Question

A father and his three children decide to hold a vote to select an after dinner activity. They will either see a play or a movie. If the majority of the family agrees on a preferred activity, then they will do that activity. If two family members vote to see a play and the other two family members vote to watch a movie, then (since the father will pay for the activity), the father's vote will be used to break the tie. (a) How many winning coalitions are there in this situation? (b) Find one of the children's Banzhaf power index.

Answer 1

Solution:

A)

Let F=father

A=children 1

B= children 2

C=children 3

List all the winning coalitions

FABC, FAB, FAC, FBC, FA, FB, FC, ABC

Therefore there are 8 winning coalitions in this situation

B)

Let F=father

A=children 1

B= children 2

C=children 3

List all the winning coalitions

FABC, FAB, FAC, FBC, FA, FB, FC, ABC

Now in each coalition identify the critical voters.

A critical voter is a member of the coalition such that, if he changed their vote

from yes to no, the coalition would not be a winning coalition anymore.

In this case the critical voters of each coalition are in bold.

FABC, FAB, FAC, FBC, FA, FB, FC, ABC

There are a total of 12 swing votes. Then the Banzhaf power indices are:

F=6/12 = 1/2

A=2/12 =1/6

B= 2/12 =1/6

C= 2/12 =1/6

The Banzhaf power index. of one of the children is 1/6

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