Let's tackle each part step by step, with explanations and a clear final answer. --- **Given payoff matrix (Widgetway's profit, Acme's profit):** | | **Acme: Low Price** | **Acme: High Price** | |---------------------------|---------------------|----------------------| | **Widgetway: Low Price** | 60, 60 | 250, -20 | | **Widgetway: High Price** | -20, 250 | 130, 130 | --- ### a. Does either firm have a dominant strategy? What strategy should each firm follow? **Definition:** A dominant strategy is one that yields a higher payoff for a player no matter what the other player does. Let's analyze for each firm. #### **Widgetway's Best Response:** - If **Acme chooses Low Price**: - If Widgetway chooses **Low Price**: 60 - If Widgetway chooses **High Price**: -20 → **60 > -20**, so **Low Price is better**. - If **Acme chooses High Price**: - If Widgetway chooses **Low Price**: 250 - If Widgetway chooses **High Price**: 130 → **250 > 130**, so **Low Price is better**. **Conclusion:** **Widgetway's dominant strategy is Low Price** (regardless of Acme's choice). --- #### **Acme's Best Response:** - If **Widgetway chooses Low Price**: - If Acme chooses **Low Price**: 60 - If Acme chooses **High Price**: -20 → **60 > -20**, so **Low Price is better**. - If **Widgetway chooses High Price**: - If Acme chooses **Low Price**: 250 - If Acme chooses **High Price**: 130 → **250 > 130**, so **Low Price is better**. **Conclusion:** **Acme's dominant strategy is also Low Price**. --- #### **What strategy should each firm follow?** Since both have dominant strategies, **each firm should choose Low Price**. #### **Final Answer (a):** - **Both Widgetway and Acme have a dominant strategy: Low Price. Both firms should choose Low Price.** - **With both choosing Low Price, the outcome is (60, 60) (Widgetway: $60,000; Acme: $60,000).** --- ### b. Assume the game is played an infinite number of times. Would the tit-for-tat strategy be reasonable? Explain this strategy. #### **Tit-for-Tat Strategy:** - **Definition:** Start by cooperating (choose High Price), then do whatever your opponent did in the previous round. If they cooperate, continue cooperating; if they defect (choose Low Price), retaliate by doing the same. #### **Why might tit-for-tat be reasonable in infinite games?** - If the game is repeated infinitely (or with an unknown end point), **cooperation can be sustained**, because the threat of future retaliation can outweigh the short-term benefit of cheating. - If both firms play tit-for-tat, and begin with High Price, they both get (130, 130) every period—better than the Nash Equilibrium of (60, 60). - If a firm defects (chooses Low Price), the other will punish by also choosing Low Price in the next period, reducing profits to (60, 60). **Rational firms may prefer to cooperate to sustain higher profits over time, rather than defect and trigger a price war.** #### **Final Answer (b):** - **Yes, tit-for-tat can be a reasonable strategy.** In an infinitely repeated game, each firm can cooperate (choose High Price) as long as the other does. If one cheats, the other punishes by choosing Low Price in the next round. This threat can sustain collusive (high price) outcomes and higher long-run profits. - **Tit-for-tat**: Start by choosing High Price; then, in each subsequent period, choose whatever price your rival chose in the previous period. --- ### c. The game is played a very large (but finite) number of times. What is the appropriate strategy if both firms are always rational? #### **Explanation:** - In a finitely repeated game, **backward induction** applies: In the last round, there is no future retaliation, so both firms will play their dominant strategy (Low Price). - Knowing this, in the second-to-last round, both will also choose Low Price, since cooperation can't be enforced in the next (last) round. - This logic applies recursively to every round: **defection (Low Price) is the rational choice in every round**. #### **Final Answer (c):** - **If both firms are always rational and the game is played a finite number of times, both will always choose the dominant strategy (Low Price) in every round.** - **Cooperation is not sustainable in a finitely repeated game with known end point; the outcome reverts to the Nash Equilibrium (Low Price, Low Price) in every round.** --- ## **Summary Table:** | Part | Answer | |------|--------| | a | Both have a dominant strategy: Low Price. Each should play Low Price; outcome is (60, 60). | | b | Yes, tit-for-tat is reasonable in an infinite game; it sustains cooperation and higher profits by punishing defections. | | c | In a finite game, both will play Low Price every round; cooperation is not sustainable because of backward induction. |