Math 115

Quiz 2 Name:
__ key __

**YOU MUST SHOW ALL WORK TO RECEIVE CREDIT**

In family voting, Mom (M) gets 3 votes, Dad (D) gets 2 votes and Junior (J) gets 1 vote.

1. Find the smallest and largest possible quotas for this weighted voting system.

*Since there are a
total of 4 votes, a majority must be greater than 3, so quota must be at least
4. The largest possible quota is when
all votes are required, so 6.*

2. List all possible coalitions for the family voting system, and give the weight of each.

*There are 7 possible
coalitions: {M}, {D}, {J}, {M,D}, {M,J}, {D,J}, {M,D,J}*

3. List all sequential coalitions for the family voting system.

*There are 6: {M,D,J}, {M,J,D}, {D,M,J}, {D,J,M}, {J,M,D},
{J,D,M}*

4. Find the * Banzhaf*
Power Index for each family member if the quota is set to 4.

*{M} = 3 votes*

*{D} = 2 votes*

*{J} = 1 vote*

*{ M,D} = 5 votes*

*{ M,J} = 4 votes *

*{D,J}=3
votes*

*{ M,D,J}=6 votes*

*Only the coalitions in
red are winning coalitions, so they are the only ones which can have critical
members.*

*The critical members
are underlined. *

*M is critical 3 times,
while D and J are critical once each. So
M has 3/5 = 60% of the power, while D and J each have 1/5 = 20% of the power.*

**Extra Credit**: Is it possible to pick a
quota that results in Junior being a dummy? If so, find one. If not, explain why not. In either case, carefully explain your answer
in 1 or 2 complete English sentences.

*Yes, if the quota is 5 there
are only 2 winning coalitions: {M,D} and {M,D,J}. J is
not a member of the first winning coalition, and is not critical in the 2 ^{nd},
so he is never critical, making him a dummy.*