MBA 211 Game Theory
Problem Set 4: Repeated Games
1. Consider the following “modified” prisoner’s dilemma:
Cooperate
Defect
Nuclear
Cooperate
4, 4
5, 1
0, 0
Defect
1, 5
2, 2
0, 0
Nuclear
0, 0
0, 0
0, 0
This is simply a Prisoner’s dilemma with a “nuclear option” added to the mix
a. Suppose this game is played once. Find all the Nash equilibria
b. Suppose this game is player three times. Show that there is an equilibrium
where cooperation is sustained early in the game.
c. Explain why our earlier conclusion about the impossibility of cooperation
in Prisoner’s dilemmas doesn’t continue to hold for the modified
Prisoner’s dilemma.
2. Consider the following “battle of the sexes” game
Opera
Fights
Opera
3, 1
0, 0
Fights
0, 0
1, 3
a. We know that all the pure strategy Nash equilibria are “unfair” in the
sense that one party gets less than the other. Construct an equilibrium in an
infinitely repeated version of the game that is fair (equal payoffs) to both
parties.
b. Suppose that the game has a 50% chance of ending each period. Show
that, even in this scenario, there is still a fair (equal payoffs in expectation)
equilibrium.
c. Suppose that the game is played only once but the two parties can look at
the outcome of a publicly revealed coin flip prior to choosing their moves.
Show that there is an equilibrium---with strategies contingent on the coin
flip---that is still fair and that doesn’t lead to miscoordination.