please answer question 1 (a,b,c,d,e,f)

Consider a legislature constituted of three (3) individuals, A,B, and C. The legislature is considering a bill. If the bill doesn't pass, payoffs are (0,0,0) (where A's payoff is listed first, B 's second, and C 's third). If the bill does pass, payoffs are (1,1,1). The outcome is determined by a vote, where each player indicates "for" or "against" ("abstain" is not a choice). The voting rule is the simple majority rule, that is, the bill passes if two or more individuals indicates "for." 1. Suppose that players cast their votes simultaneously. (a) (3 points) Identify the (pure) strategy sets for each player. (b) (3 points) Draw the extensive form of the game. (c) (3 points) Depict the normal form of the game. (Hint: with three players, you can't depict the normal form in a two dimensional matrix. Remember, however, that the normal form is defined as a mapping from strategy profiles into payoff vectors. The object here is simply to depict that mapping.) (d) (5 points) Identify all of the pure strategy Nash equilibria for this game. For each, indicate whether the bill passes. (e) (5 points) Identify all of the pure strategy Nash equilibria for which no player chooses a weakly dominated strategy. For each, indicate whether the bill passes. (f) (5 points) Modify the game as follows. Before voting, players must decide simultaneously whether to show up. Only those showing up can vote. Showing up involves some very tiny cost, >0. Players can observe who has shown up before voting. A strict majority is required for passage (ties defeat the bill). Is there a pure strategy subgame perfect Nash equilibrium? Justify your answer completely. 2. Suppose that players cast their votes sequentially in the order A,B,C, and that they can observe the votes cast by those who vote before them. (a) (3 points) Identify the (pure) strategy sets for each player. (b) (3 points) Draw the extensive form of the game. (c) (5 points) Is there a pure strategy subgame perfect Nash equilibrium in which the bill passes? If so, identify one. If not, explain why not. Consider a legislature constituted of three (3) individuals, A,B, and C. The legislature is considering a bill. If the bill doesn't pass, payoffs are (0,0,0) (where A's payoff is listed first, B 's second, and C 's third). If the bill does pass, payoffs are (1,1,1). The outcome is determined by a vote, where each player indicates "for" or "against" ("abstain" is not a choice). The voting rule is the simple majority rule, that is, the bill passes if two or more individuals indicates "for." 1. Suppose that players cast their votes simultaneously. (a) (3 points) Identify the (pure) strategy sets for each player. (b) (3 points) Draw the extensive form of the game. (c) (3 points) Depict the normal form of the game. (Hint: with three players, you can't depict the normal form in a two dimensional matrix. Remember, however, that the normal form is defined as a mapping from strategy profiles into payoff vectors. The object here is simply to depict that mapping.) (d) (5 points) Identify all of the pure strategy Nash equilibria for this game. For each, indicate whether the bill passes. (e) (5 points) Identify all of the pure strategy Nash equilibria for which no player chooses a weakly dominated strategy. For each, indicate whether the bill passes. (f) (5 points) Modify the game as follows. Before voting, players must decide simultaneously whether to show up. Only those showing up can vote. Showing up involves some very tiny cost, >0. Players can observe who has shown up before voting. A strict majority is required for passage (ties defeat the bill). Is there a pure strategy subgame perfect Nash equilibrium? Justify your answer completely. 2. Suppose that players cast their votes sequentially in the order A,B,C, and that they can observe the votes cast by those who vote before them. (a) (3 points) Identify the (pure) strategy sets for each player. (b) (3 points) Draw the extensive form of the game. (c) (5 points) Is there a pure strategy subgame perfect Nash equilibrium in which the bill passes? If so, identify one. If not, explain why not