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14.12 Game Theory — Midterm I
10/19/2010
Prof. Muhamet Yildiz
Instructions.
This is an open book exam; you can use any written material. You have one
hour and 20 minutes. Each question is 25 points. Good luck!
1. Consider the following game.
(a) Using backward induction
f
nd an equilibrium.
Answer:
At the last node, Player 1 chooses
;a
tther
igh
tnode
,P
layer2then
chooses
. At the left node, she chooses
. Hence, at the beginning, Player 1
chooses
. The equilibrium is
(
)
.
(b) Write the game in normal form.
Answer:
The game in normal form is
1
\
2
2,1
2,1
1,2
1,2
2,1
2,1
1,2
1,2
2,1
1,0
2,1
1,0
2,1
0,1
2,1
0,1
2. Compute a Nash equilibrium of the following game. (This is a version of RockScissors
Paper with preference for Paper.)
RS
P
R
0
0
2
−
2
−
2
3
S
−
2
2
0
0
2
−
1
P
3
1
−
1
2
1
1
Answer:
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View Full DocumentThis game has a unique Nash equilibrium, which is in mixed strategies. Write
,
,
and
for the probabilities with which player
plays strategies R, S, and P, respectively.
Of course,
+
+
=1
(1)
Now,
(
2
2
2
)
must make Player 1 indi
f
erent between his strategies. Indi
f
erence
betweenRandSy
ie
lds
2(
2
−
2
)=2(
2
−
2
)
,i
.e
.
,
2
2
=
2
+
2
Since
2
+
2
=1
−
2
by (1), this yields
2
=1
3
On the other hand, the indi
f
erence between S and P yields
2(
2
−
2
)=1+2(
2
−
2
)
.
Substituting
2
=1
3
,weobta
in
4
2
−
2
2
=
−
1
3
.T
oge
the
rw
i
th
2
+
2
=2
3
,th
is
yields
2
=1
6
2
=1
2
Similarly, indi
f
erence between R and S for player 2 yields
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 Spring '11
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