QuestionDetermine the election winner using Borda count. Check if majority and head-to-head criteria are met.

Studdy Solution

STEP 1

Assumptions1. The preference table is given as follows\begin{tabular}{|l|c|c|c|c|c|} \hline $\begin{array}{l}\text { Number } \\ \text { of Votes }\end{array}$ &48 &42 &36 &12 &6 \\ \hline $\begin{array}{l}\text {1st } \\ \text { Choice }\end{array}$ & & C & C & B & D \\ \hline $\begin{array}{l}\text {nd } \\ \text { Choice }\end{array}$ & A & A & A & & \\ \hline $\begin{array}{l}\text {3rd } \\ \text { Choice }\end{array}$ & B & & D & A & C \\ \hline $\begin{array}{l}\text {4th } \\ \text { Choice }\end{array}$ & C & B & B & C & A \\ \hline $\begin{array}{l}\text {5th } \\ \text { Choice }\end{array}$ & D & D & & D & B \\ \hline\end{tabular}
. The Borda count method assigns points to each candidate based on their position in the voters' preference list. The candidate with the highest total points wins.
3. The majority criterion is satisfied if the candidate who is the first choice of more than half of the voters wins.
4. The head-to-head criterion is satisfied if the candidate who would win a two-candidate election against each of the other candidates wins.

STEP 2

First, we need to calculate the Borda count for each candidate. The Borda count is calculated by assigning points to each candidate based on their position in the voters' preference list. The candidate with the highest total points wins.
The points are assigned as follows1st choice5 points2nd choice4 pointsrd choice points4th choice2 points5th choice1 point

STEP 3

Calculate the Borda count for each candidate.
For candidate $Borda\, count\, for\, = (1st\, choice\, votes \times5\, points) + (2nd\, choice\, votes \times\, points) + (3rd\, choice\, votes \times3\, points) + (th\, choice\, votes \times2\, points) + (5th\, choice\, votes \times1\, point)$ For candidate C $Borda\, count\, for\, C = (1st\, choice\, votes \times5\, points) + (2nd\, choice\, votes \times\, points) + (3rd\, choice\, votes \times3\, points) + (th\, choice\, votes \times2\, points) + (5th\, choice\, votes \times1\, point)$ Repeat this process for candidates A, B, and D.

STEP 4

Plug in the given values for the number of votes and calculate the Borda count for each candidate.

STEP 5

The candidate with the highest Borda count is the winner.

STEP 6

To check if the majority criterion is satisfied, compare the number of first choice votes for the winning candidate with half of the total number of votes. If the number of first choice votes for the winning candidate is more than half of the total number of votes, then the majority criterion is satisfied.

STEP 7

To check if the head-to-head criterion is satisfied, compare the number of votes for the winning candidate with the number of votes for each of the other candidates in a two-candidate election. If the winning candidate would win a two-candidate election against each of the other candidates, then the head-to-head criterion is satisfied.

STEP 8

Provide the answers to the questionsA. Using the Borda count method, the winner is .
B. The majority criterion is (satisfied/not satisfied) because .
C. The head-to-head criterion is (satisfied/not satisfied) because __.

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