Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

For the first 6 games, please state all Nash equilibria and draw the best response graphs.

In many cases there will be a range of possible strategies. As a reminder, here are examples of how you should describe these (replace any “x” or “p” with the appropriate values):

[0,1]U, R   [0,x]U, L   [x,1]U,R

D, [0,1]R   U, [0,p]R   D, [p,1]R

Reminder that 0 Up is the same as a pure strategy of DOWN and 1 Up is the same as purely UP.

1               L             R

U      7, 4         9, 8

D      3, 6         9, 1

L: 4x + 6 (1-x)

R: 8x + 1 (1-x)

x: L=R -> 4x + 6 – 6x = 8x + 1 – x

6 – 2x = 1 + 7x

5 = 9x

x = 5/9

Equilibria: [5/9, 1] U, R

Note, it is acceptable to also include the pure NE separately (U,R)

2      L                  R

U    1, 1              1, 4

D    2, 10            3, 10

Equilibria: D, [0,1]R

Note, it is acceptable to also include the pure NE separately: (D,L) and (D,R)

3               L                R

U             6, 0            8, 11

D             6, 12          2, 12

Equilibria: (U,R) and (D,L)

4            L                R

U        17, 99          4, 99

D        15, 53          4, 41

Equilibria: U, [0,1] R

Note, it is acceptable to also include the pure NE separately: (U,L) and (U,R)

5             L            R

U           1, 1        6, 4

D          10, 2       5, 2

U: 1 (1-p) + 6p

D: 10 (1-p) + 5p

P: U=D -> 1 – p + 6p = 10 – 10p + 5p

1 + 5p = 10 – 5p

10p = 9

p = 9/10

Equilibria: (U,R) and ( D, [0,9/10] R )

Note it is acceptable to include D,L separately.

6              L              R

U           12, 5         6, 6

D            7, 3          6, 7

Equilibria: [0,1] U, R

Note it is acceptable to include the pure NE separately: U,R and D,R

Problems 7 – 14: For each game, list:

a) All pure NE

b) Any mixed strategy NE

c) The name of the game (if it is one of our named games, write “N/A” if it is not)

d) All Pareto optimal outcomes

(You are encouraged to practice your best response graphs, but you are not required to submit them for this homework set)

7.

a) U,R

b) None

c) Prisoners Dilemma

d) (10,1), (8,9), (2,10)

8.

a) None

b) (1/2 U, ½ D), (1/2 R, ½ L)

c) Matching Pennies

d) (6,1), (2,5)

Algebra:

Pick x so L = R:

1x + 5 – 5x = 5x + 1 – 1x

5 – 4x = 4x + 1

4 = 8x

X = ½

Pick p so U = D

6 – 6p + 2p = 2 – 2p + 6p

6 – 4p = 2 + 4p

4 = 8p

p = 1/2

9.

a) U,L and D,R

b) (8/13 U, 5/13 D), (2/3 R, 1/3 L)

c) Chicken

d) (15,5), (7,8), (4,16)

Algebra:

Pick x so L = R

5x + 8 – 8x = 0x + 16 – 16x

8 – 3x = 16 – 16x

13x = 8

x = 8/13

Pick p so U = D

15 – 15p + 0p = 7 – 7p + 4p

15 – 15p = 7 – 3p

8 = 12p

p = 8/12 = 2/3

10.

a) D,R

b) None

c) N/A

d) 13,10

11.

a) (U,L), (U,R), (D,L), (D,R)

b) All/everything

c) Trivial Game

d) 3,1

12.

a) (U,L), (D,R)

b) (5/19U, 14/19D),(12/19 R, 7/19 L)

c) Stag Hunt

d) 20,20

Algebra:

Pick x so L = R

20x + 1 – 1x = 6

1 + 19x = 6

19x = 5

x = 5/19

Pick p so U = D

20 – 20p + 1p = 8

20 – 19p = 8

12 = 19p

p = 12/19

13.

a) (U,L), (D,R)

b) (11/14 U, 3/14 D), (2/3 R, 1/3 L)

c) Battle of the Sexes

d) (11,7), (6,12)

Algebra:

Pick x so L = R

7x + 1 – x = 4x + 12 – 12x

1 + 6x = 12 – 8x

14x = 11

x = 11/14

Pick p so U = D

11 – 11p + 2p = 3 – 3p + 6p

11 – 9p = 3 + 3p

8 = 12p

p = 8/12 = 2/3

14.

a) U,R and D,L

b) (8/23 U, 15/23 D), (1/3 R, 2/3 L)

c) Coordination

d) (15,17)

Algebra:

Pick x so L = R

2x + 11 – 11x = 17x + 3 – 3x

11 – 9x = 3 + 14x

8 = 23x

x = 8/23

Pick p so U = D

4 – 4p + 15p = 10 – 10p + 3p

4 + 11p = 10 – 7p

18p = 6

p = 6/18 = 1/3