Question

In a runoff election, if no candidate receives a majority of votes in the first round of voting, the top two candidates face each other in a second round. Let's say that people voting on Candidates A, B, \(C,\) and \(D\) in a runoff election have the following preferences. [LO 22. 2] 12 voters: \(\quad \mathrm{A}>\mathrm{B}>\mathrm{C}>\mathrm{D}\) 8 voters: \(\quad C>B>D>A\) 10 voters: \(\quad \mathrm{D}>\mathrm{B}>\mathrm{C}>\mathrm{A}\) 4 voters: \(\quad \mathrm{B}>\mathrm{D}>\mathrm{A}>\mathrm{C}\) a. Does anyone receive an outright majority in the first round? If so, which candidate? If not, which two candidates move on to the second round, and which of them wins? b. Suppose Candidate A drops out of the race. Does any candidate now receive an outright majority in the first round? If so, which candidate? If not, which two candidates move on to the second round, and which of them wins? c. Does this situation violate the independence of irrelevant alternatives?

Short answer

a. No majority; D wins runoff. b. No majority; B wins runoff. c. Yes, it violates independence of irrelevant alternatives.

Step-by-step solution

Calculate Total Votes

First, we determine the total number of voters. This is the sum of all votes cast for all preference orders given.\[12 + 8 + 10 + 4 = 34 \]

Check Majority in First Round (Part a)

A candidate needs more than half the total votes to win an outright majority. This means a candidate needs more than \( \frac{34}{2} = 17 \) votes to win immediately.- Candidate A: 12 votes- Candidate B: 4 votes- Candidate C: 8 votes- Candidate D: 10 votesSince no candidate has more than 17 votes, no one wins in the first round.

Select Top Two Candidates for Runoff (Part a)

The top two candidates with the most votes proceed to the second round. - Candidate A: 12 - Candidate D: 10 These are the two highest, so Candidates A and D move to the second round.

Predict Runoff Winner (Part a)

In the runoff, we determine preferences between A and D for those whose top choices were B or C. - 8 voters (C>B>D>A) prefer D over A. - 4 voters (B>D>A>C) prefer D over A. Thus, D gets 10 original votes plus 12 additional votes (8 from C, 4 from B), totaling 22 votes. A keeps its 12 votes.

Eliminate Candidate A (Part b)

With Candidate A out, adjust the votes: - 12 voters: B becomes primary, with new order: B>C>D - 8 voters: Unchanged, as A was last - 10 voters: Unchanged, as A was last - 4 voters: Unchanged, as A was below B and D.

Recalculate First-Round Majority (Part b)

Now, check if any candidate has more than 17 votes: - Candidate B: 12 (from A) + 4 (original) = 16 - Candidate C: 8 (original) - Candidate D: 10 (original) No one has more than 17 votes, so no outright majority.

Determine New Top Two (Part b)

Since A is eliminated, top two by votes: - Candidate B: 16 - Candidate D: 10 Candidates B and D proceed to the runoff.

Runoff Winner with A Eliminated (Part b)

In the new runoff: - 12 voters (B>C>D) prefer B - 8 voters (C>B>D) prefer B over D - 10 voters (D>B>C) prefer D - 4 voters (B>D>A) prefer B B receives 12+8+4 = 24 votes, while D receives 10 votes.

Check Independence of Irrelevant Alternatives (Part c)

Initially, D won, but after A's drop out, B won. The outcome changes with the removal of a candidate not in the runoff, showing a violation of independence of irrelevant alternatives.

Key concepts

Runoff Election

A runoff election is a method to determine a clear winner when no candidate receives a majority of votes in an initial round. This voting system ensures that the elected candidate has the support of more than half of the voters, which offers a level of legitimacy.
In a runoff, if no candidate wins over 50% of the votes in the first round, the candidates with the two highest vote totals proceed to a second round of voting. The subsequent runoff is then held to decide the winner solely between these two top candidates.

• This method prevents a candidate from winning just because the opposition votes were split among multiple other candidates.
• It ultimately allows voters to choose between the top two contenders, ensuring the winner has a broad base of support.

In our exercise, Candidates A and D moved forward to the runoff as they were the top two in the first round. The runoff ensures that the final decision is between the preferred candidates of the larger segments of the electorate.

Majority Vote

The concept of a majority vote is key to understanding voting systems like the runoff election. A majority vote means that a candidate receives more than half of the votes. This threshold ensures that the winning candidate has more support than any other option.
For a candidate to win in the first round of a voting process by majority, they must receive over 50% of the total votes cast. For example, in the exercise with 34 total votes, a candidate needed more than 17 votes to achieve a majority.

• No candidate secured these numbers initially, necessitating a runoff between the top two.
• The majority vote requirement ensures that elections reflect the preference of the largest percentage of voters possible.

Through this approach, it aims to enhance the democratic representation by ensuring the elected individual is preferred by the majority.

Independence of Irrelevant Alternatives

The principle of independence of irrelevant alternatives (IIA) suggests that the outcome of an election should not be affected by the removal or introduction of candidates who cannot win. This means that the choice between two candidates should not change by the presence or absence of additional irrelevant choices.
In the exercise, removing Candidate A altered the election outcome, demonstrating a violation of the IIA. Initially, Candidate D was the winner, but when A was removed, Candidate B won in a new runoff against D.

• This situation highlights that the presence of other options influenced the order of voter preferences for Candidates B and D, affecting the final outcome.
• Such a violation indicates potential flaws in the voting process, as the result could depend on irrelevant or non-competitive choices.

Understanding violations of the IIA helps in the evaluation and design of fairer voting systems, ensuring genuine preferences are always respected.