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.

{J} = 1 vote

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 2nd, so he is never critical, making him a dummy.