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Election Year Mathematics

Election Year Mathematics. Michael Buescher Hathaway Brown School mbuescher@hb.edu http://www.mbuescher.com/professional. Majority vs. Plurality. Majority: More than 50%. Plurality: More than any other candidate. Plurality Voting. Vote for one candidate.

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Election Year Mathematics

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  1. Election Year Mathematics Michael Buescher Hathaway Brown School mbuescher@hb.edu http://www.mbuescher.com/professional

  2. Majority vs. Plurality • Majority: More than 50%. • Plurality: More than any other candidate.

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

  4. The “Problem” with Plurality Voting Minnesota Gubernatorial Election, 1998 (Reform) Jesse “The Body” Ventura: 37% (Republican) Norm Coleman 35% (Democrat) Hubert Humphrey III 28%

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

  6. 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%

  7. 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

  8. Alabama Alaska Arizona California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri 1992 Presidential ElectionStates where winning candidate did not receive a majority of the vote • 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

  9. Arrow’s Criteria • Pareto Criterion • Condorcet Criterion • Monotonicity Criterion • Independence of Irrelevant Alternatives

  10. 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

  11. 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

  13. 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%

  14. Independence of Irrelevant Alternatives • Adding or removing a non-winning candidate should not change the results. • Plurality: fails • Electoral College: fails

  15. Arrow’s 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.

  16. Criterion: Equality of Votes • Every person’s 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

  18. 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

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

  20. 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).

  21. 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).

  22. 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.

  23. 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!

  24. Approval Voting • Voters either approve or disapprove of each candidate. • The candidate with the most “approve” votes is the winner.

  25. The Trouble with Ranking • 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.

  26. 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.

  27. 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_new.asp • Ventura (wrestling): http://www.secondaryenglish.com/WWF%20Table%20of%20Contents.html • Ventura (portrait) Minnesota Historical Society, http://www.mnhs.org/index.htm

  28. 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 Clinton43.0% George H. W. Bush 37.5% Ross Perot 18.9%

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