Political
Science 272: Solutions for Problem
Set 1
Spring 2005
1.
Strategic-form games.
Find and circle all Nash equilibria in pure strategies
and cross out all dominated strategies.
Note: In this solution set (*) in a cell means that the outcome in
that cell is NEQ.
a.)
|
|
|
Column |
|
|
|
|
C |
D |
|
|
C |
5,5* |
3,3 |
|
Row |
D |
0,0 |
6,7* |
|
|
|
|
|
In this
game there is no dominated strategy. Why?
First, let
us check whether row player has a dominated strategy. When row player plays C
he gets a payoff of 5 if column player plays C and he gets a payoff of 3 if the
column player plays D. Similarly, when row player plays D he gets a payoff of 0
if column player plays C and he gets a payoff of 6 if column player plays D. So
playing C is better than playing D for row player only if column player
plays C. Similarly playing D is better than playing C for row player only
if column player plays D. So neither C nor D is a dominated strategy for the
row player.
Second,
let us check whether column player has a dominated strategy. When column player
plays C he gets a payoff of 5 if row player plays C and he gets a payoff of 0
if the row player plays D. Similarly, when column player plays D he gets a
payoff of 3 if row player plays C and he gets a payoff of 7 if row player plays
D. So playing C is better than playing D for column player only if row
player plays C. Similarly playing D is better than playing C for column player only
if row player plays D. So neither C nor D is a dominated strategy for the
column player.
Thus,
there is no dominated strategy in this game.
There are
two pure strategy Nash equilibria in this game.
To find
the Nash equilibria we should check whether anyone has incentive to deviate
from a certain outcome.
Is (C, D)
a NEQ?
No. Given
that column player plays D row player prefers playing D to playing C. Also given that row player plays C
column player prefers playing C to playing D.
Is (D, C)
a NEQ?
No. Given
that column player plays C row player prefers playing C to playing D. Also
given that row player plays D column player prefers playing D to playing
C.
Is (C, C)
a NEQ?
Yes. Given
that column player plays C row player prefers playing C to playing D. Also
given that row player plays C column player prefers playing C to playing D.
None of the players has incentive to deviate from this outcome.
Is (D, D)
a NEQ?
Yes. Given
that column player plays D row player prefers playing D to playing C. Also
given that row player plays D column player prefers playing D to playing C.
None of the players has incentive to deviate from this outcome.
b.)
|
|
|
Column |
|
|
|
|
C |
D |
|
|
C |
2,-2* |
3,-3 |
|
Row |
|
1,-1 |
5,-5 |
|
|
|
|
|
In this
game D is a dominated strategy for column player. Why? When column player plays
C he gets a payoff of -2 if row player plays C and he gets a payoff of -1 if the
row player plays D. But, when column player plays D he gets a payoff of -3 if
row player plays C and he gets a payoff of -5 if row player plays D. Playing C
is better than playing D for column player no matter what row player plays. In other words,
U2(C|C)>
U2 (D|C) and
U2(C|D)> U2 (D|D).
–2>-3 and –1>-5
So, D is a
dominated strategy for the column player. Given that D becomes a dominated
strategy for row player because 2>1.
Now, given
column player will always play C and row player will play C. (C, C) is a NEQ.
c.)
|
|
|
Column |
|
|
|
|
|
D |
|
|
C |
2,2 |
0,4 |
|
Row |
D |
4,0 |
1,1* |
|
|
|
|
|
In this
game C is a dominated strategy for row player. Why? When row player plays C he gets
a payoff of 2 if column player plays C and he gets a payoff of 0 if the column
player plays D. But, when row player plays D he gets a payoff of 4 if column
player plays C and he gets a payoff of 1 if column player plays D. Playing D is
better than playing C for row player no matter what column player plays.
U1 (D|C)>
U1 (C|C) and U1(D|D)>
U1(C|D)
4>2 and
1>0
So, C is a dominated strategy for the
row player. Given that C is a dominated strategy for the column player. So,
given row player will always play D, column player will always play D. Because
1>0. (D, D) is a NEQ.
d.)
|
|
|
Column |
|
|
|
|
C |
D |
|
|
C |
3,3 |
1,5* |
|
Row |
D |
4,2* |
0,0 |
|
|
|
Column |
|
In this game
there is no dominated strategy. There are two NEQ in pure strategies; (D,C)
and (C,D).
e.)
|
|
|
Column |
|
|
|
|
C |
D |
|
|
C |
1,5* |
3,3 |
|
|
D |
0,0 |
2,7 |
|
|
|
Column |
|
In this game
D is a dominated strategy for row player. Because, U1 (C|C)>
U1 (D|C) and U1(C|D)>
U1(D|D) (1>0 and
3>2). So, D is a dominated strategy for the row player.
Now, given
row player will always play C, we can easily find column player’s best response
and NEQ. Given that row player always plays C column player will play C.
Because 5>3. D is dominated strategy for column player . (C, C) is a NEQ.
f.)
|
|
|
Column |
|
|
|
|
C |
D |
|
|
C |
0,1 |
1,0 |
|
Row |
D |
1,0 |
0,1 |
|
|
|
|
|
In this game
there is no dominated strategy. There is no NEQ in pure strategies.
2.
Dominated strategies
|
|
1 |
2 |
3 |
4 |
5 |
6 |
|
A |
2, 4 |
4, 5 |
2, 2 |
0, 3 |
-2, -6 |
1, 1 |
|
|
1, 3 |
0, 0 |
0, 0 |
0, 4 |
0, 0 |
0, 0 |
|
|
-2, -6 |
1, 1 |
1, 1 |
2, 2 |
-2, 5 |
-2, 5 |
|
D |
0, 4 |
4, 1 |
0, 0 |
0, 0 |
3, 3 |
0, 0 |
|
E |
3, 3 |
0, 4 |
3, 3 |
3, 2 |
3, 2 |
-5, 4 |
|
F |
3, 3 |
3, 3 |
2, 4 |
3, 2 |
3, 3 |
2, 4 |
|
|
1, -5 |
0, 0 |
1, 1 |
-6, 6 |
0, 4 |
1, 1 |
|
H |
3, 3 |
2, 4 |
5, 3 |
0, 0 |
1, 1 |
3, 4 |
|
|
2, 4 |
-2, 0 |
-3, 4 |
-2, 5 |
-2, 3 |
1, 3 |
|
J |
4, 4 |
5, 3 |
3, 2 |
-3, 2 |
5, 3 |
5, 4 |
F strictly
dominates B. Cross out B.
F strictly
dominates G. Cross out G.
F strictly
dominates C. Cross out C.
H strictly
dominates I. Cross out I.
The game takes
the following form:
|
|
1 |
2 |
3 |
|
|
6 |
|
|
2, 4 |
4, 5 |
2, 2 |
0, 3 |
-2, -6 |
1, 1 |
|
|
0, 4 |
4, 1 |
0, 0 |
0, 0 |
