–1–
Prof. William H. Sandholm
Department of Economics
University of Wisconsin
March 8, 2001
Midterm Exam Solutions
1. Behavior strategies are more appropriate here.
In the extensive form drawn,
player 1 is supposed to forget his initial behavior before choosing an action at his
second information set.
However, by playing a mixed strategy (e.g.,
1
2
Ll
+
1
2
Rr
),
player 1 can correlate his behavior at the two information sets.
Doing so enables
him to remember his choice between
L
and
R
when deciding between
l
and
r
.
Since
he is not supposed to be able to remember this choice, behavior strategies, which
utilize independent randomizations at each information set, are preferable.
2. Each player's unique rationalizable strategy is zero.
Thus, the prediction that all
players choose zero can be justified by common knowledge of rationality.
(In fact, if
all long enough (length
≤
1000 is far more than enough) statements of the form "
i
knows that
j
knows that …. that
k
is rational" are true, that is actually sufficient.)
The proof is as follows.
.
Since the highest possible entry is 1000, the highest
possible target integer is 900.
Therefore, because choosing the target integer is the
only way to obtain a positive payoff, it is never a best response to choose an integer
higher than 900.
So no one does this.
Hence, under CKR, everyone realizes that no one will choose an integer higher
than 900.
Once these strategies are removed, the target integer can not exceed 810,
and so no one chooses an integer higher than this.
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 Spring '08
 SANDHOLM
 Economics, Game Theory, PBE, best response, sequential equilibrium

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