continuation game is the one-shot Prisoner’s Dilemma.
• In the one-shot Prisoner’s Dilemma, $D$ strictly dominates $C$:
  • if the opponent plays $C$, $6>4$,
  • if the opponent plays $D$, $0>-2$.
• Therefore both players choose $D$ in round 2 after every history.
• The round-2 continuation value is $0$ regardless of what happened in round 1.
• Round 1 then becomes the one-shot Prisoner’s Dilemma again, so both players choose $D$.

Derivation (Nash Equilibrium)

• Backward induction gives:
  • round 2: $(D,D)$ after every history,
  • round 1: $(D,D)$.
• The resulting strategy profile is optimal in every subgame, so it is subgame perfect.

Nash Equilibrium

Result:

The unique subgame perfect Nash equilibrium is:

• play $D$ in round 1,
• play $D$ in round 2 after every possible round-1 history. The realised outcome is $(D,D)$ in both rounds.

Social Optimum

• If both players cooperated in both rounds, each would obtain $4+4=8$.
• The equilibrium gives each player $0+0=0$.
• Therefore the socially optimal path is $(C,C)$ in both rounds, but it is not sustainable in a finite horizon.

Insights

Insight:

• Repetition alone does not create cooperation when the horizon is finite and commonly known.
• The endpoint of the game is what drives the unraveling.