Question

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

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

b. Find each player’s secure strategy.

c. Find the Nash equilibrium.

Short answer

1. No player will have an optimal one shot strategy that generates a highest payoff.
2. The secure strategy for Player 1 is option B and for Player 2 is option E.
3. TheNash equilibrium is the combination that maximizes the benefit of both players is the BE strategy, in which both players will obtain a payoff of 400.

Step-by-step solution

Finding the dominant strategy

Each player's dominant strategy is the one that results in the highest payoff. This decision made by each player will be simultaneous with the decision of the second player, so that a player will not be able to make a decision based on that of the first.

In the case of player 1's strategy, when player 2 selects D, the best strategy will be option B, which generates a positive payoff. When player 2 selects E, the strategy will also be option B. However, in option F of player 2 player 1 will select option C, since 750 is much higher than the 350 and -50 that they would obtain with option B and A respectively. In the case of player 2's strategy, when player 1 selects A, the best strategy will be F, while for player 1's option B, option E will be the one that generates the highest payoff. Finally, if player 1 selects C, player 2 will also select E.

Thus, player 1's combinations will be (B, B, C) and player 2's (F, E, E). No player will have an optimal one shot strategy that generates a highest payoff.

 Step 2: Finding the secure strategy of each player

The secure strategy is that strategy applied by one of the two players that guarantees the highest possible payoff with the worst possible combination scenario. The secure strategy for Player 1 is option B, since Player 2 selects D, E or F. Player 1 will get a positive payoff (200, 400, 300), which does not happen with option A, which generates (-200,350 and -50) and with option C (-150,-250,750).

The secure strategy for player 2 is option E, since Player 1 selects A,B,C player 2 will get a positive payoff (100, 400 , 550), which does not happen with option D, which generates a payoff of (150,-300,200) and with option F(600, 100, -350).

 Step 3: Finding the Nash equilibrium

b.

Nash equilibrium is a strategy that is generated when the two players have no incentive to change the strategy based on the other player's decision. In the table there are only three combinations with positive results: AE(300,100),BE(400,400) and BF(300,100).

As can be seen, the combination that maximizes the benefit of both players is the BE strategy, in which both players will obtain a payoff of 400. This is the Nash equilibrium strategy.