Questions: Three candidates are running for the mayor of a small town. The candidates are Hu(H), Aponi (A), Ben (B) and Makayla (M). A vote of all town residents was held and the results are summarized in the preference table below. Number of Votes 71 61 57 75 First Choice B M M H Second Choice A H H A Third Choice M B A B Fourth Choice H A B M Using the pairwise comparison method, determine winner of the mayoral race. - Aponi - Ben - Makayla Suppose that, prior to the announcement of the winner, Aponi and Ben decide to drop from the mayoral race. Using the pairwise comparison method, determine winner of the mayoral race after Aponi and Ben are no longer in the running. - Aponi - Ben - Makayla Determine if the irrelevant alternatives criterion is met in this case. Explain why or why not. - Yes, the irrelevant alternatives criterion is satisfied, since the winner of the first election also won the second election, after one of the initial non-winning candidates dropped from the race. - No, the irrelevant alternatives criterion is not satisfied, since the winner of the first election also won the second election, after one of the initial non-winning candidates dropped from the race. - No, the irrelevant alternatives criterion is not satisfied, since the winner of the first election did not win the second election, after one of the initial non-winning candidates dropped from the race.

Three candidates are running for the mayor of a small town. The candidates are Hu(H), Aponi (A), Ben (B) and Makayla (M). A vote of all town residents was held and the results are summarized in the preference table below. Number of Votes 71 61 57 75 First Choice B M M H Second Choice A H H A Third Choice M B A B Fourth Choice H A B M Using the pairwise comparison method, determine winner of the mayoral race. - Aponi - Ben - Makayla Suppose that, prior to the announcement of the winner, Aponi and Ben decide to drop from the mayoral race. Using the pairwise comparison method, determine winner of the mayoral race after Aponi and Ben are no longer in the running. - Aponi - Ben - Makayla Determine if the irrelevant alternatives criterion is met in this case. Explain why or why not. - Yes, the irrelevant alternatives criterion is satisfied, since the winner of the first election also won the second election, after one of the initial non-winning candidates dropped from the race. - No, the irrelevant alternatives criterion is not satisfied, since the winner of the first election also won the second election, after one of the initial non-winning candidates dropped from the race. - No, the irrelevant alternatives criterion is not satisfied, since the winner of the first election did not win the second election, after one of the initial non-winning candidates dropped from the race.
Transcript text: Three candidates are running for the mayor of a small town. The candidates are $\mathrm{Hu}(\mathrm{H})$, Aponi $(\mathrm{A})$, Ben (B) and Makayla ( $M$ ). A vote of all town residents was held and the results are summarized in the preference table below. \begin{tabular}{|c|c|c|c|c|} \hline Number of Votes & 71 & $\mathbf{6 1}$ & $\mathbf{5 7}$ & $\mathbf{7 5}$ \\ \hline First Choice & B & M & M & H \\ \hline Second Choice & A & H & H & A \\ \hline Third Choice & M & B & A & B \\ \hline Fourth Choice & H & A & B & M \\ \hline \end{tabular} Using the pairwise comparison method, determine winner of the mayoral race. Aponi Ben Makayla Suppose that, prior to the announcement of the winner, Aponi and Ben decide to drop from the mayoral race. Using the pairwise comparison method, determine winner of the mayoral race after Aponi and Ben are no longer in the running. Aponi Ben Makayla Determine if the irrelevant alternatives criterion is met in this case. Explain why or why not. Yes, the irrelevant alternatives criterion is satisfied, since the winner of the first election also won the second election, after one of the initial non-winning candidates dropped from the race. No, the irrelevant alternatives criterion is not satisfied, since the winner of the first election also won the second election, after one of the initial non-winning candidates dropped from the race. No, the irrelevant alternatives criterion is not satisfied, since the winner of the first election did not win the second election, after one of the initial non-winning candidates dropped from the race.
failed

Solution

failed
failed

Solution Steps

Solution Approach
  1. Pairwise Comparison Method: For each pair of candidates, compare the number of votes where one candidate is preferred over the other. The candidate with the most pairwise wins is the winner.
  2. Dropping Candidates: After Aponi and Ben drop out, repeat the pairwise comparison with the remaining candidates.
  3. Irrelevant Alternatives Criterion: Check if the winner changes when candidates drop out. If the winner remains the same, the criterion is satisfied.
Step 1: Initial Pairwise Comparison

Using the pairwise comparison method among the candidates \( H \), \( A \), \( B \), and \( M \), we find that the winner is \( H \). This is determined by comparing the number of votes each candidate receives against each other in head-to-head matchups.

Step 2: Candidates After Drop

After Aponi \( (A) \) and Ben \( (B) \) drop out of the race, we are left with candidates \( H \) and \( M \). A new pairwise comparison is conducted between these two candidates.

Step 3: Winner After Drop

In the pairwise comparison between \( H \) and \( M \), the winner is \( M \). This indicates that when the non-winning candidates are removed, the outcome of the election changes.

Step 4: Irrelevant Alternatives Criterion

The irrelevant alternatives criterion is not satisfied in this case because the winner of the initial election \( (H) \) did not win the second election \( (M) \) after the drop of candidates \( A \) and \( B \).

Final Answer

  • Winner of the initial election: \( H \)
  • Winner after Aponi and Ben drop out: \( M \)
  • Irrelevant alternatives criterion satisfied: No

Thus, the final answers are:

  • Winner of the first election: \\(\boxed{H}\\)
  • Winner after drop: \\(\boxed{M}\\)
  • Irrelevant alternatives criterion: \\(\text{No}\\)
Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
What is the first step of the accounting cycle?
failedAnswered
x + 2y = 6 3x - y = -10
failedAnswered
A 0.490 -m-long guitar string, of cross-sectional area 1.00 x 10^-6 m^2, has Young's modulus Y=2.00 GPa. By how much must you stretch the string to obtain a tension of 16.1 N?
failedAnswered
Use an algebraic approach to find the measure of the two angles described below. Begin by letting x represent the degree measure of the angle complement. What is the measure of the complement? What is the measure of the angle? The measure of the angle is 76 degrees greater than its complement.
failedAnswered
Determine what the key terms refer to in the following survey. To determine the average number of classes Oxnard College students were taking, a group of 35 Oxnard College students were given and answered the following survey. How many classes you are taking this semester. 01 02 - 3 04 Sample Population Statistic Parameter a. In average, all students in Oxnard College were taking 2 classes b. The 35 Oxnard College students who were surveyed c. All students in Oxnard College d. In average, those 35 students were taking 2.6 classes