Question

social choice: do we all agree assignment

1. suppose an election is held to determine which bag of candy will be opened. the choices (candidates) are hershey’s miniatures (m), nestle crunch (c), and mars’ snickers (s). use the preference schedule from the candy election to find the winner using the borda count method. third choice receives one point, second choice receives two points, and first choice receives three points.

preference schedule for the candy election
number of voters: 3 1 4 1 9
1st choice: m m c c s
2nd choice: c s m s m
3rd choice: s c s m c

1. is the winner in question 1 also the majority rule winner? why or why not justify your answer.

Explanation:

Question 1 Solution (Step-by-Step Format)

Step1: Identify Voter Groups

We have 5 groups of voters with counts: 3, 1, 4, 1, 9. Let's list their preferences:

• Group 1 (3 voters): 1st: M, 2nd: C, 3rd: S
• Group 2 (1 voter): 1st: M, 2nd: S, 3rd: C
• Group 3 (4 voters): 1st: C, 2nd: S, 3rd: M
• Group 4 (1 voter): 1st: C, 2nd: S, 3rd: M (Wait, no, original 2nd choice for group 4? Wait, the preference schedule: 1st Choice: M, M, C, C, S (for 5 groups? Wait, the number of voters: 3,1,4,1,9. So 5 groups. 1st Choice: M (3), M (1), C (4), C (1), S (9). 2nd Choice: C (3), S (1), S (4), S (1), M (9). 3rd Choice: S (3), C (1), M (4), M (1), C (9).

Step2: Calculate Borda Points for M

For M:

• 1st choice points: (3 + 1) 3 = 4 3 = 12
• 2nd choice points: 9 * 2 = 18 (from the last group where M is 2nd)
• 3rd choice points: (4 + 1) 1 = 5 1 = 5

Total for M: 12 + 18 + 5 = 35

Step3: Calculate Borda Points for C

For C:

• 1st choice points: (4 + 1) 3 = 5 3 = 15
• 2nd choice points: 3 * 2 = 6 (from first group where C is 2nd)
• 3rd choice points: 9 * 1 = 9 (from last group where C is 3rd)

Total for C: 15 + 6 + 9 = 30

Step4: Calculate Borda Points for S

For S:

• 1st choice points: 9 * 3 = 27
• 2nd choice points: (1 + 4 + 1) 2 = 6 2 = 12 (groups 2,3,4)
• 3rd choice points: 3 1 = 3 (from first group where S is 3rd) + 1 1 = 1 (from group 2 where S is 3rd? Wait no, group 2: 3rd choice is C, so S's 3rd choice? Wait, let's re-express:

Wait, correct way: For each candidate, count how many times they are 1st, 2nd, 3rd, multiply by points (3,2,1) and sum.

Group 1 (3 voters): 1st: M (3), 2nd: C (3), 3rd: S (3)
Group 2 (1 voter): 1st: M (1), 2nd: S (1), 3rd: C (1)
Group 3 (4 voters): 1st: C (4), 2nd: S (4), 3rd: M (4)
Group 4 (1 voter): 1st: C (1), 2nd: S (1), 3rd: M (1)
Group 5 (9 voters): 1st: S (9), 2nd: M (9), 3rd: C (9)

Now, M:

• 1st: 3 + 1 = 4 voters, 4 * 3 = 12
• 2nd: 9 voters, 9 * 2 = 18
• 3rd: 4 + 1 = 5 voters, 5 * 1 = 5

Total M: 12 + 18 + 5 = 35

C:

• 1st: 4 + 1 = 5 voters, 5 * 3 = 15
• 2nd: 3 voters, 3 * 2 = 6
• 3rd: 1 + 9 = 10 voters, 10 * 1 = 10

Total C: 15 + 6 + 10 = 31

S:

• 1st: 9 voters, 9 * 3 = 27
• 2nd: 1 + 4 + 1 = 6 voters, 6 * 2 = 12
• 3rd: 3 voters, 3 * 1 = 3

Total S: 27 + 12 + 3 = 42

Wait, wait, I made a mistake earlier. Let's recalculate S:

Group 1: 3rd choice S: 3*1=3

Group 2: 2nd choice S: 1*2=2

Group 3: 2nd choice S: 4*2=8

Group 4: 2nd choice S: 1*2=2

Group 5: 1st choice S: 9*3=27

Wait, no, the 2nd choice for each group:

Group 1 (3 voters): 2nd choice C → so S is 3rd: 3*1

Group 2 (1 voter): 2nd choice S → 1*2

Group 3 (4 voters): 2nd choice S → 4*2

Group 4 (1 voter): 2nd choice S → 1*2

Group 5 (9 voters): 2nd choice M → so S is 1st: 9*3

3rd choice for S: In group 1, S is 3rd (3*1). In group 2, S is 2nd (no, 3rd choice for group 2 is C, so S is 2nd. Wait, the 3rd choice is the last. So for each group:

Group 1: 1st M, 2nd C, 3rd S → S gets 1 point (3 voters)

Group 2: 1st M, 2nd S, 3rd C → S gets 2 points (1 voter)

Group 3: 1st C, 2nd S, 3rd M → S gets 2 points (4 voters)

Group 4: 1st C, 2nd S, 3rd M → S gets 2 points (1 voter)

Group 5: 1st S, 2nd M, 3rd C → S gets 3 points (9 voters)

So S's points:

1st choice: 9 * 3 = 27

2nd choice: (1 + 4 + 1) 2 = 6 2 = 12

3rd choice: 3 * 1 = 3

Total: 27 + 12 + 3 = 42

M's points:

1st choice: (3 + 1) 3 = 4 3 = 12

2nd choice: 9 * 2 = 18 (group 5, M is 2nd)

3rd choice: (4 + 1) 1 = 5 1 = 5 (groups 3 and 4, M is 3rd)

Total: 12 + 18 + 5 = 35

C's points:

1st choice: (4 + 1) 3 = 5 3 = 15 (groups 3 and 4, C is 1st)

2nd choice: 3 * 2 = 6 (group 1, C is 2nd)

3rd choice: (1 + 9)…

To determine Majority Rule winner, we need more than 50% of first - choice votes. Total voters: 3 + 1 + 4 + 1 + 9 = 18. Majority threshold: $\lceil\frac{18}{2}
ceil= 9 + 1=10$ (since 18/2 = 9, need at least 10).

First - choice votes:

• M: 3 + 1 = 4
• C: 4 + 1 = 5
• S: 9

S has 9 first - choice votes, which is less than 10. So no candidate has a majority. The Borda winner is S, but since no candidate has a majority (S has 9 < 10, M has 4, C has 5), the “Majority Rule winner” doesn't exist (or no candidate has a majority). So the Borda winner (S) is not a Majority Rule winner because no candidate received a majority (more than 9 votes, since total voters are 18, majority is at least 10) of first - choice votes.

Answer:

Mars’ Snickers (S) is the winner using the Borda Count Method.

Question 2 Solution (Answer-Explanation Format)