Problem 19 - Thinking Mathematically

Question

The preference table shows the results of an election among three candidates, A, B, and C. $$ \begin{array}{|l|c|c|c|} \hline \text { Number of Votes } & \mathbf{7} & \mathbf{3} & \mathbf{2} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { A } & \text { A } \\ \hline \end{array} $$ 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 \(\mathrm{C}\) 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

Answer

A is the winner by the plurality method. The majority criterion and head-to-head criterion are both satisfied. Even after changing voters' preferences or removing candidate C, A is still the winner, satisfying the irrelevant alternatives criterion. No contradiction to Arrow's impossibility theorem is found here.

Step 1: Find the Plurality Winner

Count the first choice votes for each candidate: 7 votes for A, 3 votes for B and 2 votes for C. Candidate A has the most votes, hence A is the winner in the plurality method.

Step 2: Check the Majority Criterion

The majority criterion is satisfied if the candidate who gets the most first-place votes (the plurality winner) also has more than half of the total votes. Here, the total number of votes is 7+3+2 = 12, half of which is 6. Candidate A has received 7 votes, which is more than half. Hence, the majority criterion is satisfied.

Step 3: Check the Head-to-Head Criterion

The head-to-head criterion is satisfied if the plurality winner would also win in a one-on-one vote against each of the other candidates. If we arrange one-on-one contests, Candidate A wins against B and C always comes last. Therefore, the head-to-head criterion is satisfied

Step 4: Change in Voters' Preferences

Two voters change their preference from having A last to having A first. The new vote distribution becomes: A has 9, B has 3 and C has 2. Candidate A still has the most votes, hence remains the winner.

Step 5: Check the Irrelevant Alternatives Criterion

The irrelevant alternatives criterion is satisfied if the winner doesn't change when a non-winning candidate drops out. If C drops out, the votes don't change for A and B, thus A will still win. Therefore, the irrelevant alternatives criterion is satisfied.

Step 6: Check against Arrow's Impossibility Theorem

No, these results do not contradict Arrow's impossibility theorem. This theorem states that no method of voting can satisfy all fair voting criteria simultaneously for every possible election. However, in this specific election, all the criteria have been satisfied.

Key Concepts

Majority Criterion

The Majority Criterion is fundamental in determining whether a candidate truly has the support of the majority in an election. According to this criterion, if a candidate receives more than half of the first-place votes among all voters, that candidate should be declared the winner. In our exercise, the candidate A received 7 out of a total of 12 votes.

This count is more than half of the total votes, which is 6, satisfying the Majority Criterion.

This means that most voters preferred candidate A over any other, highlighting A's popularity and significant support in this election. When a candidate satisfies this criterion, it ensures that the winner chosen genuinely represents the preference of the majority.

Head-to-Head Criterion

The Head-to-Head Criterion evaluates whether a candidate would still emerge victorious in direct one-on-one contests against each of the other candidates. This idea supports the notion that the winning candidate should be able to outperform each opponent individually.

For our scenario, candidate A was tested in separate head-to-head contests with candidates B and C. In both of these matchups, A consistently surpassed the other candidates.

This fulfillment shows that A is not only the most favored in a grouped ballot but also individually stronger against each candidate, confirming the robustness of A's win.

Irrelevant Alternatives Criterion

The Irrelevant Alternatives Criterion is aimed at understanding how the elimination of non-winning candidates affects the election outcome. A fair voting system should maintain the winner even if an irrelevant or non-winning candidate is removed.

In our exercise, candidate C was removed from the running, leaving only candidates A and B. The vote counts did not change significantly enough to affect A's victory.

This criterion is satisfied because candidate A continues to have more votes than candidate B, even after C is excluded. This shows that the presence of candidate C was irrelevant to the final election result, preserving fairness and consistency.

Arrow's Impossibility Theorem

Arrow's Impossibility Theorem is a cornerstone of social choice theory, proposing that no electoral system can satisfy all desired fairness criteria simultaneously across all potential elections. The criteria include non-dictatorship, unrestricted domain, the independence of irrelevant alternatives, and more.

In the exercise, all criteria were satisfied, yet this specific outcome does not inherently contradict Arrow's theorem.

• The theorem asserts the challenge of designing a perfect voting system universally, not in a singular election.
• It reminds us that trade-offs exist in real-world voting and elections.

Despite achieving fairness in this particular instance, the theorem suggests that such alignment across all elections is impossible, shedding light on the complexities of voting methodology.