CHAPTER 10

GAME THEORY AND STRATEGIC BEHAVIOR

1. A market has only two sellers. They are both trying to decide on a pricing strategy. If both firms charge a high price, then each firm will experience a 10 percent increase in profits. If both firms charge a low price, then each firm will experience a 5 percent decrease in profits. If Firm 1 charges a low price and Firm 2 charges a high price, then Firm 1 will experience a 6 percent increase in profits and Firm 2 will experience a 2 percent decrease in profits. If Firm 2 charges a low price and Firm 1 charges a high price, then Firm 2 will experience a 7 percent increase in profits and Firm 1 will experience a 3 percent decrease in profits.

(I) Construct a payoff matrix for this game.

(ii) Determine whether each firm has a dominant strategy and, if it does, identify the strategy.

(iii) Determine the optimal strategy for each firm.

(iv) Determine the Nash equilibrium.

(v) Is this a prisoners’ dilemma? How do you know?

Solution:

(i)                Payoff matrix:

(ii) Both firms have a dominant strategy. Both firms will charge a high price.

(iii) The optimal strategy for each firm is to charge a high price.

(iv) The Nash equilibrium occurs when each firm charges a high price.

(v) This a not a prisoners’ dilemma because neither firm can be made better off by cooperating.

2. A market has only two sellers. They are both trying to decide on a pricing strategy. If both firms charge a high price, then each firm will experience a 5 percent decrease in profits. If both firms charge a low price, then each firm will experience a 2 percent increase in profits. If Firm 1 charges a high price and Firm 2 charges a low price, then Firm 1 will experience a 1 percent increase in profits and Firm 2 will experience a 4 percent increase in profits. If Firm 2 charges a high price and Firm 1 charges a low price, then Firm 2 will experience a 3 percent increase in profits and Firm 1 will experience a 4 percent increase in profits.       

(i) Construct a payoff matrix for this game.

(ii) Determine whether each firm has a dominant strategy and, if it does, identify the strategy.

(iii) Determine the optimal strategy for each firm.

(iv) Determine the Nash equilibrium.

(v) Is this a prisoners’ dilemma? How do you know?

Solution:

(i)                Payoff matrix:

(ii) Firm 1 has a dominant strategy. It should charge a low price. Firm 2 does not have a dominant strategy.

(iii) Firm 2 should charge a high price if Firm 1 charges a low price and should charge a low price if Firm 1 charges a high price. Firm 1 should choose its dominant strategy.

(iv) The Nash equilibrium exists when Firm 1 charges a low price and Firm 2 charges a high price.

(v) This is not a prisoners’ dilemma because cooperation cannot make the firms better off.

3. A market has only two sellers. They are both trying to decide on a pricing strategy. If both firms charge a high price, then each firm will experience a 5 percent increase in profits. If both firms charge a low price, then each firm will experience a 3 percent increase in profits. If Firm 1 charges a high price and Firm 2 charges a low price, then Firm 1 will experience a 1 percent increase in profits and Firm 2 will experience a 6 percent increase in profits. If Firm 2 charges a high price and Firm 1 charges a low price, then Firm 2 will experience a 2 percent increase in profits and Firm 1 will experience a 7 percent increase in profits.

(i) Construct a payoff matrix for this game.

(ii) Determine whether each firm has a dominant strategy and, if it does, identify the strategy.

(iii) Determine the optimal strategy for each firm.

(iv) Determine the Nash equilibrium.

(v) Is this a prisoners’ dilemma? How do you know?

Solution:

(i)                Payoff matrix:

(ii) The dominant strategy for each firm is to charge a low price.

(iii) The dominant strategy is optimal for each firm.

(iv) The Nash equilibrium will hold when both firms choose their dominant strategy.

(v) This is a prisoners’ dilemma because both of the firms would be better off if they cooperated in choosing to charge a high price.

                                                         

4. A market has only two sellers. They are both trying to decide on a pricing strategy. If both firms charge a high price, then each firm will experience a 5 percent increase in profits. If both firms charge a low price, then each firm will experience a 3 percent decrease in profits. If Firm 1 charges a high price and Firm 2 charges a low price, then Firm 1 will experience a 4 percent decrease in profits and Firm 2 will experience a 6 percent increase in profits. If Firm 2 charges a high price and Firm 1 charges a low price, then Firm 2 will experience a 5 percent decrease in profits and Firm 1 will experience a 7 percent increase in profits.

(i) Construct a payoff matrix for this game.

(ii) Determine whether each firm has a dominant strategy and, if it does, identify the strategy.

(iii) Determine the optimal strategy for each firm.

(iv) Determine the Nash equilibrium.

(v) Is this a prisoners’ dilemma? How do you know?

Solution:

(i)                Payoff matrix:

(ii) The dominant strategy for each firm is to charge a low price.

(iii) The dominant strategy is optimal for each firm.

(iv) The Nash equilibrium will hold when both firms choose their dominant strategy.

(v) This is a prisoners’ dilemma because both of the firms would be better off if they cooperated in choosing to charge a high price.

5. Firm A and Firm B operate retail malls. Both are trying to determine whether to increase advertising budgets during the holiday season or to keep them constant. Because of contract cycles, Firm A will have to make a commitment regarding an advertising budget before Firm B. Their alternatives and payoffs are displayed in the decision tree diagram.

Use this information to determine each firm’s optimal strategy and anticipated payoff.

Solution:

Firm A will choose Increase and receive a payoff of 1. Firm B will choose Increase and receive a payoff of 1.

      

6. Firm A and Firm B operate retail malls. Both are trying to determine whether to increase advertising budgets during the holiday season or to keep them constant. Because of contract cycles, Firm A will have to make a commitment regarding an advertising budget before Firm B. Their alternatives and payoffs are displayed in the decision tree diagram.

Use this information to determine each firm’s optimal strategy and anticipated payoff.

Solution:

Firm A will choose Increase and receive a payoff of 6. Firm B will choose Constant and

receive a payoff of −1.

7. Firm A and Firm B operate retail malls. Both are trying to determine whether to increase advertising budgets during the holiday season or to keep them constant. Because of contract cycles, Firm A will have to make a commitment regarding an advertising budget before Firm B. Their alternatives and payoffs are displayed in the decision tree diagram.     

Use this information to determine each firm’s optimal strategy and anticipated payoff.

Solution:

Firm A will choose Constant and receive a payoff of 0. Firm B will choose Constant and receive a payoff of 0.

   

8. Firm A and Firm B operate retail malls. Both are trying to determine whether to increase advertising budgets during the holiday season or to keep them constant. Because of contract cycles, Firm A will have to make a commitment regarding an advertising budget before Firm B. Their alternatives and payoffs are displayed in the decision tree diagram.

Use this information to determine each firm’s optimal strategy and anticipated payoff.

Solution:

Firm A will choose Constant and receive a payoff of 4. Firm B will choose Increase and receive a payoff of 5.          

9. A pair of duopolists, Firm A and Firm B, manufacture seasonal toys. Both are planning their pricing strategies for the coming holiday season. The firms will choose between a low price (Low) and a high price (High). Firm A has a more flexible production and distribution system than Firm B and is therefore able to begin the season with one pricing strategy and then adopt a different strategy in midseason. While Firm B is unable to change its strategy in midseason, it can delay making a choice until after Firm A has announced its prices for the first part of the season. The firms’ alternatives and payoffs are displayed in the decision tree diagram. Use this information to determine each firm’s

optimal strategy and anticipated payoff.

Solution:

Firm A will choose Low first. Firm B will choose Low. Firm A will then choose High and will have a payoff of 80. Firm B will have a payoff of 50.

      

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10. A pair of duopolists, Firm A and Firm B, manufacture seasonal toys. Both are planning their pricing strategies for the coming holiday season. The firms will choose between a low price (Low) and a high price (High). Firm A has a more flexible production and distribution system than Firm B and is therefore able to begin the season with one pricing strategy and then adopt a different strategy in midseason. While Firm B is unable to change its strategy in midseason, it can delay making a choice until after Firm A has announced its prices for the first part of the season. The firms’ alternatives and payoffs

are displayed in the decision tree diagram. Use this information to determine each firm’s optimal strategy and anticipated payoff.       ­­­­

Solution:

Firm A will choose High first. Firm B will choose Low. Firm A will then choose Low and will have a payoff of 70. Firm B will have a payoff of 75.

11. A pair of duopolists, Firm A and Firm B, manufacture seasonal toys. Both are planning their pricing strategies for the coming holiday season. The firms will choose between a low price (Low) and a high price (High). Firm A has a more flexible production and distribution system than Firm B and is therefore able to begin the season with one pricing strategy and then adopt a different strategy in midseason. While Firm B is unable to

change its strategy in midseason, it can delay making a choice until after Firm A has announced its prices for the first part of the season. The firms’ alternatives and payoffs are displayed in the decision tree diagram. Use this information to determine each firm’s optimal strategy and anticipated payoff.

Solution:

Firm A will choose High first. Firm B will choose High. Firm A will then choose Low and will have a payoff of 120. Firm B will have a payoff of 100.

12. A pair of duopolists, Firm A and Firm B, manufacture seasonal toys. Both are planning their pricing strategies for the coming holiday season. The firms will choose between a low price (Low) and a high price (High). Firm A has a more flexible production and distribution system than Firm B and is therefore able to begin the season with one pricing strategy and then adopt a different strategy in midseason. While Firm B is unable to change its strategy in midseason, it can delay making a choice until after Firm A has

announced its prices for the first part of the season. The firms’ alternatives and payoffs are displayed in the decision tree diagram. Use this information to determine each firm’s optimal strategy and anticipated payoff.

Solution:

Firm A will choose High first. Firm B will choose High. Firm B will then choose High and will have a payoff of 120. Firm B will have a payoff of 100.

13. Two grocery stores compete against each other in a community. Both are considering an increase in advertising expenditures. Their interdependent alternatives are described by the payoff matrix.

(i) Determine whether each firm has a dominant strategy and, if it does, identify the strategy.

(ii) Determine the optimal strategy for each firm.

(iii) Determine the Nash equilibrium.

(iv) Is this a prisoners’ dilemma? How do you know?

Solution:

(i) Firm 1 has a dominant strategy: Advertise. Firm 2 has a dominant strategy:

Advertise.

(ii) The firms’ dominant strategies are their optimal strategies.

(iii) The optimal strategies define a Nash equilibrium.

(iv) This is a prisoners’ dilemma because both firms would be better off with a

cooperative solution in which neither advertises.

14. Two grocery stores compete against each other in a community. Both are considering an increase in advertising expenditures. Their interdependent alternatives are described by the payoff matrix.

(i) Determine whether each firm has a dominant strategy and, if it does, identify the strategy.

(ii) Determine the optimal strategy for each firm.

(iii) Determine the Nash equilibrium.

(iv) Is this a prisoners’ dilemma? How do you know?

Solution:

(i) Firm 1 has a dominant strategy: Advertise. Firm 2 does not have a dominant

strategy.

(ii) Firms 1 has a dominant strategy that is optimal regardless of what Firm 2 chooses to do. It is optimal for Firm 2 to choose Advertise if Firm 1 chooses Don’t Advertise and to choose Don’t Advertise if Firm 1 chooses Advertise.

(iii) The Nash equilibrium is for Firm 1 to choose Advertise and for Firm 2 to choose Don’t Advertise.

(iv) This is not a prisoners’ dilemma.

15. Two grocery stores compete against each other in a community. Both are considering an increase in advertising expenditures. Their interdependent alternatives are described by the payoff matrix.

(i) Determine whether each firm has a dominant strategy and, if it does, identify the strategy.

(ii) Determine the optimal strategy for each firm.

(iii) Determine the Nash equilibrium.

(iv) Is this a prisoners’ dilemma? How do you know?

Solution:

(i) Firm 1 does not have a dominant strategy. Firm 2 does have a dominant strategy: Don’t Advertise.

(ii) Firm 2 has a dominant strategy that is optimal regardless of what Firm 1 chooses to do. It is optimal for Firm 1 to choose Advertise if Firm 2 chooses Advertise and to choose Don’t Advertise if Firm 2 chooses Don’t Advertise.

(iii) The Nash equilibrium is for both firms to choose Don’t Advertise.

(iv) This is a prisoners’ dilemma because both firms would be better off with the cooperative solution, which is to choose Advertise.

16. Two grocery stores compete against each other in a community. Both are considering an increase in advertising expenditures. Their interdependent alternatives are described by the payoff matrix.

(i) Determine whether each firm has a dominant strategy and, if it does, identify the strategy.

(ii) Determine the optimal strategy for each firm.

(iii) Determine the Nash equilibrium.

(iv) Is this a prisoners’ dilemma? How do you know?

Solution:

(i) Firm 1 does not have a dominant strategy. Firm 2 does have a dominant strategy: Advertise.

(ii) Firm 2 has a dominant strategy that is optimal regardless of what Firm 1 chooses to do. It is optimal for Firm 1 to choose Advertise if Firm 2 chooses Don’t Advertise and to choose Don’t Advertise if Firm 2 chooses Advertise.

(iii) The Nash equilibrium is for Firm 1 to choose Don’t Advertise and for Firm 2 to choose Advertise.

(iv) This is not a prisoners’ dilemma.