Answer to Question #242361 in Economics of Enterprise for Princess

Question

Consider the following game: 

Strategy 

Player 2 

C D 

Player 

1 

A 3, 7-x 2,3 

B 4,4 7-x,5 

a) For what values of x is strategy D strictly dominant for player 2? 

b) For what values of x is strategy B strictly dominant for player 1? 

c) For what values of x is (B,D) the only Nash equilibrium of the game? 

Answer 1

a) In this game, if the Player 1 chose to play strategy B, the best response of Player 2 is to play the strategy D as it provides it a payoff of 5 which is higher than had it played the strategy C with the payoff of 4.

For Player 2 to have the best response in the form of strategy D, if the Player 1 chose to play strategy A, it is necessary to ensure that strategy D provides greater payoff to Player 2 than strategy C. That means:

$3>7-x \\ 7-x<3 \\ x>7-3 \\ x> 4$

Hence, the value of x should be greater than 4 for the strategy D to be strictly dominant for Player 2.

b) In this game, if the Player 2 chose to play strategy C, the best response of Player 1 is to play the strategy B as it provides it a payoff of 4 which is higher than had it played the strategy A with the payoff of 3.

For Player 1 to have the best response in the form of strategy B, if the Player 2 chose to play strategy D, it is necessary to ensure that strategy B provides greater payoff to Player 1 than strategy A. That means:

$7-x>2 \\ x<5$

Hence, the value of x should be less than 5 for the strategy B to be strictly dominant for Player 1.

c) Since, for the value x>4, the strategy D is strictly dominant for Player 2 and for the values x<5, the strategy B is strictly dominant for Player 1, the values of x for which (B,D) is the only Nash equilibrium of the game is 4<x<5.