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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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