14.12 Game Theory °Midterm II Solution
11/18/2010
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. Find all subgameperfect equilibria of the following game in which Player 2 plays a
pure strategy:
Answer:
There are two subgames in this game: a proper subgame starting from the
node in which Player 3 makes a move, and the whole game itself. The normal form
representation of the proper subgame is given by
1
n
3
L
R
a
1,0
0,1
b
0,1
1,0
:
This game has the unique Nash equilibrium
°
1
2
a
+
1
2
b;
1
2
L
+
1
2
R
±
.
Therefore, we can
replace this subgame with the payo/ vector
1
4
0
@
1
1
0
1
A
+
1
4
0
@
0
2
1
1
A
+
1
4
0
@
0
1
1
1
A
+
1
4
0
@
1
2
0
1
A
=
0
@
1
2
3
2
1
2
1
A
:
The normal form representation of the new reduced form game is given by
1
n
2
x
y
A
1
2
;
3
2
1
2
;
3
2
B
2
,1
0,0
C
1,1
0,0
:
Clearly this game has two Nash equilibria in which Player 2 plays a pure strategy,
namely,
(
B; x
)
and
(
A; y
)
.
In conlusion, there are two subgameperfect equilibria
1
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of the game in which Player 2 plays a pure strategy:
°
1
2
Ba
+
1
2
Bb; x;
1
2
L
+
1
2
R
±
and
°
1
2
Aa
+
1
2
Ab; y;
1
2
L
+
1
2
R
±
. Note that we cannot eliminate strategy
y
of Player 2 be
cause it is not strictly dominated (strategies
x
and
y
give the same payo/ to Player 2
if Player 1 plays
A
). (Those who found only one NE got 15 points out of 25 points.)
2. Alice is a small business owner. She has two workers, named Bob and Colin. The three
play the in°nitely repeated game with the following stage game:
In every payo/ vector in the °gure, the °rst entry is for Alice, the second entry is for
Bob, and the third entry is for Caroline. Each player±s payo/ in the repeated game
is the discounted some of his or her payo/s in the stage game with discount factor
°
2
(0
;
1)
, and all the previous moves are observable. For each strategy pro°le below
°nd the range of
°
under which the strategy pro°le is a subgameperfect equilibrium.
(The range may be empty.) Show your work.
(a) (10 Points) Alice Hires and Bob and Colin both Work until any of the workers
Shirk; Alice Hires and Bob and Colin both Shirk thereafter.
Answer:
We will show that the range for which the proposed strategy pro°le
is a SPE is empty. There are two states which we call "Good" and "Bad". In
"Good" state, Alice hires, Bob and Colin work. In "Bad" state, Alice hires, Bob
and Colin shirk. We never transit away from "Bad" state. Using singledeviation
principle, Alice does not have an incentive to deviate (Day o/ instead of Hire) in
"Bad" state if
°
2
°
2
1
X
t
=1
°
t
±
1
°
2
1
X
t
=1
°
t
;
which cannot be satis°ed for any
°
2
(0
;
1)
. Therefore, Alice has an incetive to
deviate and take a day o/ if anyone shirked in the past.
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 Game Theory, Alice Hires, strategy pro

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