Questions: Number of voters 40 32 22 10 28 18 1st choice C C A B D A 2nd choice B A B C A D 3rd choice A B D A B B 4th choice D D C D C C Find the winner of the election using the Pairwise Comparison method. For each comparison, enter the number of times each candidate was preferred to the other. Avs. B Votes where A is preferred to B : Votes where B is preferred to A : Avs. C Votes where A is preferred to C : Votes where C is preferred to A : Avs. D Votes where A is preferred to D : Votes where D is preferred to A : B vs. C Votes where B is preferred to C : Votes where C is preferred to B : B vs. D Votes where B is preferred to D : Votes where D is preferred to B : C vs. D Votes where C is preferred to D : Votes where D is preferred to C :

Number of voters 40 32 22 10 28 18
1st choice C C A B D A
2nd choice B A B C A D
3rd choice A B D A B B
4th choice D D C D C C

Find the winner of the election using the Pairwise Comparison method. For each comparison, enter the number of times each candidate was preferred to the other.

Avs. B Votes where A is preferred to B : Votes where B is preferred to A :

Avs. C Votes where A is preferred to C : Votes where C is preferred to A :

Avs. D Votes where A is preferred to D : Votes where D is preferred to A :

B vs. C Votes where B is preferred to C : Votes where C is preferred to B :

B vs. D Votes where B is preferred to D : Votes where D is preferred to B :

C vs. D Votes where C is preferred to D : Votes where D is preferred to C :

Transcript text: Number of voters & 40 & 32 & 22 & 10 & 28 & 18 \\ 1st choice & C & C & A & B & D & A \\ 2nd choice & B & A & B & C & A & D \\ 3rd choice & A & B & D & A & B & B \\ 4th choice & D & D & C & D & C & C \\ Find the winner of the election using the Pairwise Comparison method. For each comparison, enter the number of times each candidate was preferred to the other. Avs. B Votes where $A$ is preferred to $B$ : $\square$ Votes where $B$ is preferred to $A$ : $\square$ Avs. C Votes where $A$ is preferred to $C$ : $\square$ Votes where $C$ is preferred to $A$ : $\square$ Avs. D Votes where $A$ is preferred to $D$ : $\square$ Votes where $D$ is preferred to $A$ : $\square$ B vs. C Votes where $B$ is preferred to $C$ : $\square$ Votes where $C$ is preferred to $B$ : $\square$ B vs. D Votes where $B$ is preferred to $D$ : $\square$ Votes where $D$ is preferred to $B$ : $\square$ C vs. D Votes where $C$ is preferred to $D$ : $\square$ Votes where $D$ is preferred to $C$ : $\square$

Solution

Solution Steps

Step 1: Count Preferences for Each Pair

We conducted pairwise comparisons between the candidates A, B, C, and D. The results of the comparisons are as follows:

For $ A $ vs $ B $:
- Votes where $ A $ is preferred to $ B $: $ 100 $
- Votes where $ B $ is preferred to $ A $: $ 50 $
For $ A $ vs $ C $:
- Votes where $ A $ is preferred to $ C $: $ 68 $
- Votes where $ C $ is preferred to $ A $: $ 82 $
For $ A $ vs $ D $:
- Votes where $ A $ is preferred to $ D $: $ 122 $
- Votes where $ D $ is preferred to $ A $: $ 28 $
For $ B $ vs $ C $:
- Votes where $ B $ is preferred to $ C $: $ 78 $
- Votes where $ C $ is preferred to $ B $: $ 72 $
For $ B $ vs $ D $:
- Votes where $ B $ is preferred to $ D $: $ 104 $
- Votes where $ D $ is preferred to $ B $: $ 46 $
For $ C $ vs $ D $:
- Votes where $ C $ is preferred to $ D $: $ 82 $
- Votes where $ D $ is preferred to $ C $: $ 68 $

Step 2: Tally Overall Preferences

Next, we tally the overall preferences for each candidate based on the pairwise results:

Total preferences for $ A $:
- Preferred over $ B $: $ 100 $
- Preferred over $ C $: $ 68 $
- Preferred over $ D $: $ 122 $
- Total: $ 100 + 68 + 122 = 290 $
Total preferences for $ B $:
- Preferred over $ A $: $ 50 $
- Preferred over $ C $: $ 78 $
- Preferred over $ D $: $ 104 $
- Total: $ 50 + 78 + 104 = 232 $
Total preferences for $ C $:
- Preferred over $ A $: $ 82 $
- Preferred over $ B $: $ 72 $
- Preferred over $ D $: $ 68 $
- Total: $ 82 + 72 + 68 = 222 $
Total preferences for $ D $:
- Preferred over $ A $: $ 28 $
- Preferred over $ B $: $ 46 $
- Preferred over $ C $: $ 68 $
- Total: $ 28 + 46 + 68 = 142 $