To solve this problem using the pairwise comparison method, we need to compare each option against every other option to determine which one is preferred based on the ballots. Here's the process:

1. List all options:

   • D (Dining facilities)
   • A (Athletic facilities)
   • C (Campus security)
   • S (Student union building)

2. Compare each pair of options using the table:

   • For each pair, count how many ballots rank one option higher than the other. The option with more votes in the comparison scores 1 point. If there's a tie, both score 0.5 points.

3. Step-by-step pairwise comparison:

   • Compare D vs A
   • Compare D vs C
   • Compare D vs S
   • Compare A vs C
   • Compare A vs S
   • Compare C vs S

4. Add up the total scores for each option.

I'll process these calculations now.### Step 1: Pairwise Comparisons Using the preference table, compare the number of ballots ranking one option higher than another for each pair.

1. D vs A

• D ranks higher on: 24 (2nd place) + 9 (1st place) + 7 (2nd place) = 40 ballots.
• A ranks higher on: 19 (1st place) + 12 (1st place) = 31 ballots.
  Winner: D (1 point).

2. D vs C

• D ranks higher on: 19 (2nd place) + 9 (1st place) + 7 (2nd place) = 35 ballots.
• C ranks higher on: 24 (1st place) + 12 (3rd place) = 36 ballots.
  Winner: C (1 point).

3. D vs S

• D ranks higher on: 24 (1st place) + 12 (3rd place) + 9 (2nd place) = 45 ballots.
• S ranks higher on: 19 (2nd place) + 7 (4th place) = 26 ballots.
  Winner: D (1 point).

4. A vs C

• A ranks higher on: 19 (1st place) + 12 (1st place) + 7 (2nd place) = 38 ballots.
• C ranks higher on: 24 (1st place) + 9 (3rd place) = 33 ballots.
  Winner: A (1 point).

5. A vs S

• A ranks higher on: 19 (1st place) + 12 (1st place) + 24 (3rd place) = 55 ballots.
• S ranks higher on: 9 (2nd place) + 7 (4th place) = 16 ballots.
  Winner: A (1 point).

6. C vs S

• C ranks higher on: 24 (1st place) + 19 (2nd place) + 12 (3rd place) = 55 ballots.
• S ranks higher on: 9 (2nd place) + 7 (4th place) = 16 ballots.
  Winner: C (1 point).

Step 2: Total Scores

• D: Wins against A (1), loses to C (0), wins against S (1) → Total = 2 points.
• A: Loses to D (0), wins against C (1), wins against S (1) → Total = 2 points.
• C: Wins against D (1), loses to A (0), wins against S (1) → Total = 2 points.
• S: Loses to all → Total = 0 points.

Step 3: Decision

Since there is a tie between D, A, and C, additional tiebreaking criteria may be required. However, the pairwise comparison method alone does not directly resolve ties. Based on the table and votes, the decision could focus on Campus Security (C) since it ranks high in direct comparisons.

Let me know if you'd like clarification!

Related Questions:

1. How do you use other voting methods like the Borda count to resolve ties?
2. Why is pairwise comparison preferred in decision-making processes?
3. Can ties be resolved by considering 2nd-place votes in pairwise comparisons?
4. What are practical applications of pairwise comparisons in real life?
5. How can alternative rankings affect the outcome in this voting method?

Tip: Always double-check total ballots in pairwise comparison to ensure no miscount in larger datasets.