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Study Guides > Mathematics for the Liberal Arts

Putting It Together: Voting Theory

There are four candidates for senior class president, Garcia, Lee, Nguyen, and Smith.  Using a preference ballot, 7575 ballots were cast, and the votes are shown below.
2020 33 88 1616 2828
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 2828 first place votes, while Garcia received 20+3=2320+3=23, Lee 88, and Nguyen 1616.  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) 23×4=9223\times4=92 8×4=328\times4=32 16×4=6416\times4=64 28×4=11228\times4=112
2nd choice (3 pts) 16×3=4816\times3=48 48×3=14448\times3=144 11×3=3311\times3=33 0×3=00\times3=0
3rd choice (2 pts) 36×2=7236\times2=72 19×2=3819\times2=38 20×2=4020\times2=40 0×2=00\times2=0
4th choice (1 pt) 0×1=00\times1=0 0×1=00\times1=0 28×1=2828\times1=28 47×1=4747\times1=47
Total Points 212212 214214 165165 159159
  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 88 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=2320+3=23 8+16=248+16=24 2828
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=4723+24=47 2828
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 7575 votes would be at least 3838 (majority votes are highlighted in blue).
Garcia Lee Nguyen Smith
Garcia 3636 2424 2828
Lee 3939 1919 2828
Nguyen 5151 5656 2828
Smith 4747 4747 4747
Garcia is the Condorcet winner with 3939, 5151, and 4747 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!

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