Three legislators are set to vote on a bill to raise the salary of legislators. The majority wins, so all three will receive the raise if at least two of them vote in favor of the bill. The raise is valued at R by each legislator. Voting in favor of the bill comes with political backlash from constituents, though, even if the bill fails. Let C be the cost of backlash for anyone voting in favor of the bill. Finally, suppose that 0 < C < R. There are four possible payoffs for each legislator: 0 if they vote against the bill and at least one other legislator votes against it (so the bill fails) R if they vote against the bill and the others vote for the bill (so the bill passes) -C if they vote for the bill and no one else votes for the bill (so the bill fails) R-C if they vote for the bill and at least one other legislator votes for it (so the bill passes). The three legislators are named X, Y, and Z, and voting happens sequentially and orally. So X announces their vote (to the world), followed by Y announcing their vote, followed by Z announcing their vote. Once voting ends, the payoffs above are realized. a) Draw the sequential game for this sequential voting scheme, with the order of voting being X, then Y, and then Z. This is a 3-player game, so you need to identify who is voting at each of the three stages of the voting (X at the beginning, Y second, and Z third) and you need 3 payoffs (for X,Y,Z) at the end of each branch of the sequential game.. b) Use the backward induction approach to determine your game’s SPNE. i) Does the bill pass in your equilibrium? ii) What is the equilibrium payoff for each player? iii) Is there an advantage for X (first mover), or Y (middle mover) or Z (last mover) in this voting game?
Three legislators are set to vote on a bill to raise the salary of legislators. The majority wins, so all three will receive the raise if at least two of them vote in favor of the bill.
The raise is valued at R by each legislator. Voting in favor of the bill comes with political backlash from constituents, though, even if the bill fails. Let C be the cost of backlash for anyone voting in favor of the bill. Finally, suppose that 0 < C < R. There are four possible payoffs for each legislator:
0 if they vote against the bill and at least one other legislator votes against it (so the bill fails)
R if they vote against the bill and the others vote for the bill (so the bill passes)
-C if they vote for the bill and no one else votes for the bill (so the bill fails)
R-C if they vote for the bill and at least one other legislator votes for it (so the bill passes).
The three legislators are named X, Y, and Z, and voting happens sequentially and orally. So X announces their vote (to the world), followed by Y announcing their vote, followed by Z announcing their vote. Once voting ends, the payoffs above are realized.
a) Draw the sequential game for this sequential voting scheme, with the order of voting being X, then Y, and then Z. This is a 3-player game, so you need to identify who is voting at each of the three stages of the voting (X at the beginning, Y second, and Z third) and you need 3 payoffs (for X,Y,Z) at the end of each branch of the sequential game..
b) Use the backward induction approach to determine your game’s SPNE.
i) Does the bill pass in your equilibrium?
ii) What is the equilibrium payoff for each player?
iii) Is there an advantage for X (first mover), or Y (middle mover) or Z (last mover) in this voting game?
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