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Solution to Problem Set 2
ECON414: Game Theory
Spring 2010
1 Question 1
Let’s take one ﬁrm, say ﬁrm A, as an example. If no ﬁrms enter, ﬁrm A should produce and
earn a (net) proﬁt of 700.
So ”no ﬁrms enter” is not a Nash Equilibrium.
Then suppose there’s
one ﬁrm producing. If ﬁrm A enters, it can make a positive proﬁt instead of zero, so
only one
ﬁrm producing is not a Nash Equilibrium either.
Now consider when there’re two ﬁrms producing. Each incumbent ﬁrm earns a positive proﬁt
but the potential entrants will suffer a loss if they enter as net proﬁt is negative. Therefore to
achieve a Nash Equilibrium in this game, there should be only 2 ﬁrms producing. As there’re
5 different ﬁrms in this game, we will have
(
5
2
)
= 10
different combinations, hence there will
be 10 Nash Equilibria.
2 Question 2
Suppose initially there are
1
≤
m
≤
n

1
thieves (interior solution). In order for this to be a
Nash equilibrium, it needs to satisfy BOTH best reply arguments:
1. For the thieves, they do not have incentives to abide the law, i.e.
m

1
m
W
+
1
m
Z
≥
0
(2a)
2. For the residents abiding the law, they do not have incentives to steal, i.e.
m
m
+ 1
W
+
1
m
+ 1
Z
≤
0
(2b)
Given
Z <
0
< W
, then
m
m
+1
W >
m

1
m
W
and
1
m
+1
Z >
1
m
Z
(as
Z
is negative). Therefore,
if equation (2a) holds, (2b) cannot hold because the left hand side of (2b) is strictly greater
than (2a) and hence greater than 0. We have a contradiction here so there’re no interior Nash
Equilibria for
1
≤
m
≤
n

1
.
1
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View Full Document Now consider the corners. If
m
= 0
, if a resident deviates and steals, she will get a payoff
of Z (which is negative) because the probability that she will be caught is one. Therefore, no
residents have incentives to deviate and it is a Nash Equilibrium.
If
m
=
n
, it will be a Nash Equilibrium if and only if
n

1
n
W
+
1
n
Z
≥
0
(2c)
To conclude, there could be one or two Nash Equilibria at the corners depending on whether
(2c) holds or not.
3 Question 3
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This note was uploaded on 04/30/2010 for the course ECON 412 taught by Professor Gribbin,j during the Spring '08 term at UMBC.
 Spring '08
 Gribbin,J
 Macroeconomics, Game Theory

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