Question

Use the following normal-form game to answer the questions below.

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?

Short answer

a. X must be greater than 4(x>4).

b. The value of X for player 1 must be less than 5 for the result to be greater than 2 and strategy B to be dominant

c. The Nash equilibrium with strategy (B, D) the value of X must be greater than 4 and less than 5.

Step-by-step solution

Finding values of x is strategy D dominant for player 2

a.

Analyzing the scenario of strategy D for player 2, if player 1 decides to choose option B, player 2 will choose option D since the payoff will be 5 vs 4 if he chooses option C. On the other hand, we must obtain the value of x in strategy D for the movement of player 1 , that is, 7-x must be greater or equal than 2 for him to choose option B.

We can determine this value in the following way:

$\begin{array}{l}3>7-\text{x}\\ \text{x}>7-3\\ \text{x}>4\end{array}$

Thus, for player 1 to choose option B and at the same time generate the dominant strategy of player 2 with option D, X must be greater than 4(x>4).

Finding values of x is strategy B dominant for player 1

b.

In the case of player l, we must find the value of X that generates a payoff equal to the combination C, B(4) that will be greater than the combination C, A(3). On the other hand, player 2 should choose option D if the value of X is greater than 2. Therefore, it can be determined as follows:

$\begin{array}{r}7-x>2\\ 7-2<\text{x}\\ \text{x}<5\end{array}$

Thus, when player 2 chooses option D, the value of X for player 1 must be less than 5 for the result to be greater than 2 and strategy B to be dominant.

 Step 3:To find values of x is (B,D)the only Nash equilibrium of the game

c.

Given that the dominant strategy for player 2 where X>4 and the dominant strategy for player 2 where X<5 (obtained in exercises a and b) the Nash equilibrium with strategy (B, D) the value of X must be greater than 4 and less than 5

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