–1–
Prof. William H. Sandholm
Department of Economics
University of Wisconsin
March 11, 2004
Midterm Exam Solutions – Economics 713
1. (i)
G
and
′
G
need not have the same perfect equilibria.
For example,
removing one of player 1’s strategies can cause one of player 2’s strategies to become
weakly dominated, so that it can no longer be a part of a perfect equilibrium.
(ii)
G
and
′
G
must have the same value:
the games have the same set of
Nash equilibria, and the value of any zero sum game is the payoff achieved by
player 1 in all of its Nash equilibria. (This payoff is unique by the Minmax
Theorem.)
2.
(i)
The two extreme cases occur when Johnny likes
p
and
r
equally and less
than
q
, and when Johnny likes
q
and
r
equally and strictly less than
p
.
It is easily
verified by drawing a picture of the possible indifference curves that if Johnny likes
r
at least as much as
s
in these two cases, then he likes
r
at least as much as
s
under
any preferences satisfying the constraints stated in the question.
Now an NM utility
function for the first extreme case is (
u
a
,
u
b
,
u
c
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 Spring '08
 SANDHOLM
 Economics, Game Theory, Johnny, best response, NM utility function

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