Question

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 IV: (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.   

Answer 1

The outcome will be Proposal V.

Explanation

a) Suppose Z chooses X to be the proposal raiser.

Now, X can choose any proposal. But, if he chooses proposals where Y get the minimum share, Y will reject it and no one gets anything. So, X should not choose Proposal II, III. He can choose one among Proposals I,IV, V and VI. The payoffs for X is given below,

Choosing Proposal I will yield X a share of 50

Choosing Proposal IV will yield X a share of 20

Choosing Proposal V will yield X a share of 30

Choosing Proposal VI will yield X a share of 20

So, X will choose only Proposal 1 which will give him the maximum share. Therefore the shares of (X,Y,Z) will be (50,40,10)

Thus, if Z chooses X, Z will get a payoff of only 10. Z knows this.

b) Suppose Z choose Y to be the proposal raiser.

Now, Y can choose any proposal. But, if he chooses proposals where X get the minimum share, X will reject it and no one gets anything. So, Y should not choose Proposal IV, VI. He can choose one among Proposals I,II, III and V. The payoffs for Y is given below,

Choosing Proposal I will yield Y a share of 40

Choosing Proposal II will yield Y a share of 10

Choosing Proposal III will yield Y a share of 20

Choosing Proposal V will yield Y a share of 50

So, Y will choose only Proposal V which will give him the maximum share. Therefore the shares of (X,Y,Z) will be (30,50,20)

Thus, if Z chooses Y, Z will get a payoff of 20. Z knows this.

Thus, Z is better off choosing Y to be the proposal raiser. And Y will choose Proposal V