E
CONOMICS
S1050
Summer 2009
P
RACTICE
P
ROBLEMS
II
1)
Consider the following Prisoner’s Dilemma game:
Defect
Cooperate
Defect
1,1
5,0
Cooperate
0,5
4,4
a)
What is the unique subgame perfect Nash Equilibrium (NE) of this game, if:
i)
the game is repeated twice?
ii)
the game is repeated 1,984 times?
b)
If the game is indefinitely repeated:
i)
Is an average payoff of 2 for each player a possible NE payoff?
ii)
Construct a pair of strategies that would yield an average payoff of 2 for each player.
c) If payoffs are discounted by a discount parameter,
δ
:
i)
For what range of
δ
can (4,4) be achieved in a NE?
ii) For what range of
δ
can (2,2) be achieved in a NE?
2)
Consider the indefinitely repeated game in 1), above.
a)
According to the Folk Theorem, what is the set of payoffs that can be achieved in a SPNE?
b)
What two conditions must hold for this to be true?
c)
Is the likelihood of cooperation higher or lower if:
i)
the discount parameter is increased?
ii)
the temptation payoff is increased?
iii) players have short memories and bad facial recognition skills?
3)
Bargaining game.
Alex and Matt are deciding to divide a pot of gold, with S number of gold coins
(i.e., if S = 10, there are 10 gold coins.)
The discount parameter is 1.
a) Assume the game is not repeated.
Alex gets to make 1 offer, and Matt only gets to decide
whether he accepts the offer or not.
If he does not, they both get zero.
i)
What is the NE outcome?
ii)
Is this what happens in reallife? Why or why not?
Does this depend on the size of the pot?
Does this violate any assumptions of Homo Economicus?
b)
Assume now that the discount parameter is less than 1, and the game is repeated four times such
that players make alternating offers (Alex makes offers in rounds 1 and 3; Matt makes offers in
rounds 2 and 4)
i)
Would you rather be in Alex or Matt’s position?
ii)
When will the game end?
Who will be on top?
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c)
If the game is repeated indefinitely, and the discount parameter is less than one:
i)
According to game theory, when will the game end?
ii)
What is the likelihood of this result, if people are characterized by bounded rationality?
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 Summer '09
 NEUGEBOREN
 Economics, Game Theory, Alex, payoﬀs, Hawk Dove, discount parameter

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