Repeated games
Finitely repeated prisoner’s dilemma
● There is unravelling towards the end of the game
○ Punishment is not effective at the end
○ The SPE of the game means it unravels to the 1st period
● Always defecting is the only SPE for a finite PD (via backwards induction)
● Empirical evidence shows cooperation is more likely at the start of the game
and when players are less experienced
○ The SPE of complete unravelling does not occur
● If the end of the game is in sight, cooperation becomes unstable
Infinitely repeated games
● Model games as indefinitely repeated
● Players can discount future payoffs, with a discount factor 𝛿<1
● Grim trigger strategy - cooperate at T=1 and until either player defects. If
either player has defected at t’<T, then defect
○ Can sustain cooperation if 𝛿 is sufficiently high
○ Grim trigger is an SPE - an outcome which is not self-enforceable in
the 1-period case can become self enforceable
○ Strict punishment means cooperation can break down and not return
○ Players could renegotiate after the cooperation has broken down
○ May be other possible cooperative strategies
1
● The sum of payoffs would be 1−𝛿
○ Cooperate if > payoff from cheating
Strategy tournaments
● Strategies in repeated games can be complex
○ Long / infinite and conditional on what others did in the past
● Infinitely many strategies can be chosen
○ Some strategies are more successful than others
○ Axelrod’s tournament shows tit for tat and tit for two tats are most
successful
■ Grim trigger and random perform the worst
● High performing strategies share common features:
○ Niceness - tried to cooperate, were never the first to defect
○ Forgiveness - allow cooperation after defection
The Folk theorem
Finitely repeated prisoner’s dilemma
● There is unravelling towards the end of the game
○ Punishment is not effective at the end
○ The SPE of the game means it unravels to the 1st period
● Always defecting is the only SPE for a finite PD (via backwards induction)
● Empirical evidence shows cooperation is more likely at the start of the game
and when players are less experienced
○ The SPE of complete unravelling does not occur
● If the end of the game is in sight, cooperation becomes unstable
Infinitely repeated games
● Model games as indefinitely repeated
● Players can discount future payoffs, with a discount factor 𝛿<1
● Grim trigger strategy - cooperate at T=1 and until either player defects. If
either player has defected at t’<T, then defect
○ Can sustain cooperation if 𝛿 is sufficiently high
○ Grim trigger is an SPE - an outcome which is not self-enforceable in
the 1-period case can become self enforceable
○ Strict punishment means cooperation can break down and not return
○ Players could renegotiate after the cooperation has broken down
○ May be other possible cooperative strategies
1
● The sum of payoffs would be 1−𝛿
○ Cooperate if > payoff from cheating
Strategy tournaments
● Strategies in repeated games can be complex
○ Long / infinite and conditional on what others did in the past
● Infinitely many strategies can be chosen
○ Some strategies are more successful than others
○ Axelrod’s tournament shows tit for tat and tit for two tats are most
successful
■ Grim trigger and random perform the worst
● High performing strategies share common features:
○ Niceness - tried to cooperate, were never the first to defect
○ Forgiveness - allow cooperation after defection
The Folk theorem
