Discussion Forum

Let's Vote! That's Easy, Right? (Weekly Brain Potion Discussion)

Picture of John Lensmire
Let's Vote! That's Easy, Right? (Weekly Brain Potion Discussion)
by John Lensmire - Friday, November 2, 2018, 11:59 AM

It's almost time to vote in the US. People often spend a lot of time debating who or what to vote for, but less time on how the process works. In this week's brain potion we explore some different ways of deciding who wins an election.

In our hypothetical election we have four candidates, label them $A$, $B$, $C$, and $D$. For simplicity, we'll assume there are $8$ voters.

In our first mock election, voters are asked for their preferred candidate, giving the results below: $$\begin{array}{|c|c|} \hline \text{Voter #} & \text{Preferred Candidate} \\ \hline 1 & B \\ \hline 2 & B \\ \hline 3 & C \\ \hline 4 & A \\ \hline 5 & A \\ \hline 6 & B \\ \hline 7 & B \\ \hline 8 & A \\ \hline \end{array}$$ Based on these results, which candidate should win? Explain your reasoning.

In our second mock election, we collect more information from voters. This time, each voter is asked if they find each candidate acceptable or not. In this way, each voter can choose multiple candidates as acceptable. These results are found below (so for example, Voter $1$ thinks candidates $B$, $C$, and $D$ are all acceptable). $$\begin{array}{|c|c|} \hline \text{Voter #} & \text{Preferred Candidates} \\ \hline 1 & B, C, \text{ and } D \\ \hline 2 & B \text{ and } D\\ \hline 3 & A, C, \text{ and } D \\ \hline 4 & A \text{ and } D \\ \hline 5 & A, C, \text{ and } D \\ \hline 6 & B \text{ and } D \\ \hline 7 & B \text{ and } D\\ \hline 8 & A \text{ and } C\\ \hline \end{array}$$ With the information provided here, which candidate should win? Explain your reasoning.

In our final election, we collect even more information from voters. This time, each voter is asked to rank each candidate from $1$ (low approval) to $10$ (high approval) based on their preferences. Full results are shown below: $$\begin{array}{|c|c|c|c|c|} \hline \text{Voter\Candidate} & A & B & C & D \\ \hline 1 & 3 & 9 & 8 & 6 \\ \hline 2 & 4 & 7 & 4 & 6 \\ \hline 3 & 7 & 2 & 9 & 7 \\ \hline 4 & 9 & 2 & 4 & 7 \\ \hline 5 & 10 & 3 & 9 & 6 \\ \hline 6 & 2 & 7 & 4 & 6 \\ \hline 7 & 1 & 8 & 5 & 6 \\ \hline 8 & 8 & 3 & 7 & 2 \\ \hline \end{array}$$ With the information provided here, which candidate should win? Explain your reasoning.

With all three mock elections completed, compare and contrast the methods used to determine the winning candidate. Did you get different results based on the methods used for each election? Which method do you think is most effective? Why or why not might different methods be used in the real world?

Please share any thoughts or questions you have below. We'll monitor the responses and give our thoughts as well! Have your own request, idea, or feedback for the Brain Potion series? Share with us in our Request and Idea Thread available here.

Picture of Darren Tung
Re: Let's Vote! That's Easy, Right? (Weekly Brain Potion Discussion)
by Darren Tung - Monday, November 5, 2018, 12:54 PM

Incoming flashbacks to my CTY course on Probability and Game Theory. 

This is a classic study of what I call election theory: the study of electoral systems’ legitimacy. The above represent three kinds of elections: plurality, approval voting, and score voting.

Plurality is simple: each voter chooses a single candidate, and the one with the most votes wins. The first mock election is a clear plurality system, with 3 voters voting A, 4 voters voting B, 1 voter voting C, and none for D. In such a system, B would win, and become what is known as the Condorcet Candidate. The plurality system satisfies Pareto Efficiency, where because every voter prefers B, B wins. In addition, irrelevant candidates for each voter do not affect the final result. However, there is plenty of room for dictators, or people who have swing vote power, to form.

Approval voting is a similar idea, only with the ability to vote for more than one candidate per voter. The second mock election shows approval voting, and since A received 4 votes, B received 4 votes, C received 4 votes, and D received 7 votes, D would win. Dictators can’t form in such a system because no one person can swing vote here. Furthermore, candidates that are not considered by voters are irrelevant to the final result. However, Pareto Efficiency is not satisfied because not every person prefers candidate D as their number one choice judging by the other elections, and yet D still won. 

Lastly, score voting (or cardinal voting) is when people score candidates on a certain scale. In the case of the third mock election, with A obtaining 45 points, B obtaining 41 points, C obtaining 50 points, and D obtaining 46 points, C would win. Score voting prevents swing votes again, while also providing a more complete picture behind voting. However, Pareto Efficiency fails because 4 voters prefer C and B alike in such a case and yet C won by a large margin. In addition, changing around opinions of “irrelevant candidates” (not your top two choices) can significantly affect such an election. 

Many people claim that there exists a perfect election method, especially with recent outrage against elections such as the 2016 Presidential Election and the UK Referendum. Mathematically, such an election would satisfy all three criteria I alluded to above: No dictators, Pareto Efficiency, and irrelevant candidates not playing a role in voting. That simply is not the case; in fact, it was actually proven by Nobel Laureate Kenneth Arrow in what is known as Arrow’s Impossibility Theorem.