MBA 211 Game Theory
Problem Set 4: Repeated Games
Answers
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
(Defect, defect) and (Nuclear, nuclear) are both Nash equilibria. Note, however, that
nuclear is a dominated strategy.
b. Suppose this game is player three times. Show that there is an equilibrium
where cooperation is sustained early in the game.
Both players play the following strategy: Cooperate in periods 1 and 2 and defect in
period 3. If any player deviates before period 3, play nuclear for the remainder of 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.
The multiplicity of equilibria are a “blessing” in the repeated game since the nuclear
outcome provides a credible threat to sustain cooperation early in the game.
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.
In odd numbered periods, both players go to the opera. In even numbered periods, both
players play go to the fights. Indeed, this alternation scheme is how many households
solve this type of “game.”
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.
In period 1, go to opera. In all other periods, go to the fights. This yields an expected
payoff equal to 4 for both players (To see this, either simulate in Excel or recall the
finance formula for discounting a perpetuity at a 50% discount rate. Hidden lesson:
finance is really useful in game theory!)
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.
If heads, go to opera. If tails, go to the fights.