Questions: The preference table for an election is given. Use the table to answer the questions below: A. Using the Borda count method, who is the winner? B. Is the majority criterion satisfied? Why or why not? Number of Votes: 120, 96, 60, 24 1st Choice: B, A, A, C 2nd Choice: C, C, D, D 3rd Choice: D, D, B, B 4th Choice: A, B, C, A

Solution Steps

To solve the problem using the Borda count method, we need to assign points to each candidate based on their ranking in each column of the preference table. The candidate ranked first receives the most points, and the candidate ranked last receives the least. We then sum the points for each candidate across all columns to determine the winner. For the majority criterion, we check if any candidate has more than half of the first-choice votes.

To determine the winner using the Borda count method, we assign points to each candidate based on their ranking in each column of the preference table. The candidate ranked first receives 4 points, the second receives 3 points, the third receives 2 points, and the fourth receives 1 point. We then sum the points for each candidate across all columns.

• For candidate \( A \): \[ \text{Points} = 3 \times 120 + 4 \times 96 + 4 \times 60 + 1 \times 24 = 768 \]

• For candidate \( B \): \[ \text{Points} = 4 \times 120 + 1 \times 96 + 2 \times 60 + 2 \times 24 = 744 \]

• For candidate \( C \): \[ \text{Points} = 2 \times 120 + 3 \times 96 + 1 \times 60 + 4 \times 24 = 804 \]

• For candidate \( D \): \[ \text{Points} = 1 \times 120 + 2 \times 96 + 3 \times 60 + 3 \times 24 = 684 \]

The candidate with the highest Borda score is the winner. From the calculations:

• \( A: 768 \)
• \( B: 744 \)
• \( C: 804 \)
• \( D: 684 \)

The candidate with the highest score is \( C \) with 804 points.

The majority criterion is satisfied if a candidate receives more than half of the first-choice votes. The total number of votes is:

\[ \text{Total Votes} = 120 + 96 + 60 + 24 = 300 \]

The first-choice votes are:

• \( A: 96 + 60 = 156 \)
• \( B: 120 \)
• \( C: 24 \)
• \( D: 0 \)

To satisfy the majority criterion, a candidate must have more than \( \frac{300}{2} = 150 \) first-choice votes. Candidate \( A \) has 156 first-choice votes, which satisfies the majority criterion.

Final Answer