The Method of Plurality with Elimination

This method is a preferential voting method and is a variation of the Plurality Method.
Plurality with Elimination is carried out in rounds. After each round of voting the candidate (or alternative) with the fewest first place votes is eliminated and a new round of voting is done with the remaining candidates. When only two candidates remain in a round, the candidate with the most votes wins the election. For an N candidate election, the Method of Plurality with Elimination requires N-1 rounds.
The mayor of Smallville is being chosen in a Plurality with Elimination election. The four candidates are Paul (the town barber), Rita (head of the town council), Sarah (Superintendent of Education), and Tim (former District Attorney).

How many rounds will it take to determine a winner? answer

500 registered voters cast their preference ballots. The results are summarized in the preference schedule below.
   

# of

Voters

  

Place

 130 120 100 150
1st P T T S
2nd R R R R
3rd S S P P
4th T P S T

Using the Method of Plurality with Elimination, which candidate is eliminated in the first round? answer

Since Round 1 eliminated a candidate, Round 2 begins with the 3 remaining candidates.

IMPORTANT POINT!!
Since each voter completed a preference ballot (all of which are summarized in the preference schedule above), we do not have to hold a new election. We simply remove the eliminated candidate from the preference schedule and adjust the remaining candidates appropriately. For instance, if a column of a preference schedule was B (1st), A (2nd), C (3rd), D (4th) and candidate A got eliminated, the column would be B (1st), C (2nd), D (3rd) in the preference schedule for the next round. In other words, for each column of a preference schedule, each candidate below an eliminated candidate moves up one place while the positions of candidates above an eliminated candidate remain unchanged.

Write out the preference schedule for Round 2 (the round with the 3 candidates left after Round 1). answer

Which candidate is eliminated in the second round? answer

Write out the preference schedule for Round 3 (the round with the 2 candidates left after Round 2). answer

Which candidate is eliminated in the third (and final) round? answer

Who wins the election using the Method of Plurality with Elimination? answer

 

see another example

The Method of Plurality with Elimination uses only the first place votes in each round. However, since lower order preferences (2nd, 3rd, etc.) can make it onto the first place row in later rounds, the Plurality with Elimination Method uses much more of the information contained in the preference schedule compared to the Plurality Method. When using the Plurality Method, all of the information in the preference schedule not related to first place is ignored. This is a powerful theoretical argument in favor of the Method of Plurality with Elimination over the Plurality Method.

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