Math
115

T1
practice Questions

*Note: These questions are all taken from tests that
I have prepared previously. Don’t expect
to see exactly the same questions (or even necessarily exactly the same set of
topics) on this test, but these are good representatives of the types of
problems you can expect to see, and do make a good self-test on these
particular topics. The answers are
included at the end (but make sure you could show the work necessary to find
these answers!).*

If
a problem asks you to EXPLAIN, it should be answered in the context of the
question, using complete English sentences.

*Questions 1 and 2 refer to the *

*preference** schedule shown to the
right:*

1)
a. Find the winner using the __Borda____ count__
method.

b. Find the winner using the __plurality-with-elimination__
method.

2) a. Rank the
candidates using the __extended plurality__ method.

b. Rank the candidates using the __recursive
plurality__ method.

c.
Do any of the methods used in question 1 or 2 violate the majority
criterion for this preference schedule?
Carefully EXPLAIN your answer.

3) For the weighted voting system [100:70,50,30,10] , determine *with
justification *who (if anyone) is a dictator, who (if anyone) has veto
power, and who (if anyone) is a dummy.

4) Find the Banzhaf
power index of each of the players in the previous problem.

5)
Matt and Terry buy a dozen donuts - 6 chocolate, 6 jelly. Matt likes the chocolate donuts twice as much
as jelly. Find *2 different ways* that Matt could fairly (to him) divide the donuts.

(Note: "*different
ways*"* *means a different
number of each type of donut in each pile)

6) Players A, B, C, and D use the method
of markers to divide these 15 objects.

Determine
which objects each player gets.

7) A brother and sister inherit a house, a
cabin, and a vintage car. They agree to
divide the items by the method of sealed bids.
They bid as shown:

a. What does each sibling consider
to be their fair share?

b. What is the final outcome of
this fair division problem?

8) A small town has 4 schools, and 25 school
buses. The schools' enrollments are:

A - |
336 |

B - |
536 |

C - |
236 |

D - |
184 |

The
school buses are apportioned to the schools based on the number of students.

a. Find the standard divisor.

b. EXPLAIN what the standard divisor
means in the context of this problem.

c. How many buses would each
school get by Hamilton's method?

9)
If Jefferson's method were used to apportion the town's buses (from the
previous problem):

a. Find a suitable modified
divisor:

b. Find the modified quota for each
school, and determine how many buses each school gets.

c. Does this violate the quota rule? Carefully EXPLAIN why it does or does not.

10) (continuing the
previous problem)

a. Find a suitable modified divisor
for Webster’s Method.

b. Find the modified quota for each
school, and determine how many buses each school gets.

1) and 2) Borda Count Winner = B

Plurality
with elimination: Eliminate B first, then A.
At that point C has a majority so C wins.

Extended
plurality: D/C/A/B

Recursive
Plurality: D/A /B/C

None
of these violate the majority criterion, as no candidate won a majority of the
1^{st} place votes.

3) and 4) No one has 100+ votes, so there is no
dictator. Players 2, 3 and 4 combined do
not reach 100 votes, so player 1 has veto power.

Player
4 is never critical, so it is a dummy.

The
6 winning coalitions are (naming the players A,B,C,
and D) :

__A, B__

* A, B*, D

__A, C__

* A, C*, D

* A*, B, C

* A*, B, C, D

The
critical players are underlined and shown in italics. A’s power is 6/10 = 60%, B and C are 2/10 =
20% each (D is 0%)

5)
6C and 6J to each, or 5C and 8J to one, 7C 4J to the other. There are 2 other possibilities, too (can you
find them?).

6)C
gets 1 and 2

D gets 5,6,7

B gets 9,10,11

A gets 13,14,15

The leftovers, 3,4,8
and 12 can be divided in any way whatsoever and it will still be fair.

7) a) Bro: 170000,
Sis: 162000.

b)The brother
gets the house and car, pays 96000. The
sister gets the cabin and 96000.

8) Std divisor = 51.68. For every 51.68 students attending a school,
it should get one bus.

Standard quotas: 6.50/10.37/4.57/3.56. Apportionment: A-6 / B-10 / C-5 / D-4.

9) 48 works (there are other choices).

7/
11/ 4/ 3

No,
each gets either its upper standard quota or its lower standard quota (see #8
for the standard quotas).

10) 52 works(there are
other choices).

6 /
10 / 5 /4