Department of Economics
The Ohio State University
Econ 805—Homework #4 Answers
Prof James Peck
Winter 2008
1.
MasColell, exercise 8.E.1.
Assume that
0
<M<
2
s
holds.
Answer:
First, let us convert the verbal description of the game into a
Bayesian game in matrix form.
When nature chooses the types of the two
armies, it selects one of four payo
f
matrices. Army 1 either observes that it is
strong, so that one of the top matrices is the correct one, or it observes that it
is weak, so that one of the bottom matrices is the correct one. Army 2 either
observes that it is strong, so that one of the left matrices is the correct one, or
it observes that it is weak, so that one of the right matrices is the correct one.
army 2: strong
army 2: weak
attack
no
attack
no
army 1: strong
attack
−
s,
−
sM
,
0
M
−
s,
−
wM
,
0
no
0
,M
0
,
00
0
,
0
army 1: weak
attack
−
w,M
−
,
0
−
w,
−
,
0
no
0
0
,
0
,
0
In order to reduce the number of possible equilibria to consider, notice
f
rst
that, no matter what strategy the opponent is playing, if a weak army is better
o
f
attacking than not attacking, thenastrongarmyisa
lsobettero
f
attacking
than not attacking. Letting the strategy of an army be denoted by a vector of
what the army does when it is strong, followed by what the army does when it
is weak, the only possible Bayesian Nash equilibria are:
1. {(A,A) , (N,N)}
2. {(A,A) , (A,N)}
3. {(A,A) , (A,A)}
4. {(A,N) , (N,N)}
5. {(A,N) , (A,N)}
6. {(A,N) , (A,A)}
7. {(N,N) , (N,N)}
8. {(N,N) , (A,N)}
9. {(N,N) , (A,A)}
Because it is always optimal to attack when the rival army never attacks,
we can eliminate possibilities (4), (7), and (8) above. Because possibilities (1)
and (9) are mirror images of each other in this symmetric problem, and (2) and
1
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View Full Document(6) are mirror images of each other, we can treat them as the same possibility,
so we are essentially down to (1), (2), (3), and (5).
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 Spring '08
 PECK
 Economics, Game Theory, player, Bayesian Nash equilibrium

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