Question

number of voters | 12 | 9 | 10 | 25 | 9
1st choice | b | d | b | c | a
2nd choice | c | a | d | b | b
3rd choice | a | b | c | a | c
4th choice | d | c | a | d | d

based on the sample of votes recorded:

who is the winner of the election using the plurality method?

winner is

is there a condorcet winner of the election? enter none if there is no condorcet winner.

winner is

Explanation:

First Sub - Question: Plurality Method Winner

Step 1: Count 1st - choice votes for each candidate

• For candidate A: The number of voters with A as 1st choice is 9.
• For candidate B: The number of voters with B as 1st choice is \(12 + 10=22\).
• For candidate C: The number of voters with C as 1st choice is 25.
• For candidate D: The number of voters with D as 1st choice is 9.

Step 2: Determine the maximum number of 1st - choice votes

We compare the number of 1st - choice votes: \(9\) (A), \(22\) (B), \(25\) (C), \(9\) (D). The maximum number of 1st - choice votes is 25, which corresponds to candidate C.

A Condorcet winner is a candidate who would win a head - to - head election against every other candidate.

1. Compare C vs A:

• To find the number of voters who prefer C over A: We look at the ballots where C is ranked higher than A.
• In the first group (12 voters): 1st choice B, 2nd choice C, 3rd choice A. So C > A here.
• In the second group (9 voters): 1st choice D, 2nd choice A, 3rd choice B, 4th choice C. So A > C here.
• In the third group (10 voters): 1st choice B, 2nd choice D, 3rd choice C, 4th choice A. So C > A here.
• In the fourth group (25 voters): 1st choice C, 2nd choice B, 3rd choice A, 4th choice D. So C > A here.
• In the fifth group (9 voters): 1st choice A, 2nd choice B, 3rd choice C, 4th choice D. So A > C here.
• Number of voters preferring C over A: \(12 + 10+25 = 47\)
• Number of voters preferring A over C: \(9 + 9=18\)
• Since \(47>18\), C beats A.

1. Compare C vs B:

• Voters preferring C over B:
• First group (12 voters): 2nd choice C, 1st choice B. So B > C here.
• Second group (9 voters): 3rd choice B, 1st choice D, 2nd choice A. So A > B > C? No, 1st choice D, 2nd choice A, 3rd choice B, 4th choice C. So B > C here.
• Third group (10 voters): 3rd choice C, 1st choice B, 2nd choice D. So B > C here.
• Fourth group (25 voters): 2nd choice B, 1st choice C. So C > B here.
• Fifth group (9 voters): 2nd choice B, 1st choice A, 3rd choice C. So B > C here.
• Number of voters preferring C over B: \(25\)
• Number of voters preferring B over C: \(12 + 9+10 + 9=40\)
• Since \(40>25\), B beats C.

Since C does not beat B in a head - to - head comparison, there is no Condorcet winner.

Answer:

C

Second Sub - Question: Condorcet Winner