Consider a game in which two players, Fred and Barney, take turns removing matchsticks from a pile. They start with 21 matchsticks, and Fred goes first. On each turn, each player may remove either 1, 2, 3, or 4 matchsticks. The player to remove the last matchstick wins the game. 1. Suppose there are only 6 matchsticks left, and it is Barney's turn. What move should Barney make to guarantee himself victory? Explain your reasoning. 2. Suppose there are 12 matchsticks left, and it is Barney's turn. What move should Barney make to guarantee himself victory? (Hint: Use your answer to part (a) and roll back.) 3. Now start from the beginning of the game. If both players play optimally, who will win? 4. What are the optimal (complete) strategies for each player? 6. Consider the game in the previous exercise. Suppose the players have reached a point where it is Fred's move and there are just 5 matchsticks left. 1. Draw the game tree for the game starting with 5 matchsticks. 2. Find the rollback equilibria for this game starting with 5 matchsticks. 3. Would you say this 5-matchstick game has a first-mover advantage or a second-mover advantage? 4. Explain why you found more than one rollback equilibrium. How is your answer related to the optimal strategies you found in part (c) of the previous exercise?