Election Year Mathematics

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Election Year Mathematics

• Michael Buescher
• Hathaway Brown School
• mbuescher_at_hb.edu
• http//www.mbuescher.com/professional

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Majority vs. Plurality

• Majority More than 50.
• Plurality More than any other candidate.

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Plurality Voting

• Vote for one candidate.
• The candidate with more votes than any other
  candidate wins the election.

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The Problem with Plurality Voting
Minnesota Gubernatorial Election, 1998 (Reform)
Jesse The Body Ventura 37 (Republican)
Norm Coleman 35 (Democrat) Hubert Humphrey
III 28
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Voting for the President

• Each state determines a winner through Plurality
  voting.
• State results are combined in the Electoral
  College.

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QUIZ!

• Who was the last president who won a majority of
  the popular vote?
• George H. W. Bush (1988)

1988 George H. W. Bush 53.4 Michael
Dukakis 45.7
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2000 Presidential ElectionStates where winning
candidate did not receive a majority of the vote

• Florida
• Iowa
• Maine
• Minnesota
• Nevada
• New Hampshire
• New Mexico
• Ohio
• Oregon
• Wisconsin

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1992 Presidential ElectionStates where winning
candidate did not receive a majority of the vote

• Alabama
• Alaska
• Arizona
• California
• Colorado
• Connecticut
• Delaware
• Florida
• Georgia
• Hawaii
• Idaho
• Illinois

• Indiana
• Iowa
• Kansas
• Kentucky
• Louisiana
• Maine
• Maryland
• Massachusetts
• Michigan
• Minnesota
• Mississippi
• Missouri

• Montana
• Nebraska
• Nevada
• New Hampshire
• New Jersey
• New Mexico
• New York
• North Carolina
• North Dakota
• Ohio
• Oklahoma
• Oregon

• Pennsylvania
• Rhode Island
• South Carolina
• South Dakota
• Tennessee
• Texas
• Utah
• Vermont
• Virginia
• Washington
• West Virginia
• Wisconsin
• Wyoming

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Arrows Criteria

• Pareto Criterion
• Condorcet Criterion
• Monotonicity Criterion
• Independence of Irrelevant Alternatives

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Pareto (Majority) Criterion

• If a majority NOT plurality! of voters prefers
  candidate A over all others, then A should win
  the election.
• Plurality voting passes
• Electoral College fails

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Condorcet Criterion

• 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

12
Monotonicity Criterion

• If voters change their mind and rank a candidate
  higher than they used to, it should not hurt that
  candidate.
• Plurality voting passes
• Electoral college passes

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Monotonicity Fails France 2002
The Rules Vote for your favorite candidate. If
no candidate receives a majority, there is a
runoff between the top two vote-getters.
First Round Results Jacques Chirac 19.9
Jean-Marie le Pen 16.9 Lionel Jospin 16.2
First Round Results Jacques Chirac 20.9
Jean-Marie le Pen 15.9 Lionel Jospin 16.2
Jacques Chirac
Lionel Jospin
The Polls Widely expected runoff between
Jacques Chirac (incumbent) and Lionel Jospin
Jospin heavily favored to win the runoff.
Second Round Chirac 82, LePen 18
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Independence of Irrelevant Alternatives

• Adding or removing a non-winning candidate should
  not change the results.
• Plurality fails
• Electoral College fails

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Arrows Theorem

• The only voting system that satisfies all of
  these criteria when there are more than two
  candidates is
• A DICTATORSHIP
• Only one person votes.
• For this, Arrow wins the Nobel Prize in Economics.

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Criterion Equality of Votes

• Every persons vote should carry the same weight.
• Plurality passes
• Electoral College fails

17
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

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Inequality of votes Electoral College

• Number of votes per electoral vote
• (2000 presidential election)
• Wyoming (3) 84,893
• Hawaii (4) 91,189
• Alaska (3) 91,716
• Wisconsin (11) 234,031
• Florida (25) 236,901
• Minnesota (10) 240,462
• Nationwide 194,300

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Voting Alternatives

• Run-Off Election
• Instant Run-Off
• Borda (rank-order voting)
• Condorcet
• Approval Voting

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Run-Off Election

• 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).

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Instant Run-Off

• Voters rank all candidates.
• If no candidate receives a majority, the
  candidate receiving the fewest first-place 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).

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Borda (Weighted) Voting

• Voters rank all n candidates.
• First place receives n points second place (n -
  1) third (n - 2)
• Used in college football and basketball polls.

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Condorcet Voting

• 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!

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Approval Voting

• Voters either approve or disapprove of each
  candidate.
• The candidate with the most approve votes is
  the winner.

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The Trouble with Ranking

• Its 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.

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Some Sources

• 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.

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Photo Credits

• 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_ne
  w.asp
• Ventura (wrestling) http//www.secondaryenglish.c
  om/WWF20Table20of20Contents.html
• Ventura (portrait) Minnesota Historical Society,
  http//www.mnhs.org/index.htm

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No Candidate with a Majority
2000 George W. Bush 47.9 Al Gore 48.4
1996 Bill Clinton 49.2 Robert
Dole 40.7 Ross Perot 8.4
1992 Bill Clinton 43.0 George H. W.
Bush 37.5 Ross Perot 18.9