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6. Consider a sequential game with 3 players. Player 1 can choose A or B. Player...

6. Consider a sequential game with 3 players. Player 1 can choose A or B. Player 2 can choose C, D, E, or F (depending on what player 1 chooses). Player 3 can choose G, H, I, J, K, L, M, or N (depending on what player 1 and 2 choose). Player 1 (P1) goes first, player 2 (P2) goes second, and player 3 (P3) goes third. Payoffs are written as the payoffs for P1, P2, and the for P3.

(a) Given the following structure of the game and payoffs, setup the sequential game. Be sure to label all the necessary parts.

For this game there are 8 possible outcomes. Here are the payoffs for those outcomes: If P1 picks A, P2 picks C, P3 picks G then the payoffs (5,6,4). If P1 picks A, P2 picks C, P3 picks H then the payoffs (7,9,6). If P1 picks A, P2 picks D, P3 picks I then the payoffs (11,3,10). If P1 picks A, P2 picks D, P3 picks J then the payoffs (4,8,9). If P1 picks B, P2 picks E, P3 picks K then the payoffs (3,7,4). If P1 picks B, P2 picks E, P3 picks L then the payoffs (6,2,5). If P1 picks B, P2 picks F, P3 picks M then the payoffs (8,4,3). If P1 picks B, P2 picks F, P3 picks N then the payoffs (2,5,2). (5 points)

(b) How many subgames are there in the sequential game you wrote for part a? What are they? Rewrite each subgame in the space below. (4 points)

(c) What is the subgame perfect equilibrium of this game? Show or explain how you know this set of strategies is the subgame perfect equilibrium. (6 points)

(d) Is the subgame perfect equilibrium Pareto optimal? How do you know? State the definition for Pareto optimality in your explanation of you answer. (3 points)​​​​​​​

(e) Identify all of the Pareto optimal outcomes. How many are there in total and what outcomes are they? To specify an outcome, you should list the actions that lead to that particular payoff. (4 points)

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