Module 8: Repeated Games
What is a PD?
• Remember, a Prisoners’ Dilemma has the following characteristics:
• There is a single Nash Equilibrium
• Both players have a dominant strategy (in a generic format, this strategy is
labelled as Defect)
• In the Nash Equilibrium, both players receive a strictly lower payoff than the
situation where they play their dominated strategies (known as Cooperate).
• An example of a Prisoners’ Dilemma is below:
Cooperate or Defect?
• Players in a PD game have got two strategies: Defect and Cooperate.
• In general, for a PD, the cooperative strategy is the strategy that, when played by
both players, leads to the optimal outcome for the society …
• … while the defecting strategy is the strategy that is the best response for the
individual that when played by both players leads to the worst outcome for society.
,PART 1:
Repetition as a solution to the Prisoners’ Dilemma
Repeated games
• The first solution to the Prisoners’ Dilemma we will look at is that of repetition.
• Why might repeated play of the game be able to sustain cooperation?
o Well, perhaps players care about the future relationship they have with their
opponent and the reputation they are building.
o Players may also be concerned that defecting now will lead to the collapse of
the cooperative outcome in the future.
• So, we can say that if the value of future cooperation is large and exceeds what can
be gained by defecting in the short-term, then long-term benefits of cooperation can
lead to players automatically and tacitly maintaining the cooperative outcome.
Repetition through example
• Let’s imagine that Xolani and Yvonne own competing restaurants and must decide
whether to set a high or low price for their dishes.
• Imagine that the interaction is represented as follows, where the payoffs represent
profits in hundreds of Rands per month.
Although the analysis we are doing can be done for any PD, it makes the most sense when
you can assign cardinal value to payoffs. Hence, we look at example where the payoffs have
cardinal significance.
, The Restaurant Pricing PD
• You should be able to identify the game alongside as a Prisoners’ Dilemma.
• We can also see that the cooperative strategy is to Price High while the defecting
strategy is to Price Low.
• Now, imagine that Xolani and Yvonne begin by Cooperating – i.e. by Pricing High.
• In the case of mutual cooperation, both players are receiving payoffs of 324.
• Of course, this is not the Nash Equilibrium, but assume this is where we begin.
• Imagine now that Xolani deviates from the cooperative strategy.
• Note that the logic is exactly symmetric for Yvonne since the game is
symmetric.
• Xolani notices that if he deviates from the cooperative outcome (i.e. if he chooses to
Price Low), then he could earn a payoff of 360 for one month.
• This means that if Xolani were to defect this month, his profits or payoff would
increase by 36 (360 – 324).
So, what this tells us is that the short-term gain from defecting is 36
What is a PD?
• Remember, a Prisoners’ Dilemma has the following characteristics:
• There is a single Nash Equilibrium
• Both players have a dominant strategy (in a generic format, this strategy is
labelled as Defect)
• In the Nash Equilibrium, both players receive a strictly lower payoff than the
situation where they play their dominated strategies (known as Cooperate).
• An example of a Prisoners’ Dilemma is below:
Cooperate or Defect?
• Players in a PD game have got two strategies: Defect and Cooperate.
• In general, for a PD, the cooperative strategy is the strategy that, when played by
both players, leads to the optimal outcome for the society …
• … while the defecting strategy is the strategy that is the best response for the
individual that when played by both players leads to the worst outcome for society.
,PART 1:
Repetition as a solution to the Prisoners’ Dilemma
Repeated games
• The first solution to the Prisoners’ Dilemma we will look at is that of repetition.
• Why might repeated play of the game be able to sustain cooperation?
o Well, perhaps players care about the future relationship they have with their
opponent and the reputation they are building.
o Players may also be concerned that defecting now will lead to the collapse of
the cooperative outcome in the future.
• So, we can say that if the value of future cooperation is large and exceeds what can
be gained by defecting in the short-term, then long-term benefits of cooperation can
lead to players automatically and tacitly maintaining the cooperative outcome.
Repetition through example
• Let’s imagine that Xolani and Yvonne own competing restaurants and must decide
whether to set a high or low price for their dishes.
• Imagine that the interaction is represented as follows, where the payoffs represent
profits in hundreds of Rands per month.
Although the analysis we are doing can be done for any PD, it makes the most sense when
you can assign cardinal value to payoffs. Hence, we look at example where the payoffs have
cardinal significance.
, The Restaurant Pricing PD
• You should be able to identify the game alongside as a Prisoners’ Dilemma.
• We can also see that the cooperative strategy is to Price High while the defecting
strategy is to Price Low.
• Now, imagine that Xolani and Yvonne begin by Cooperating – i.e. by Pricing High.
• In the case of mutual cooperation, both players are receiving payoffs of 324.
• Of course, this is not the Nash Equilibrium, but assume this is where we begin.
• Imagine now that Xolani deviates from the cooperative strategy.
• Note that the logic is exactly symmetric for Yvonne since the game is
symmetric.
• Xolani notices that if he deviates from the cooperative outcome (i.e. if he chooses to
Price Low), then he could earn a payoff of 360 for one month.
• This means that if Xolani were to defect this month, his profits or payoff would
increase by 36 (360 – 324).
So, what this tells us is that the short-term gain from defecting is 36
