QuestionCreate a preference table for candidates A, B, C. Show how the plurality-with-elimination method violates monotonicity. Votes: A=12, B=8, C=6; after eliminating C: A=14, B=12.
Studdy Solution
STEP 1
Assumptions1. The initial votes are split as follows Candidate A gets12 votes, Candidate B gets8 votes and Candidate C gets6 votes.
. In the second round, with Candidate C eliminated, the votes are redistributed, resulting in14 votes for Candidate A and12 votes for Candidate B.
3. The plurality-with-elimination method is used, which means that the candidate with the least votes is eliminated in each round until a winner is determined.
4. The monotonicity criterion states that if a voter changes their vote to increase the ranking of a candidate, it should not harm that candidate's chance of winning.
STEP 2
First, we need to understand the initial voting distribution. The votes are distributed as follows\begin{tabular}{|c|c|c|c|} \hlineFirst Choice & A & B & C \\ \hlineNumber of Votes &12 &8 &6 \\ \hline\end{tabular}
STEP 3
In the second round, Candidate C is eliminated as they have the least votes. The votes are then redistributed, resulting in the following distribution\begin{tabular}{|c|c|c|} \hlineFirst Choice & A & B \\ \hlineNumber of Votes &14 &12 \\ \hline\end{tabular}
STEP 4
The winner by the plurality-with-elimination method is Candidate A as they have the most votes in the final round.
STEP 5
However, let's consider the monotonicity criterion. If a voter changes their vote to increase the ranking of a candidate, it should not harm that candidate's chance of winning.
STEP 6
Looking at the initial voting distribution, we can see that if two voters who initially voted for Candidate B as their first choice changed their votes to Candidate A, Candidate A would have14 votes and Candidate B would have6 votes in the first round.
STEP 7
In the second round, Candidate B would be eliminated, and their votes would be redistributed. Assuming that all of Candidate B's second choice votes went to Candidate C, Candidate C would then have12 votes.
STEP 8
This would mean that Candidate A, despite receiving more votes, would lose to Candidate C in the final round. This violates the monotonicity criterion, as increasing the ranking of Candidate A resulted in them losing the election.
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