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212.501 Studies in Microeconomics
In Ho Lee
Homework 4
(due on 2012, November 20)
1. Find the Nash equilibrium in mixed strategy in Prisoner’s Dilemma, Battle of Sexes, Coordination Game, and Stag Hunt Game. Use payoffs described in the lecture. (Remember
that some payoffs were modified from the lecture note. There may be no mixed strategy
equilubrium for some game.)
2. There are n witnesses to a crime. All of the witnesses would like to see the perpetrator
apprehended. This will happen only if the crime is reported to the police. Since all of
the witnesses are busy people, they would rather have someone else report the crime than
reporting it themselves. To be specific, assume that the utility that the players get when the
criminal is caught is v (and the criminal is caught if and only if a report is made) and zero if
not caught. The disutility from reporting is c (Assume that v > c). All the witnesses decide
simultaneously on whether to report.
(a) Is there a Nash equilibrium in pure strategy?
(b) Find the Nash equilibrium in mixed strategies.
(c) What is the probability that, at least, 1 witness reports the crime? How does it change
as n increases?
3. Each of two people has one unit of resource. Each person chooses how much of the resource
to use in fighting the other individual and how much to use productively. If each person i
devotes yi to fighting, then the total amount produced is 2 − y1 − y2 , and person i obtains the
fraction pi (y1 , y2 ) of the output, where
1, if yi > yj ,
1/2, if yi = yj ,
pi (y1 , y2 ) =
0, if yi < yj .
Each person cares only about the amount of output she receives, and prefers to receive as
much as possible.
(a) Find the set of Nash equilibria of this game.
(b) How much is produced at a Nash equilibrium and could the two individuals do better
at some other outcome?
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