–1–
Prof. William H. Sandholm
Department of Economics
University of Wisconsin
Spring 2000
Final Exam Solutions
1.
Since we are only assuming common knowledge of rationality, we need to look
for the rationalizable strategies.
Since this is a two player game, it is equivalent to
look for strategies which survive iterated removal of strictly dominated strategies.
It
turns out to be easiest to first use iterated strict dominance to get rid of pure
strategies, and then to get rid of mixed strategies which are never a best response.
We can first remove
c
, which is strictly dominated by
d
.
After this, we can
remove
A
, which is then strictly dominated by
B
.
Then, we can remove
a
, which is
strictly dominated by
2
5
b
+
3
5
d
.
No other pure strategies can be removed.
Since pure strategies
B
and
C
are both best responses to
b
, these strategies and all
mixtures between them are rationalizable.
If player 1 plays
B
, then
d
and
e
are both
best responses for player 2; if
σ
1
(
B
)
∈
(
3
5
, 1), then
d
is the unique best response; if
1
(
B
) =
3
5
, then
b
and
d
are both best responses; if
1
(
B
)
∈
[0,
3
5
),
b
is the unique best
response.
Thus, it is never a best response for player 2 to put positive weight on
both
b
and
e
; but all other mixtures between
b
,
d
, and
e
are best responses.
Thus:
Rationalizable strategies for 1:
All mixtures of
B
and
C
.
Rationalizable strategies for 2:
All mixtures of
b
,
d
, and
e
which do not put
positive weight on both
b
and
e
.
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 Spring '08
 SANDHOLM
 Economics, Game Theory, player, payoff stream, C. Rationalizable

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