Let's break down the problem step by step as requested. --- ### **Payoff Matrix** Let's write the matrix more clearly: | | **United: Increase** | **United: Don't Increase** | |-----------------------|---------------------|---------------------------| | **Metro: Increase** | (25, 15) | (30, 10) | | **Metro: Don't Inc.** | (15, 20) | (40, 5) | Each cell is (Metro, United). --- ## **a. Does either firm have a dominant strategy? If yes, identify and explain it.** A **dominant strategy** is one that gives a player a higher payoff than any other strategy, regardless of what the other player does. Let's check for each firm. --- ### **Metro Limo's Choices:** - **If United Increases:** - Metro Increase: 25 - Metro Don't Increase: 15 - **25 > 15:** Metro prefers to Increase. - **If United Doesn't Increase:** - Metro Increase: 30 - Metro Don't Increase: 40 - **40 > 30:** Metro prefers to Don't Increase. **Conclusion for Metro:** Metro Limo does **not** have a dominant strategy, because the best choice depends on what United does. --- ### **United Limo's Choices:** - **If Metro Increases:** - United Increase: 15 - United Don't Increase: 10 - **15 > 10:** United prefers to Increase. - **If Metro Doesn't Increase:** - United Increase: 20 - United Don't Increase: 5 - **20 > 5:** United prefers to Increase. **Conclusion for United:** United Limo **does** have a dominant strategy: **Increase advertising**. No matter what Metro does, United gets a higher payoff by Increasing. --- **Summary for (a):** - **Metro Limo:** No dominant strategy. - **United Limo:** Yes, dominant strategy is to **Increase advertising**. --- ## **b. What is the Nash equilibrium of this game? Identify the strategy pair and explain why it qualifies as a Nash equilibrium using the payoff matrix.** A **Nash equilibrium** is a set of strategies where each player's choice is the best response to the other player's choice; neither has an incentive to unilaterally deviate. Let’s check each possible outcome: --- ### **1. (Increase, Increase):** (25, 15) - **Metro:** If United Increases, Metro gets 25 by Increasing, 15 by Not Increasing. Sticking with Increase is best. - **United:** If Metro Increases, United gets 15 by Increasing, 10 by Not Increasing. Sticking with Increase is best. - **Conclusion:** **Neither wants to deviate.** **This is a Nash equilibrium.** --- ### **2. (Increase, Don't Increase):** (30, 10) - **Metro:** If United Doesn't Increase, Metro gets 30 by Increasing, 40 by Not Increasing. Metro would want to switch to Not Increase. - **United:** If Metro Increases, United gets 10 by Not Increasing, 15 by Increasing. United would want to switch to Increase. - **Conclusion:** **Not a Nash equilibrium.** --- ### **3. (Don't Increase, Increase):** (15, 20) - **Metro:** If United Increases, Metro gets 15 by Not Increasing, 25 by Increasing. Metro would want to switch to Increase. - **United:** If Metro Doesn't Increase, United gets 20 by Increasing, 5 by Not Increasing. United sticks with Increase. - **Conclusion:** **Not a Nash equilibrium.** --- ### **4. (Don't Increase, Don't Increase):** (40, 5) - **Metro:** If United Doesn't Increase, Metro gets 40 by Not Increasing, 30 by Increasing. Metro sticks with Not Increase. - **United:** If Metro Doesn't Increase, United gets 5 by Not Increasing, 20 by Increasing. United would want to switch to Increase. - **Conclusion:** **Not a Nash equilibrium.** --- ### **Final Answer** #### **a. Dominant Strategy:** - **United Limo** has a dominant strategy: **Increase advertising**. - **Metro Limo** does **not** have a dominant strategy. #### **b. Nash Equilibrium:** - The **Nash equilibrium** is **(Increase, Increase)**: Both firms increase advertising. - **Explanation:** In this outcome, neither firm can improve its payoff by unilaterally changing its decision: - If Metro changes to Don't Increase, its profit drops from 25 to 15. - If United changes to Don't Increase, its profit drops from 15 to 10. --- **Summary Table:** | Metro | United | (Metro, United) | Nash? | |---------------|---------------|-----------------|-------| | Increase | Increase | (25, 15) | YES | | Increase | Don't Increase| (30, 10) | NO | | Don't Increase| Increase | (15, 20) | NO | | Don't Increase| Don't Increase| (40, 5) | NO | --- ## **Final Answers** ### **a.** - **United Limo** has a dominant strategy: **Increase advertising**. - **Metro Limo** does **not** have a dominant strategy. ### **b.** - The **Nash equilibrium** is **(Increase, Increase)**, with payoffs (25, 15). - This qualifies as a Nash equilibrium because neither firm can do better by unilaterally changing its strategy.