Question

2. In a two-player, one-shot, simultaneous-move game, each player can choose strategy A or strategy B. If both players choose strategy A, each earns a payoff of \(400. If both players choose strategy B, each earns a payoff of \)200. If player 1 chooses strategy A and player 2 chooses strategy B, then player 1 earns \(100 and player 2 earns \)600. If player 1 chooses strategy B and player 2 chooses strategy A, then player 1 earns \(600 and player 2 earns \)100.

a. Write the above game in normal form.

b. Find each player’s dominant strategy, if it exists.

c. Find the Nash equilibrium (or equilibria) of this game.

d. Rank strategy pairs by aggregate payoff (highest to lowest).

e. Can the outcome with the highest aggregate payoff be sustained in equilibrium? Why or why not?

Short answer

a. Possible combinations are-

		P 2
		A	B
P1	A	(400,400)	(100,600)
B	(600,100)	(200,200)

b. The dominant strategy of player 1 will be option B and dominant strategy of player 2 will be option B

c. The Nash equilibrium is the BB strategy (200,200).

d. We can rank each strategy from highest to lowest as follows:

$\begin{array}{l}AA(400,400)=\$800\\ AB(100,600)=\$700\\ BA(600,100)=\$700\\ BB(200,200)=\$400\end{array}$

e.The outcome with the highest aggregate payoff AA will not be in equilibrium

Step-by-step solution

Finding possible combinations

1. a.

In a scenario with two players in a one-shot, simultaneous-move game, the possible combinations of strategies A and B will be as follows.

		P 2
		A	B
P1	A	(400,400)	(100,600)
B	(600,100)	(200,200)

Finding the dominant strategy

b.

In order to establish the dominant strategy of player 1, we must take into account the decision made by player 2. It is important to keep in mind that the values of the first cell correspond to player 1 while the values of the second cell correspond to player two.

If player 2 selects option A, player 1 will chose option B since it generates a higher payoff (600) than option (400). Likewise, if player 2 chooses option B, player 1 will also choose option B since 200 is greater than 100 generated by option A. Therefore, the dominant strategy of player 1 will be option B, since regardless of the decision of player 2 the highest payoff will be obtained.

Regarding player 2's strategy, if player 1 selects option A, player 2 will select option B, since 600 is greater than 400 , while if player 1 selects B, player 2 will still choose option B. Therefore, the dominant strategy of player 2 will be option B, since regardless of the decision of player 1 the highest payoff will be obtained.

 Step 3: Finding the Nash equilibrium

c.

The Nash equilibrium occurs when the strategy that each player chooses and is the most optimal, that is, under another strategy, one of the players will not reach their maximum payoff.

As can be seen in table (a), the BB strategy (200,200) is the most optimal for both players. With any of the other combinations, some players will have a lower benefit.

 Step 4:To find rank strategy

1. d.

We can rank each strategy from highest to lowest as follows:

$\begin{array}{l}AA(400,400)=\$800\\ AB(100,600)=\$700\\ BA(600,100)=\$700\\ BB(200,200)=\$400\end{array}$

Finding equilibrium

In this one-shot-simultaneous game scenario the two players do not know the other player's decision and will only meet once and make a single decision regarding the price level of each one. Therefore a Nash equilibrium level scenario BB (200,200) benefits both players. However, this equilibrium represents the lowest outcome of the four scenarios. Likewise, the strategy that generates a greater outcome is AA, in which both players will have (400,400).

The dominant strategy of each is option B, which is completely independent and without knowledge of the decisions of another player, the equilibrium strategy will remain at BB. Thus, the outcome with the highest aggregate payoff AA will not be in equilibrium.