91Ó°ÊÓ

The preference table shows the results of an election among three candidates, A, B, and C. $$ \begin{array}{|l|l|l|l|} \hline \text { Number of Votes } & \mathbf{7} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Using the plurality method, who is the winner? b. Is the majority criterion satisfied? Explain your answer. c. Is the head-to-head criterion satisfied? Explain your answer. d. The two voters on the right both move candidate A from last place on their preference lists to first place on their preference lists. Construct a new preference table for the election. Using the table and the plurality method, who is the winner? e. Suppose that candidate C drops out, but the winner is still chosen by the plurality method. Is the irrelevant alternatives criterion satisfied? Explain your answer. f. Do your results from parts (b) through (e) contradict Arrow’s Impossibility Theorem? Explain your answer.

Step by step solution

01

Applying the Plurality Method

Evaluate the total number of first-choice votes each candidate received. Candidate A received 7 votes, B received 3 votes and C received 2 votes. Therefore, using the plurality method, Candidate A is the winner.

02

Assessing the Majority Criterion

A candidate satisfies the majority criterion if they are the choice of a majority of voters. In this case, there are 12 votes in total and candidate A received 7 votes. 7 is more than half of 12, so the majority criterion is satisfied.

03

Assessing the Head-to-Head Criterion

We need to check if candidate A would still win if we compared them one on one with each other candidate. By analyzing the table, we find that 7 people prefer A over B and also prefer A over C. Thus, the head-to-head criterion is satisfied.

04

Updating the Preference Table

The 2 voters move candidate A from last to first place, which results in a new table: 7 votes for A, 5 votes for B, and 0 votes for C. According to the plurality method, candidate A still wins.

05

Assessing the Irrelevant Alternatives Criterion

If a non-winning candidate withdraws or joins the election, it shouldn't affect the outcome of the election. Here, if candidate C drops out, the winner remains the same (candidate A). Therefore, the irrelevant alternatives criterion is satisfied.

06

Verifying the Arrow’s Impossibility Theorem

Arrow’s Impossibility Theorem states that it's impossible for a voting method to satisfy all fairness criteria. In this case, all criteria seem to be satisfied, which seems to defy the theorem. However, a single example isn't enough to prove or contradict this theorem, as it can show anomalies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plurality Method

The plurality method is one of the simplest voting systems used to determine a winner in an election. In this method, the candidate who receives the most first-choice votes wins. It's a straightforward approach that is easy to understand.
In the example given, Candidate A received 7 first-choice votes, Candidate B received 3, and Candidate C received 2. By simply counting the number of first-choice votes each candidate receives, Candidate A emerges as the winner using the plurality method.
This method has the advantage of being quick and simple, but it might not always reflect the best choice in terms of overall preference if the votes are spread out evenly among multiple candidates.

Majority Criterion

The majority criterion asks whether a candidate who receives more than half of the votes becomes the winner. If such a candidate exists, they should win according to this criterion. This ensures that the collective choice of more than half of the voters determines the outcome.
In the situation we are studying, there are 12 total votes. Candidate A secures 7 of these, which is more than half. As a result, the majority criterion is met, confirming Candidate A as the valid winner of the election. This criterion serves as a fairness measure, ensuring that a candidate who commands majority support doesn't lose to another with less support.

Head-to-Head Criterion

The head-to-head criterion checks whether a candidate would still win if compared directly with each other candidate, one at a time. This involves creating a series of one-on-one matchups between the candidates.
Looking at the initial voter preferences, Candidate A is preferred over Candidate B by 7 voters and over Candidate C by the same number. This means that Candidate A consistently wins when pitted one-on-one against each competitor.
Thus, Candidate A meets the head-to-head criterion. This criterion helps to ensure that the winner is not just possibly liked but is actually preferred over each individual opponent by most voters.

Irrelevant Alternatives Criterion

The irrelevant alternatives criterion states that the outcome should remain consistent even if a non-winning candidate enters or leaves the election. This means the presence or absence of these candidates should not influence who becomes the winner.
In our scenario, when Candidate C drops out, we examine whether the result changes. Candidate A still wins by plurality as the remaining votes shift, signifying that the outcome remains unaffected by the withdrawal of Candidate C.
Therefore, the election satisfies the irrelevant alternatives criterion. This criterion is aimed at ensuring that non-influential candidates do not inadvertently alter the outcome of the election, maintaining consistency.