SI 563 Homework 2 Solution
Sep 21, 2016
Chapter6
Exercise 1: (6 points)
Determine which strategies are dominated in the following normal-form games.
a)
2
L
R
A
(3,3)
(2,0)
B
(4,1) (8,-1)
1
For 1: strategy A is dominated by B.
For 2: strategy R is dominated by L.
b)
2
1
L
C
R
U
(5,9) (0,1) (4,3)
M
(3,2) (0,9) (1,1)
D
(2,8) (0,1) (8,4)
For 1: strategy M is weakly dominated by U.
For 2: strategy R is dominated by L.
Note: In the book, dominance was defined as strict dominance. You will get full grades regardless of
including the weakly dominance or not.
c)
2
1
1/4
W
X
Y
Z
U
(3,6) (4,10) (5,0) (0,8)
M
(2,6) (3,3) (4,10) (1,1)
SI 563 Homework 2 Solution
D
Sep 21, 2016
(1,5) (2,9)
(3,0) (4,6)
For 1: strategy M is dominated by a mixed strategy of 0.5U and 0.5D.
For 2: strategy Z is dominated by X.
There are no strategies dominated by other pure strategies for P1, but there is a strategy dominated by
mixed strategies.
For Player 1, M is dominated by a mix of U and D so long as U is played more than ½ the time and D
is played at least ¼ of the time. Mixing 51% U, 49% D would dominate M. A mixed strategy does not
have to be 50/50.
It’s a little tricky to find strategies that are dominated by mixes of other strategies. The first thing to do
is to look for strategies that could not possible be dominated. A player’s strategy cannot be dominated
if it yields the highest payoff to the player for at least one action by the other player. For example, for
Player 1, U cannot be dominated because it yields 5—the most of any strategy for P1—when P2 plays
Y. Once you have eliminated strategies that cannot be dominated, test the strategies that are left against
different mixes of other strategies. Note: You don’t have to mix equally between strategies.
d)
2
1
L
R
A
(1,1) (0,0)
B
(0,0) (5,5)
For 1: there is no dominated strategy.
For 2: there is no dominated strategy.
Exercise 6 (2 points)
In the game pictured here, is it ever rational for player 1 to select strategy C? Why?
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SI 563 Homework 2 Solution
Sep 21, 2016
We can see from the normal form below:
2
1
X
Y
A
(3,0) (0,3)
B
(0,3) (3,0)
C
(1,1) (1,1)
For player 1, strategy C is strictly dominated by a mix strategy of 0.5 A and 0.5 B, which can generate
an expected payoff of 1.5. In this sense, it’s irrational for player 1 to select strategy C.
Note: You don’t have to mix equally between strategies.
Exercise 10: (2 points)
Consider the game pictured here:
In this simplified “ultimatum-offer” bargaining game, the players negotiate over how to divide
two dollars. Player 1 chooses whether to offer all of the money (action H), half of the money
(M), or none of the money (L) to player 2. Player 2 observes the offer and then decides whether
to accept it. If the offer is rejected, then the players both get zero. For this game, can you find
any strategies that are weakly dominated but not dominated?
We can see the normal form below:
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SI 563 Homework 2 Solution
2
Sep 21, 2016
AHAMAL
AHAMRL
AHRMAL
AHRMRL
RHAMAL
RHAMRL
RHRMAL
RHRMRL
H
(0,2)
(0,2)
(0,2)
(0,2)
(0,0)
(0,0)
(0,0)
(0,0)
M
(1,1)
(1,1)
(0,0)
(0,0)
(1,1)
(1,1)
(0,0)
(0,0)
L
(2,0)
(0,0)
(2,0)
(0,0)
(2,0)
(0,0)
(2,0)
(0,0)
1
For 1: strategy H is weakly dominated by strategy M and L.
For 2: strategy RHRMAL and RHRMRL are weakly dominated by strategy AHAMAL, AHAMRL, AHRMAL,
AHRMRL, RHAMAL and RHAMRL.
Strategy RHAMAL and RHAMRL are weakly dominated by strategy AHAMAL and AHAMRL.
Strategy AHRMAL and AHRMRL are weakly dominated by strategy AHAMAL and AHAMRL
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