math-#

MAT1009 Exam 1_v2

MAT1009 Exam 1 Name ___________________________________ Given the following preference ballot, solve problems 1 through 4.

A math club is voting for president. There are 4 candidates: Alisha (A), Boris (B), Carmen (C), and Dave (D).

Number of voters 14 10 8 4 1

1st A B D B C

2nd B C C D D

3rd C D B A B

4th D A A C A

1. Who wins with the plurality method?

2. Who wins with the Borda count method?

MAT1009 Exam 1_v2

3. Who wins with Instant Runoff Voting (Plurality with Elimination)?

4. Who wins with the Copeland’s method (Method of pairwise comparisons)?

MAT1009 Exam 1_v2

5. Find the Banzhaf power index of [6: 4, 3, 1]

6. Find the Shapely-Shubik power index of [6: 4, 3, 1]

MAT1009 Exam 1_v2

For problems 7 thru 10, use the following information.

Parador is a small republic located in Central America and consists of 6 states: Azucar (A), Bahia (B), Café (C), Diamante (D), Esmeralda (E), and Felicidad (F). There are 250 seats in the Congress to be apportioned among the 6 states in proportion to their respective populations, given below.

State A B C D E F Total___

Population 2,000,000 7,000,000 150,000 2,000,000 700,000 1,000,000 12,850,000

7. Use Hamilton’s method to find the apportionment.

MAT1009 Exam 1_v2

8. Use Jefferson’s method to find the apportionment.

MAT1009 Exam 1_v2

9. Use Webster's Method to find the apportionment.

MAT1009 Exam 1_v2

10. Use the Huntington-Hill method to find the apportionment.

MAT1009 Exam 1_v2

11. Suppose 4 classmates are splitting equally a $16 pizza. What is each person's fair share?

12. Damian and Cleo are given a half-chocolate, half-strawberry cheesecake. Damian likes chocolate and strawberry equally well, so in his eyes the chocolate and strawberry halves are equal in value. Cleo doesn’t like chocolate, so in her eyes the value of the cake is 20% chocolate and 80% strawberry. To ensure a fair division, assume neither of them knows anything about the other's likes and dislikes. Use the Divider-Chooser method to describe a fair division. Damian volunteers to go first and be the divider. He divides the cheesecake chocolate half as shown in the diagram. (Damian cut the cheesecake so that each half contains a chocolate slice and a strawberry slice.) Which half does Cleo pick and why?

MAT1009 Exam 1_v2

13. Three brothers, say Henry, Tom , and Fred, are splitting up their partnership and dividing up a bunch of assets of unspecified value. The set of assets are divided into three shares called s1, s2, and s3. Below is a table how each of the brothers values each share. Find a fair division for each brother where Henry starts first.

s1 s2 s3

Henry 30% 30% 40%

Tom 35% 30% 35%

Fred 33 1 3 % 33

1 3 % 33

1 3 %

MAT1009 Exam 1_v2

14. Bill and Alice were given a chocolate cake and a cherry pie, and need to divide them. Bill values the chocolate cake at $5.00 and the cherry pie at $5.00. Alice values the chocolate cake at $4.00 and the cherry pie at $10.00. Describe a fair division if Alice is dividing, and specify which "half" Bill will choose.

MAT1009 Exam 1_v2

15. Al and Betty (brother and sister) have inherited their parents' house. They agree to give the method of sealed bids a try in splitting up the house. Al's bid is $300,000 and Betty's bid is $400,000. Determine the final allocation.