Answer Key: Problem Set 5
March 31, 2005
Question 1
a) From the lemma taught on Feb. 24th, we know that with
fi
xed prices
and
n
fi
rms, in any Nash Equilibrium, each peripheral
fi
rm locates at the
same position as some other
fi
rm and no
fi
rm’s market share is smaller than
any halfmarket of any other
fi
rm. Therefore in our example of two
fi
rms,
both will locate at the middle of the unit interval:
x
1
=
x
2
=
1
2
.
For proof,
fi
rst note that there is no pro
fi
table deviation for any of them
(by deviating to any location other than
1
/
2
when their opponent locates
at
1
/
2
, the
fi
rm makes less pro
fi
t). Second there can not be any other Nash
equilibrium. This can easily be shown by contradiction. Suppose there is
some Nash equilibrium in which
x
i
<
1
2
.
Then
x
j
< x
i
can not be a best
response because it would be more pro
fi
table for
fi
rm
j
to move to
x
j
≥
x
i
.
Now if
x
j
=
x
i
then
fi
rm
j
can make more pro
fi
t by moving to
x
i
+
ε
, for a
small positive number
ε.
If
x
j
> x
i
,
fi
rm
j
can increase its pro
fi
t by moving
to
x
j
−
ε,
for any
ε < x
j
−
x
i
. So
x
i
<
1
2
can not be part of an equilibrium.
The same argument applies for
x
i
>
1
2
.
Hence
x
i
=
x
j
=
1
2
is the only Nash
equilibrium.
b) Let us solve the game backward. Without loss of generality, assume
fi
rm 1 moves
fi
rst and
x
1
≤
1
2
.
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 Spring '08
 PHILIPHAILE
 Game Theory, Firm, p1

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