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 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.
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