Zoom International Math League

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: $$\begin{array}{|cc|}\hline \text{Voter \#}& \text{Preferred Candidate}\\ 1& B\\ 2& B\\ 3& C\\ 4& A\\ 5& A\\ 6& B\\ 7& B\\ 8& A\\ \hline\end{array}$$ 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). $$\begin{array}{|cc|}\hline \text{Voter \#}& \text{Preferred Candidates}\\ 1& B,C,\text{ and }D\\ 2& B\text{ and }D\\ 3& A,C,\text{ and }D\\ 4& A\text{ and }D\\ 5& A,C,\text{ and }D\\ 6& B\text{ and }D\\ 7& B\text{ and }D\\ 8& A\text{ and }C\\ \hline\end{array}$$ 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: $$\begin{array}{|ccccc|}\hline \text{VoterCandidate}& A& B& C& D\\ 1& 3& 9& 8& 6\\ 2& 4& 7& 4& 6\\ 3& 7& 2& 9& 7\\ 4& 9& 2& 4& 7\\ 5& 10& 3& 9& 6\\ 6& 2& 7& 4& 6\\ 7& 1& 8& 5& 6\\ 8& 8& 3& 7& 2\\ \hline\end{array}$$ 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?

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