QuestionFind the winner using plurality-with-elimination for candidates A, B, C. How does a vote change affect the outcome? Is monotonicity met?

Studdy Solution

STEP 1

Assumptions1. The number of votes for each preference order is given in the table. . The plurality-with-elimination method is used for voting.
3. In the plurality-with-elimination method, the candidate with the least number of first-choice votes is eliminated in each round, and their votes are redistributed according to the next choice on each voter's ballot. This process continues until a candidate has a majority of the votes.

STEP 2

First, we need to count the number of first-choice votes for each candidate in the straw vote. The number of first-choice votes for each candidate is given in the first row of the table.
$B90\, votes$ $A72\, votes$ $C63\, votes$

STEP 3

In the plurality-with-elimination method, the candidate with the least number of first-choice votes is eliminated in the first round. In this case, candidate C has the least number of first-choice votes and is therefore eliminated.

STEP 4

The votes for candidate C are redistributed according to the second choice on each voter's ballot. From the table, we can see that the second choice for the63 voters who voted for candidate C first is candidate A. Therefore, these63 votes are added to candidate A's total.
$A72\, votes +63\, votes =135\, votes$

STEP 5

Now we compare the number of votes for candidates A and B. Candidate A has135 votes and candidate B has90 votes. Since candidate A has more votes, candidate A wins the straw vote.
For part b, we need to consider the change in votes mentioned in the problem.

STEP 6

In the actual election, the36 voters who originally voted $C, B, A$ , change their votes to $B, C, A$ . This means that candidate B gains36 first-choice votes, candidate A gains36 third-choice votes (which do not count in the plurality-with-elimination method), and candidate C loses36 first-choice votes.

STEP 7

We need to update the number of first-choice votes for each candidate to reflect this change.
$B90\, votes +36\, votes =126\, votes$ $A72\, votes$ $C63\, votes -36\, votes =27\, votes$

STEP 8

Again, we eliminate the candidate with the least number of first-choice votes. In this case, candidate C has the least number of first-choice votes and is therefore eliminated.

STEP 9

The votes for candidate C are redistributed according to the second choice on each voter's ballot. From the table, we can see that the second choice for the27 voters who voted for candidate C first is candidate B. Therefore, these27 votes are added to candidate B's total.
$B126\, votes +27\, votes =153\, votes$

STEP 10

Now we compare the number of votes for candidates A and B. Candidate A has72 votes and candidate B has153 votes. Since candidate B has more votes, candidate B wins the actual election.
For part c, we need to determine whether the monotonicity criterion is satisfied.

STEP 11

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 chances of winning. In this case, when the36 voters changed their votes from $C, B, A$ to $B, C, A$ , candidate B's ranking increased. However, this did not harm candidate B's chances of winning; in fact, it helped candidate B win the election. Therefore, the monotonicity criterion is satisfied.
The winners of the straw vote and the actual election are candidate A and candidate B, respectively, and the monotonicity criterion is satisfied.

Was this helpful?

QuestionFind the winner using plurality-with-elimination for candidates A, B, C. How does a vote change affect the outcome? Is monotonicity met?

Studdy Solution

STEP 1

STEP 2

First, we need to count the number of first-choice votes for each candidate in the straw vote. The number of first-choice votes for each candidate is given in the first row of the table.
$B90\, votes$ $A72\, votes$ $C63\, votes$

STEP 3

In the plurality-with-elimination method, the candidate with the least number of first-choice votes is eliminated in the first round. In this case, candidate C has the least number of first-choice votes and is therefore eliminated.

STEP 4

STEP 5

Now we compare the number of votes for candidates A and B. Candidate A has135 votes and candidate B has90 votes. Since candidate A has more votes, candidate A wins the straw vote.
For part b, we need to consider the change in votes mentioned in the problem.

STEP 6

STEP 7

We need to update the number of first-choice votes for each candidate to reflect this change.
$B90\, votes +36\, votes =126\, votes$ $A72\, votes$ $C63\, votes -36\, votes =27\, votes$

STEP 8

Again, we eliminate the candidate with the least number of first-choice votes. In this case, candidate C has the least number of first-choice votes and is therefore eliminated.

STEP 9

STEP 10

Now we compare the number of votes for candidates A and B. Candidate A has72 votes and candidate B has153 votes. Since candidate B has more votes, candidate B wins the actual election.
For part c, we need to determine whether the monotonicity criterion is satisfied.

STEP 11

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind the winner using plurality-with-elimination for candidates A, B, C. How does a vote change affect the outcome? Is monotonicity met?

Studdy Solution

STEP 1

STEP 2

First, we need to count the number of first-choice votes for each candidate in the straw vote. The number of first-choice votes for each candidate is given in the first row of the table.B90 votesB90\, votesB90votesA72 votesA72\, votesA72votesC63 votesC63\, votesC63votes

STEP 3

In the plurality-with-elimination method, the candidate with the least number of first-choice votes is eliminated in the first round. In this case, candidate C has the least number of first-choice votes and is therefore eliminated.

STEP 4

STEP 5

Now we compare the number of votes for candidates A and B. Candidate A has135 votes and candidate B has90 votes. Since candidate A has more votes, candidate A wins the straw vote.For part b, we need to consider the change in votes mentioned in the problem.

STEP 6

STEP 7

STEP 8

Again, we eliminate the candidate with the least number of first-choice votes. In this case, candidate C has the least number of first-choice votes and is therefore eliminated.

STEP 9

STEP 10

Now we compare the number of votes for candidates A and B. Candidate A has72 votes and candidate B has153 votes. Since candidate B has more votes, candidate B wins the actual election.For part c, we need to determine whether the monotonicity criterion is satisfied.

STEP 11

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

First, we need to count the number of first-choice votes for each candidate in the straw vote. The number of first-choice votes for each candidate is given in the first row of the table.
$B90\, votes$ $A72\, votes$ $C63\, votes$

Now we compare the number of votes for candidates A and B. Candidate A has135 votes and candidate B has90 votes. Since candidate A has more votes, candidate A wins the straw vote.
For part b, we need to consider the change in votes mentioned in the problem.

Now we compare the number of votes for candidates A and B. Candidate A has72 votes and candidate B has153 votes. Since candidate B has more votes, candidate B wins the actual election.
For part c, we need to determine whether the monotonicity criterion is satisfied.