Consider a committee that consists of three people, A, B, and C, that is to exam
ID: 1249499 • Letter: C
Question
Consider a committee that consists of three people, A, B, and C, that is to examine three proposals, x, y, and z. The committee can adopt at most one of the three proposals; it also has the option of adopting none. The committee will first vote on x. If x receives at least two votes it will be adopted; if not, the committee will vote on y. If y receives at least two votes it will be adopted; if not, the committee will vote on z. If z receives at least two votes it will be adopted; if not, then none of the proposals will be adopted. The committee members’ payoffs are as follows:
A B C
x 2 1 3
y 1 4 2
z 4 3 1
None 3 2 4
Every member of the committee knows everyone’s preferences (e.g., A knows that B likes y best, z second best, and so on). Find the subgame perfect Nash equilibrium of this game.
Explanation / Answer
I would use backwards induction to solve this problem. First we consider what would happen if the result of the game is none. A B & C are all fairly happy, but both A and B would prefer z. So, when it is time to vote on z, both A and B will both vote for z. This means that none will never be the result of the game. Now, what if z is the result of the game? B and C both prefer y. So, when they vote on y, B and C will both vote for y. (If either votes against y, A and B will both vote for z, and B and C are worse off.) Thus, z will never be the result of the game. Finally, what happens when y is the result of the game? A and C both prefer x. So, to prevent y from winning, A and C will both vote to approve x.
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