Election Year Mathematics
Michael Buescher
Hathaway Brown School
mbuescher@hb.edu
http://www.mbuescher.com/professional
Majority vs. Plurality
 Majority:
 Plurality:
More than 50%.
More than any other
candidate.
Plurality Voting
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Vote for one candidate.
The candidate with more votes than any
other candidate wins the election.
The “Problem” with Plurality Voting
Minnesota Gubernatorial Election, 1998
(Reform) Jesse “The Body” Ventura:
(Republican) Norm Coleman
(Democrat) Hubert Humphrey III
37%
35%
28%
Voting for the President

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Each state determines a winner through
Plurality voting.
State results are combined in the
Electoral College.
QUIZ!

Who was the last president who won a
majority of the popular vote?

George H. W. Bush (1988)
1988:
George H. W. Bush
Michael Dukakis
53.4%
45.7%
2000 Presidential Election
States where winning candidate did not
receive a majority of the vote
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Florida
Iowa
Maine
Minnesota
Nevada
New Hampshire
New Mexico
Ohio
Oregon
Wisconsin
1992 Presidential Election
States where winning candidate did not
receive a majority of the vote
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Alabama
Alaska
Arizona
California
Colorado
Connecticut
Delaware
Florida
Georgia
Hawaii
Idaho
Illinois
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Indiana
Iowa
Kansas
Kentucky
Louisiana
Maine
Maryland
Massachusetts
Michigan
Minnesota
Mississippi
Missouri
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Montana
Nebraska
Nevada
New Hampshire
New Jersey
New Mexico
New York
North Carolina
North Dakota
Ohio
Oklahoma
Oregon
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Pennsylvania
Rhode Island
South Carolina
South Dakota
Tennessee
Texas
Utah
Vermont
Virginia
Washington
West Virginia
Wisconsin
Wyoming
Arrow’s Criteria
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Pareto Criterion
Condorcet Criterion
Monotonicity Criterion
Independence of Irrelevant Alternatives
Pareto (Majority) Criterion

If a majority [NOT plurality!] of voters
prefers candidate A over all others, then
A should win the election.
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Plurality voting: passes
Electoral College: fails
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Condorcet Criterion
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If candidate A is preferred to all other
candidates in pairwise head-to-head
comparisons, A should win the election.

Plurality voting: fails
Electoral college: fails
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Monotonicity Criterion
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If voters change their mind and rank a
candidate higher than they used to, it
should not hurt that candidate.
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Plurality voting: passes
Electoral college: passes
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Monotonicity Fails: France 2002
The Rules:
First Round
Results:
Vote for your favorite
Jacques Chirac
candidate.
Jacques Chirac
If 20.9
no candidate
19.9
%
receives a majority, there is
Jean-Mariea runoff
le Penbetween
16.9the%top
15.9
two vote-getters.
Lionel Jospin
Lionel Jospin
Second
16.2 %
The Polls:
Widely expected: runoff
between Jacques Chirac
(incumbent) and Lionel
Round:
Jospin; Jospin heavily
favored to win the runoff.
Chirac 82%, LePen 18%
Independence of Irrelevant
Alternatives
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Adding or removing a non-winning
candidate should not change the
results.
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Plurality: fails
Electoral College: fails
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Arrow’s Theorem

The only voting system that satisfies all
of these criteria when there are more
than two candidates is …
A
DICTATORSHIP
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Only one person votes.

For this, Arrow wins the Nobel Prize in Economics.
Criterion: Equality of Votes
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Every person’s vote should carry the
same weight.
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Plurality: passes
Electoral College: fails
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Inequality of votes: Electoral College
Wyoming
254,680 people voted
3 Electoral Votes
84,893 voters per electoral vote
Minnesota
2,404,621 people voted
10 Electoral Votes
240,462 voters per electoral vote
Inequality of votes: Electoral College
Number of votes per electoral vote
(2000 presidential election)
Wyoming (3)
Hawaii (4)
Alaska (3)
…
Wisconsin (11)
Florida (25)
Minnesota (10)
84,893
91,189
91,716
234,031
236,901
240,462
Nationwide
194,300
Voting Alternatives
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Run-Off Election
Instant Run-Off
Borda (rank-order voting)
Condorcet
Approval Voting
Run-Off Election
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If no candidate receives a majority of
the vote, the top two candidates meet
head-to-head in a second election.
Widely used in local elections and in
other countries.
Fringe candidates can sometimes skew
results (see France, 2002).
Instant Run-Off
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Voters rank all candidates.
If no candidate receives a majority, the
candidate receiving the fewest firstplace votes is eliminated, and votes for
the other candidates are shifted up.
Repeat as necessary.
Used in future San Francisco municipal
elections (ballot initiative, 2004).
Borda (Weighted) Voting
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Voters rank all n candidates.
First place receives n points; second
place (n - 1); third (n - 2); …
Used in college football and basketball
polls.
Condorcet Voting
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Voters rank all candidates.
Head-to-head comparisons are made.
The winner is the candidate who beats
every other candidate in a head-to-head
contest.
If voter preferences are not transitive,
there is no winner!
Approval Voting
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Voters either approve or disapprove of
each candidate.
The candidate with the most “approve”
votes is the winner.
The Trouble with Ranking
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It’s more complicated.
Voters need more information to
accurately cast their vote.
Strong incentives for insincere voting,
especially if you know how others are
likely to vote.
Some systems are more susceptible to
these weaknesses than others.
Some Sources
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Malkevitch, Joseph. “The Mathematical Theory of Elections.” COMAP,
1989.
Needham, Sam. Voting Methods course (Math 124) online at
http://voyager.dvc.edu/~sneedham/
Saari, Donald G. Chaotic Elections and Decisions and Elections by!
For a sample instant run-off vote (ice cream flavors), see
http://www.improvetherunoff.com/
Dasgupta, Partha, and Eric Maskin. “The Fairest Vote of all.” Scientific
American, vol. 290 #3, March 2004.
Historical Election Data:
 http://www.uselectionatlas.org/ -- a truly excellent site.
Photo Credits
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Chirac: http://www.rtvbih.ba/2002/vijesti/maj/04/
Jospin: http://www.newgenevacenter.org/movers/21st-cen-r.htm
Le Pen: http://www.adl.org/international/le-pen_new.asp
Ventura (wrestling):
http://www.secondaryenglish.com/WWF%20Table%20of%20Contents.html
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Ventura (portrait) Minnesota Historical Society,
http://www.mnhs.org/index.htm
No Candidate with a Majority
2000: George W. Bush
Al Gore
47.9%
48.4%
1996: Bill Clinton
Robert Dole
Ross Perot
49.2%
40.7%
8.4%
1992: Bill Clinton
George H. W. Bush
Ross Perot
43.0%
37.5%
18.9%