Dr. Huanxing Yang
Econ 601 Game Theory
Homework Assignment 6
Due May 28, Wednesday
1.
Chapter 10, Problem 5
In answering part (b), we will also provide the answer to part (a). A strategy
for player 1 is a pair of actions; an action when his type is
H
and an action
when his type is
L
. A strategy for player 2 is a single action since he has only
one information set. There are then eight possible strategy pairs, which are
shown in Table SOL10.5.1
1
3
1
2
Table SOL10.5.1
# 1's action when H 1's action when L
2's strategy
Equilibrium?
1N
o
2Y
e
s
i
f
.
3N
o
4N
o
5N
o
6N
o
7N
o
8Y
e
s
i
f
xx
a
xy
a
p
yx
a
yy
a
b
b
b
b
p
≤
≥
Let's consider each of them. Strategy pair #1 is not an equilibrium as when player
1 is type
L
, he prefers to choose
y
and get a payoff of 3 than
x
and get a payoff
of 2. As to strategy pair #2, player 1's strategy is optimal as, given player 2 is to
choose action
a
, player 1's payoff is 3 from
x
and 2 from
y
(when he is type
H
) and is 3 from
y
and 2 from
x
(when he is type L). Action
a
yields a higher
expected payoff than
b
for player 2 when:
p
1
1
−
p
1
≥
p
3
1
−
p
0
1
3
≥
p
.
Turning to strategy pair #3, player 1's strategy is not optimal as it calls for action
y
when he is type
H
and we already showed that he prefers
x
(given player 2
chooses
a
). The same argument applies to strategy pair #4. For strategy pair #5,
given player 2 chooses
b
, player 1 prefers
y
when he is type H (with a payoff
of 5) to
x
(with a payoff of 1). Hence, both strategy pairs #5 and #6 are not
equilibria. For strategy pair #7, player 1 prefers
y
to
x
when he is type L since
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View Full Documentthe former yields 4 and the latter yields 1 (given player 2 is choosing
b
). Finally,
it should be clear from the above that player 1's strategy in strategy pair #8 is
optimal for both types. Player 2's strategy is optimal when:
()
1
21
0
11
1
2
pp
p
p
p
×+ − ×≥ ×+ − ×⇒ ≥
The last column on the right in Table SOL10.5.1 answers part (b) and the answer
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 Spring '08
 YANG
 Game Theory, payoﬀ, Dr. Huanxing Yang

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