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Sagot :
To determine the winner using the Pairwise Comparison method, we need to compare each pair of candidates and count how many voters prefer one candidate over the other. Here is the step-by-step comparison and tallying:
#### Pairwise Comparisons:
1. A vs. B:
- Votes where A is preferred to B: [tex]\( 143 \)[/tex]
- Votes where B is preferred to A: [tex]\( 41 \)[/tex]
2. A vs. C:
- Votes where A is preferred to C: [tex]\( 110 \)[/tex]
- Votes where C is preferred to A: [tex]\( 74 \)[/tex]
3. A vs. D:
- Votes where A is preferred to D: [tex]\( 144 \)[/tex]
- Votes where D is preferred to A: [tex]\( 40 \)[/tex]
4. B vs. C:
- Votes where B is preferred to C: [tex]\( 64 \)[/tex]
- Votes where C is preferred to B: [tex]\( 120 \)[/tex]
5. B vs. D:
- Votes where B is preferred to D: [tex]\( 113 \)[/tex]
- Votes where D is preferred to B: [tex]\( 71 \)[/tex]
6. C vs. D:
- Votes where C is preferred to D: [tex]\( 74 \)[/tex]
- Votes where D is preferred to C: [tex]\( 110 \)[/tex]
#### Tallying the Results:
Based on the pairwise comparisons, we can assign points as follows:
- Candidate A:
- Wins against B: 1 point
- Wins against C: 1 point
- Wins against D: 1 point
- Total points for A: [tex]\( 3 \)[/tex]
- Candidate B:
- Loses against A: 0 points
- Loses against C: 0 points
- Wins against D: 1 point
- Total points for B: [tex]\( 2 \)[/tex]
- Candidate C:
- Loses against A: 0 points
- Wins against B: 1 point
- Loses against D: 0 points
- Total points for C: [tex]\( 0 \)[/tex]
- Candidate D:
- Loses against A: 0 points
- Wins against B: 1 point
- Wins against C: 1 point
- Total points for D: [tex]\( 1 \)[/tex]
#### Determining the Winner:
The candidate with the most points is declared the winner.
- Points for A: [tex]\( 3 \)[/tex]
- Points for B: [tex]\( 2 \)[/tex]
- Points for C: [tex]\( 0 \)[/tex]
- Points for D: [tex]\( 1 \)[/tex]
#### The Winner:
The winner of the election is Candidate A.
#### Pairwise Comparisons:
1. A vs. B:
- Votes where A is preferred to B: [tex]\( 143 \)[/tex]
- Votes where B is preferred to A: [tex]\( 41 \)[/tex]
2. A vs. C:
- Votes where A is preferred to C: [tex]\( 110 \)[/tex]
- Votes where C is preferred to A: [tex]\( 74 \)[/tex]
3. A vs. D:
- Votes where A is preferred to D: [tex]\( 144 \)[/tex]
- Votes where D is preferred to A: [tex]\( 40 \)[/tex]
4. B vs. C:
- Votes where B is preferred to C: [tex]\( 64 \)[/tex]
- Votes where C is preferred to B: [tex]\( 120 \)[/tex]
5. B vs. D:
- Votes where B is preferred to D: [tex]\( 113 \)[/tex]
- Votes where D is preferred to B: [tex]\( 71 \)[/tex]
6. C vs. D:
- Votes where C is preferred to D: [tex]\( 74 \)[/tex]
- Votes where D is preferred to C: [tex]\( 110 \)[/tex]
#### Tallying the Results:
Based on the pairwise comparisons, we can assign points as follows:
- Candidate A:
- Wins against B: 1 point
- Wins against C: 1 point
- Wins against D: 1 point
- Total points for A: [tex]\( 3 \)[/tex]
- Candidate B:
- Loses against A: 0 points
- Loses against C: 0 points
- Wins against D: 1 point
- Total points for B: [tex]\( 2 \)[/tex]
- Candidate C:
- Loses against A: 0 points
- Wins against B: 1 point
- Loses against D: 0 points
- Total points for C: [tex]\( 0 \)[/tex]
- Candidate D:
- Loses against A: 0 points
- Wins against B: 1 point
- Wins against C: 1 point
- Total points for D: [tex]\( 1 \)[/tex]
#### Determining the Winner:
The candidate with the most points is declared the winner.
- Points for A: [tex]\( 3 \)[/tex]
- Points for B: [tex]\( 2 \)[/tex]
- Points for C: [tex]\( 0 \)[/tex]
- Points for D: [tex]\( 1 \)[/tex]
#### The Winner:
The winner of the election is Candidate A.
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