Question

In: Economics

Q1. Player 2 e f g h a 0,0 3,5 1,6 -3,3 Player 1 b x,y...

Q1.

Player 2
e f g h
a 0,0 3,5 1,6 -3,3
Player 1 b x,y 0,7 4,y 10,5
c x,3 4,0 2,1 -4,-2
d -2,4 0,5 0,7 -7,3

a) Suppose x = -2 and y = 9. How many rationalizable strategies does Player 1 have?

b) Suppose x = -2 and y = 9. How many rationalizable strategies does Player 2 have?

c) Suppose x = -2 and y = 9. What is the sum of both players' payoffs in all pure strategy Nash equilibria of the game?

d) Suppose x = 3 and y = 9. How many rationalizable strategies does Player 1 have?

e) Suppose x = 3 and y = 9. How many rationalizable strategies does Player 2 have?

f) Suppose x = 3 and y = 9. What is the sum of both players' payoffs in all pure strategy Nash equilibria of the game?

Solutions

Expert Solution

Consider the following game as given in the question -

Player 2
Player 1 e f g h
a 0,0 3,5 1,6 -3,3
b x,y 0,7 4,y 10,5
c x,3 4,0 2,1 -4,-2
d -2,4 0,5 0,7 -7,3

Suppose x = -2 and y = 9, we get -

Player 2
Player 1 e f g h
a 0,0 3,5 1,6 -3,3
b -2,9 0,7 4,9 10,5
c -2,3 4,0 2,1 -4,-2
d -2,4 0,5 0,7 -7,3

a.)

Now, let us evaluate the game.

If Player 1 chooses a, player 2 will choose g

If Player 1 chooses b, player 2 will choose e or g

If Player 1 chooses c, player 2 will choose e

If Player 1 chooses d, player 2 will choose g

Hence, player 2 has two rationalizable strategies.

b.)

If Player 2 chooses e, player 1 will choose a

If Player 2 chooses f, player 1 will choose c

If Player 2 chooses g, player 1 will choose b

If Player 2 chooses h, player 1 will choose b

Hence, player 1 has three rationalizable strategies.

c.)

Nash equilibrium -

Reduced game (removing all non-rationalizable strategies) -

Player 2
Player 1 e g
a 0,0 1,6
b -2,9 4,9
c -2,3 2,1

If Player 2 chooses e, player 1 will choose a

If Player 2 chooses g, player 1 will choose b

Removing c for player 1

Player 2
Player 1 e g
a 0,0 1,6
b -2,9 4,9

If Player 1 chooses a, player 2 will choose g

If Player 1 chooses b, player 2 will choose e or g

Given player 2 chooses g, player 1 will choose b.

Hence, that is Nash Equilibruim, and the sum of the payoffs of both the players is 4 + 9 = 13.

Suppose x = 3 and y = 9, we get -

Player 2
Player 1 e f g h
a 0,0 3,5 1,6 -3,3
b 3,9 0,7 4,9 10,5
c 3,3 4,0 2,1 -4,-2
d -2,4 0,5 0,7 -7,3

d.)

Now, let us evaluate the game.

If Player 1 chooses a, player 2 will choose g

If Player 1 chooses b, player 2 will choose e or g

If Player 1 chooses c, player 2 will choose e

If Player 1 chooses d, player 2 will choose g

Hence, player 2 has two rationalizable strategies.

e.)

If Player 2 chooses e, player 1 will choose b or c

If Player 2 chooses f, player 1 will choose c

If Player 2 chooses g, player 1 will choose b

If Player 2 chooses h, player 1 will choose b

Hence, player 1 has two rationalizable strategies.

f.)

Nash equilibrium -

Reduced game (removing all non-rationalizable strategies) -

Player 2
Player 1 e g
b 3,9 4,9
c 3,3 2,1

If Player 1 chooses b, player 2 will choose e or g

If Player 1 chooses c, player 2 will choose e

If player 2 chooses e, player 1 chooses b or c

If Player 2 chooses g, player 1 will choose b

Given that it is a simultaneous move game, both players will want to maximise their earnings, and hence they will choose the strategy (b,e) and this will be their Nash Equilibrium. The sum of their payoffs will be 3 + 9 = 12.

Rahul Sunny answered 2 years ago
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