Math 115

Quiz 2                                                                          Name:                                                             

            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.  Is it possible to choose a quota which results in Junior being a dummy?  If so, find such a quota; if not, explain why not.  Either way, carefully justify your answer in one or more complete sentences.

 

Yes.  If q=5, then there are only 2 winning coalitions are M,D  and M,D,J.  Since the one time that J is part of a winning coalition, the coalition would still even win without J, it isn’t critical.  Never being critical is what makes a player a dummy.

 

 

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

 

M-3

D-2

J-1

M,D – 5

M,J – 4

D,J – 3

M,D,J - 6

 

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

 

MDJ

MJD

DMJ

DJM

JMD

JDM

 

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

 

The winning coalitions from above are copied, with the critical members bolded:

M,D – 5

M,J – 4

M,D,J - 6

 

 M is critical 3 times, 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:   Suppose the dog, the cat, the hamster, and the goldfish are also allowed to vote (so there are a total of 7 players). 

How many coalitions are possible now?  27-1=127

 

                                sequential coalitions?  7! = 5040