## Putting It Together: Voting Theory

There are four candidates for senior class president, Garcia, Lee, Nguyen, and Smith.  Using a preference ballot, $75$ ballots were cast, and the votes are shown below.

 $20$ $3$ $8$ $16$ $28$ 1st choice Garcia Garcia Lee Nguyen Smith 2nd choice Lee Nguyen Nguyen Garcia Lee 3rd choice Nguyen Lee Garcia Lee Garcia 4th choice Smith Smith Smith Smith Nguyen

Now that the votes are in, it should be a simple matter to find out who won the election, right?

Well that depends on which voting system you choose.

Using plurality method, Smith wins.  This is because Smith got $28$ first place votes, while Garcia received $20+3=23$, Lee $8$, and Nguyen $16$.  However, Smith was the very last choice for the majority of the students!  This seems rather unfair, so let’s explore another method.

The Borda count assigns points based on the ranking: 4 points for first place, 3 for second, 2 for third, and 1 for last.

 Garcia Lee Nguyen Smith 1st choice (4 pts) $23\times4=92$ $8\times4=32$ $16\times4=64$ $28\times4=112$ 2nd choice (3 pts) $16\times3=48$ $48\times3=144$ $11\times3=33$ $0\times3=0$ 3rd choice (2 pts) $36\times2=72$ $19\times2=38$ $20\times2=40$ $0\times2=0$ 4th choice (1 pt) $0\times1=0$ $0\times1=0$ $28\times1=28$ $47\times1=47$ Total Points $212$ $214$ $165$ $159$

This time Smith comes in last and Lee is the winner.  However the preference votes indicate that Lee is a lukewarm choice for most people.  Only $8$ students chose Lee as their first choice.  Perhaps another voting method will reflect the students’ preferences better.

Let’s try instant runoff voting (IRV).  This method proceeds in rounds, eliminating the candidate with the least number of first place votes at each round (with votes redistributed to voters’ next choices) until a majority winner emerges.  In the first round, Lee is immediately eliminated.

 $20+3=23$ $8+16=24$ $28$ 1st choice Garcia Nguyen Smith 2nd choice Nguyen Garcia Garcia 3rd choice Smith Smith Nguyen

There is still no majority winner.  Garcia is eliminated next, which gives the election to Nguyen.

 $23+24=47$ $28$ 1st choice Nguyen Smith 2nd choice Smith Nguyen

Finally, let’s see if there is a Condorcet winner.  We examine all one-on-one contests based on the original preference schedule.  The table below summarizes the results.  Each column shows the total number of ballots in which that candidate beats the candidate listed in each row.  Remember, a majority of the $75$ votes would be at least $38$ (majority votes are highlighted in blue).

 Garcia Lee Nguyen Smith Garcia $36$ $24$ $28$ Lee $39$ $19$ $28$ Nguyen $51$ $56$ $28$ Smith $47$ $47$ $47$

Garcia is the Condorcet winner with $39$, $51$, and $47$ votes against Lee, Nguyen, and Smith, respectively.

Which voting method do you think is the most fair?  The same voting preference schedule produced four different “winners.”  In a close election with many competing preferences, perhaps there is no clear winner.  However a decision must be made.

This small example serves to show why understanding voting theory helps to put the election process in perspective.  At the end of the day, one voting method must be selected and the winner decided according to those agreed-upon rules.  Try out some other voting methods and see if you can make a case for who should be the senior class president!