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 isnt 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? 2^{7}-1=127

sequential
coalitions? 7! = 5040