Let's address each part step by step: ### **Payoff Matrix:** | | **Acme: Low Price** | **Acme: High Price** | |----------------|---------------------|----------------------| | **Widgetway: Low Price** | 60, 60 | 240, 20 | | **Widgetway: High Price** | 20, 240 | 150, 150 | (Each cell is: Widgetway's profit, Acme's profit) --- ### **(a) Dominant Strategy** **Definition:** A dominant strategy is one that provides a higher payoff for a player, no matter what the other player does. Let's check for each firm: #### **For Widgetway:** - If Acme chooses **Low Price**: - Widgetway: Low Price → **60** - Widgetway: High Price → **20** - If Acme chooses **High Price**: - Widgetway: Low Price → **240** - Widgetway: High Price → **150** For **both cases**, Widgetway gets a higher payoff by choosing **Low Price** (60 > 20, 240 > 150). **Conclusion:** Widgetway's dominant strategy is **Low Price**. #### **For Acme:** - If Widgetway chooses **Low Price**: - Acme: Low Price → **60** - Acme: High Price → **20** - If Widgetway chooses **High Price**: - Acme: Low Price → **240** - Acme: High Price → **150** For **both cases**, Acme gets a higher payoff by choosing **Low Price** (60 > 20, 240 > 150). **Conclusion:** Acme's dominant strategy is **Low Price**. **Final Answer for (a):** > **Yes, both firms have a dominant strategy: Low Price.** > > Each firm should choose Low Price. The outcome will be (Low Price, Low Price) with payoffs **60, 60**. --- ### **(b) Infinite Repetition (Tit-for-Tat)** If the game is played an infinite number of times, the firms may consider cooperating (choosing High Price) to get higher profits (150, 150). **Tit-for-tat strategy:** - Start by cooperating (High Price). - In each subsequent round, do whatever the other firm did in the previous round. **Would tit-for-tat be reasonable?** - If both firms cooperate, they each get 150 per period (better than the 60 from both choosing Low Price). - If one cheats (chooses Low Price while the other chooses High Price), the cheater gets 240, and the other gets 20. - But, if tit-for-tat is used, the cheated firm will retaliate by choosing Low Price in the next round, reducing both firms' profits. **This threat of retaliation can sustain cooperation** if the value of future profits is high enough (i.e., firms care enough about the future). **Conclusion for (b):** > **Yes, tit-for-tat can be a reasonable strategy in an infinitely repeated game. It can sustain cooperation (High Price, High Price) if both firms value future profits highly (i.e., have a high discount factor).** --- ### **(c) Finite but Large Number of Rounds** If both firms know the game will end after a finite number of rounds: - In the **last round**, both will choose their dominant strategy (Low Price). - Knowing this, in the **second-to-last round**, both will also choose Low Price, and so on (backward induction). - **Result:** Both firms will always choose Low Price. **Conclusion for (c):** > **With a finite number of rounds, the only equilibrium is for both firms to choose Low Price in every round.** --- ## **Summary Table** | Part | Answer | |---------|------------------------------------------------------------------------| | (a) | Both firms have a dominant strategy: **Low Price** (payoff 60, 60). | | (b) | **Tit-for-tat** can sustain cooperation (High Price) if repeated infinitely and firms value the future. | | (c) | With a finite number of rounds, both choose **Low Price** every round. | --- **Let me know if you need any further explanation or calculations!**