The Plurality MethodThe Borda Count

Method

Notes 2 – Sections 1.2 & 1.3

Essential LearningsStudents will understand and be

able to determine the winners of elections using the Plurality and the Borda Count Methods.

Plurality Method

Candidate with the most first-place votes (called the plurality candidate) wins

Don’t need each voter to rank the candidates - need only the voter’s first choice

Plurality Method

Vast majority of elections for political office in the United States are decided using the plurality method

Many drawbacks - other than its utter simplicity, the plurality method has little else going in its favor

The Math Club ElectionUsing Plurality Method:

A gets 14 first-place votes

B gets 4 first-place votes

C gets 11 first-place votes

D gets 8 first-place votes

Winner: A (Alisha)

Majority RuleIn a democratic election

between two candidates, the candidate with a majority (more than half) of the votes should be the winner.

More than half greater than 50%

Majority Candidate

Problems with Plurality Method Two candidates: a plurality

candidate is also a majority candidate - everything works out well

Three or more candidates: there is no guarantee that there is going to be a majority candidate

The Math Club ElectionMajority of votes:

Alisha, with 14 first-place votes, had a plurality (more than any other candidate) but is not a majority candidate

The Majority CriterionIf candidate X has a majority of

the first-place votes, then candidate X should be the winner of the election.

Under plurality method, the majority candidate is guaranteed to be the winner of the election.

ViolationsA violation of the majority

criterion occurs in an election in which there is a majority candidate but that candidate does not win the election.

If this happens, then we say that the voting method itself violates the majority criterion.

The Marching Band ElectionTasmania State University has a superb

marching band. They are so good that this coming bowl season they have invitations to perform at five different bowl games: the Rose Bowl (R), the Hula Bowl (H), the Fiesta Bowl (F), the Orange Bowl (O), and the Sugar Bowl (S). An election is held among the 100 members of the band to decide in which of the five bowl games they will perform. A preference schedule giving the results of the election is shown.

The Marching Band Election

The Marching Band ElectionUnder Plurality Method:

Wait! 51 voters have the Rose Bowl as last choice!

Hula Bowl has 48 first-place votes and 52 second-place votes

The Marching Band ElectionHead to head comparison:

Hula vs. Rose: 51 to 49

Hula vs. Fiesta: 97 to 3

Hula vs. Orange: 100 votes for Hula

Hula vs. Sugar: 100 votes for Hula

Hula is the best choice to represent all voters.

The Condorcet CriterionIf candidate X is preferred by the

voters over each of the other candidates in a head-to-head comparison, then candidate X should be the winner of the election.

The Marching Band ExampleThe plurality method violates the Condorcet criterion.

Insincere VotingThe idea behind insincere voting

(also known as strategic voting) is simple: If we know that the candidate we really want doesn’t have a chance of winning, then rather than “waste our vote” on our favorite candidate we can cast it for a lesser choice who has a better chance of winning the election. In closely contested elections a few insincere voters can completely change the outcome of an election.

The Marching Band ExampleThree of the band members

realize that there is no chance that their first choice, the Fiesta Bowl, can win this election, so rather than waste their votes they decide to make a strategic move and they cast their votes for the Hula Bowl by switching the first and second choices in their ballots.

The Marching Band Example

Hula wins 51 votes.

Plurality MethodOne of the major flaws of the

plurality method: the ease with which election results can be manipulated by a voter or a block of voters through insincere voting.

Consequences of Insincere VotingInsincere voting common in real-

world elections

2000 and 2004 presidential elections: Close races, Ralph Nader lost many votes - voters did not want to “waste their vote.”

Consequences of Insincere VotingIndependent and small party

candidates never get a fair voice or fair level funding (need 5% of vote to qualify for federal funds).

Entrenched two-party system, often gives voters little real choice.

The Borda Count Method

Each place on a ballot is assigned points

With N candidates, 1 point for last place, 2 points for second from last, and so on

First-place vote is worth N points Tally points for each candidate

separately Candidate with highest total is winner

The Math Club ElectionUse the Borda Count Method to

determine the winner.Num. of voters 14 10 8 4 1

1st choice – 4 pts A C D B C

2nd choice – 3 pts B B C D D

3rd choice – 2 pts C D B C B

4th choice – 1 pt D A A A A

The Math Club ElectionUse the Borda Count Method to

determine the winner.Num. of voters 14 10 8 4 1

1st choice – 4 pts A: 56 pts C: 40 pts D: 32 pts B: 16 pts C: 4 pts

2nd choice – 3 pts B: 42 pts B: 30 pts C: 24 pts D: 12 pts D: 3 pts

3rd choice – 2 pts C: 28 pts D: 20 pts B: 16 pts C: 8 pts B: 2 pts

4th choice – 1 pt D: 14 pts A: 10 pts A: 8 pts A: 4 pts A: 1 pts

The Math Club ElectionWait! Wasn’t A (Alisha) the

winner using the Plurality method?

The Borda winner is the candidate with the best average ranking - the “best compromise candidate”.

The School Principal ElectionThe last principal at Washington

Elementary School has just retired and the School Board must hire a new principal. The four finalists for the job are Mrs. Amaro, Mr. Burr, Mr. Castro, and Mrs. Dunbar (A, B, C, and D, respectively). After interviewing the four finalists, each of the 11 school board members gets to rank the candidates by means of a preference ballot, and the Borda winner gets the job.

The School Principal ElectionPreference schedule – Use Borda

count method.Num. of voters 6 2 3

1st A B C2nd B C D3rd C D B4th D A A

Borda Count Method Flaws

The Borda Method violates two basic criteria of fairness:

Majority criterionCondorcet criterion

Experts in voting theory consider the Borda Count method one of the best, if not the very best, method for deciding elections with many candidates.

Real Life Uses of Borda Count Method

Individual sports awards (Heisman Trophy winner, NBA Rookie of the Year, NFL MVP, etc.)

College football pollsMusic industry awardsHiring of school principals,

university presidents, and corporate executives

Assignment

p. 31 – 32: 11, 14, 16, 17 a – c, 20, 22, 23, 25

Covered Textbooks

Signed Syllabus Slip