Problem 12 - Thinking Mathematically

Question

The preference table for an election is given. Use the table to answer the questions that follow it. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Number of Votes } & \mathbf{2 0} & \mathbf{1 5} & \mathbf{3} & \mathbf{1} \\ \hline \text { First Choice } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Second Choice } & \text { B } & \text { C } & \text { D } & \text { B } \\ \hline \text { Third Choice } & \text { C } & \text { D } & \text { B } & \text { C } \\ \hline \text { Fourth Choice } & \text { D } & \text { A } & \text { A } & \text { A } \\ \hline \end{array} $$ a. Using the Borda count method, who is the winner? b. Is the majority criterion satisfied? Explain your answer.

Step-by-Step Solution

Answer

a. Candidates B and C win according to the Borda count method. b. The Majority Criterion is not satisfied, as no candidate received more than half the total votes as their first choice.

Step 1: Calculate the Borda Count

To begin, calculate the Borda points for each candidate. Each first choice vote deserves 4 points, each second choice gets 3 points, each third choice receives 2 points, and each fourth choice receives 1 point. Multiply each by the number of votes for the candidate in that position and sum all the points for each candidate. So for candidate A: \[ (4 * 20) + (1 * 15) = 95 \] For candidate B: \[ (4 * 15) + (3 * 20) + (2 * 1) = 105 \] For candidate C: \[ (4 * 3) + (3 * 15) + (2 * 20) = 105 \] And for candidate D: \[ (4 * 1) + (3 * 3) + (2 * 15) + (1 * 20) = 60 \] So, based on Borda counts, candidates B and C are the joint winners.

Step 2: Evaluate the Majority Criterion

The Majority Criterion is met if a candidate who gets more than half of the votes should win. Here, candidate A has the maximum number of first choice votes, which is equal to 20. But, the total votes are 20 + 15 + 3 + 1 = 39. So candidate A does not possess the majority of the votes, receiving less than half of the 39 total votes. Thus the majority criterion is not satisfied in this case.

Key Concepts

Voting Systems in Mathematics

When it comes to making a collective decision, whether it's a group of friends deciding where to eat or a country electing a president, we rely on some form of voting system. In mathematics, a voting system is a method for aggregating the preferences of individuals into a collective decision. There are many types of voting systems, each with its own rules about how to count votes and determine the winner.

Voting systems are critical because they impact the fairness and representativeness of the outcome. For example, some systems favor majoritarian outcomes, where the majority's preference dominates, while others seek to reflect the diverse preferences of all voters. The Borda count method is one such system that attempts to give a more comprehensive picture of voter preferences.

Under the Borda count, candidates are ranked according to the preference of voters, and points are assigned inversely to their rank—more points for higher preferences. It's designed to provide a fairer assessment when there are multiple choices and avoids the 'winner-takes-all' approach of simple majority systems. In essence, it captures not only who is most preferred but also the extent to which all candidates are favored by the electorate.

Preference Table

At the heart of many voting systems is the preference table—a visual representation that clearly displays the ranked choices of all voters. A preference table lays out, in columns, how many votes each candidate receives for each ranking position. It's a crucial tool for understanding the overall landscape of voter opinion and is central to implementing methods like the Borda count.

Each row of a preference table corresponds to a choice ranking (first, second, third, and so forth), while each column represents how many voters assigned that rank to the candidates. By looking at a preference table, you can quickly see which candidates are the most and least popular, as well as appreciate the nuances of voter preferences. In our textbook exercise, the table is used to visualize the distribution of votes across four candidates, thereby laying the foundation to apply voting methods such as the Borda count.

Majority Criterion

A key principle in some voting systems is the majority criterion, which insists that if a candidate receives more than half of the first preference votes, that candidate should be declared the winner. The rationale behind this is straightforward: if the majority of voters prefer one candidate over all others as their first choice, many believe that candidate has the strongest mandate to win.

In the context of our exercise, we see that no candidate has more than half of the total votes as their first choice, signaling that the majority criterion is not met. While candidate A received the most first choice votes, they did not secure a majority, indicating no clear preference from the majority of voters. This outcome shows the majority criterion's limitations, especially in elections with multiple candidates where votes are spread out, and it highlights the importance of using a voting system, like the Borda count, that takes into account the broader preferences expressed by voters.