Please solve the following game: Assume that a total $100 grant will be shared by the three researchers, X, Y, and Z. Each person is rational and selfish. There are six proposals with different shares of (X, Y, Z) for choices as the following.
Proposal I: (X, Y, Z) = (50, 40, 10)
Proposal II: (X, Y, Z) = (60, 10, 30)
Proposal III: (X, Y, Z) = (40, 20, 40)
Proposal VI: (X, Y, Z) = (20, 30, 50)
Proposal V: (X, Y, Z) = (30, 50, 20)
Proposal VI: (X, Y, Z) = (20, 50, 30)
The rule of choosing the final proposal is simple. First, Z is the person to determine who (either X or Y) is the proposal raiser. Then the proposal raiser chooses a particular proposal. Finally, the last person has the right to pass it or reject it. If the last person’s payoff is the smallest among the three, then the proposal will be rejected and no one will get anything. The decision making process can be done by only one time. Which proposal will be the final outcome? Explain the decision briefly.