Math and Voting. October 22, 2009 Maura Bardos. Outline. Two Candidates Majority Rule Three Candidates or More Plurality Borda Condorcet Sequential Pairwise Instant Runoff Arrow’s Theorem Approval voting A better method?. 3 Properties of Fair Elections.
October 22, 2009
Can you think of an examples where these criteria fail?
Can you think of an example where all three properties are satisfied for a two candidate election?
Obama: 1,959,532 votes
McCain: 1,725,005 votes
Total Votes cast:
6 Students: Salad > Chips > Popcorn
5 Students: Popcorn > Chips > Salad
4 Students: Chips > Popcorn > Salad
Plurality: Salad Wins!
5 Students (33%): Popcorn > Chips > Salad
4 Students (27%): Chips > Popcorn > Salad
We get to the store…we see that Bloom is sold out of Popcorn.
What difference does it make? Lets Revisit our preferences
6 Students (40%): Salad > Chips
5 Students (33%): Chips > Salad
4 Students (27%): Chips > Salad
60% prefer chips to Salad.
5 Students (33%): Popcorn > Salad
4 Students (27%): Popcorn > Salad
Either way- voters prefer anything to Salad.
With majority rule- we select a “winner” that the voters don’t really want. Note that voter preferences did not change
First Place is worth n-1 points
Second Place is worth n-2 points
…Last Place is worth n-n=0 points
How many points to award?
Top Rank = n-1 points, where n is the number of candidates
….Last Ranked = 0 points
Borda Score for :
A = 3 (2 points) + 2 (0 points) = 6
B = 3 ( 1 point) + 2 (1 point) = 5
C = 3 (0 points) + 2 (2 points) = 4
Candidate A is the winner
Lets switch the rank of B and C.
Now recalculate the Borda Score
A = 6 (same as last time)
B = 3 (1 point) + 2( 2 points) = 7
C = 3 (0 points) + 2(1 point) = 2
Candidate B is the winner.
Presentation packet Problem #1:
Salad: 6 (2 points) + 5 ( 0 points) + 4 ( 0 points) = 12
Chips: 6 (1 points) + 5 ( 1 points) + 4 ( 4 points) = 27
Popcorn: 6 (0 points) + 5 ( 2 points) + 4 ( 1 points) = 14
Morneau: (15 x 14) + (8 x 9) + (3 x 8) + (2 x 7) = 320
Jeter: (12 x 14) + (14 x 9) + (1 x 7) + (1 x 5) = 306
Photo source: http://en.wikipedia.org/wiki/Minnesota_gubernatorial_election,_1998
Jesse Ventura (Reform Party)
St. Paul Mayor Norm Coleman (R)
Attorney General Skip Humphrey (D)
1998 Minnesota Governors race with Jesse Ventura (Reform Party), Attorney General Skip Humphrey (D), and St. Paul Mayor Norm Coleman (R).
Lets examine who wins the election under a variety of systems
Now try Problem 2
What about other voting Systems:
“The Condorcet Criterion shall be used to determine the results, and if there is a tie, the Adjusted Borda Count, direct paired comparisons, the Borda Count, and a deciding vote by the Dean, are to be used sequentially, until the tie is broken.”
a vs. b: a a vs. c: c c vs. d: dAgenda: BCAD
b vs. c: b a vs. b: a a vs. d: a
a vs. c: c b vs. c: b b vs. d: bAgenda: ABDC
a vs. b: a a vs. d: a a vs. d: c
But a Condorcet winner doesn’t always exist. In these situations, the result is contingent in the agenda.
In general, the later an alternative is introduced, the better its chances of winner.
Obviously not applicable for elections
Used in single elimination tournaments, such as tournaments where teams are ‘seeded’
Original Procedure (for awards 1936-2008)
Minimum number of votes a candidate must receive to be the winner
For our example, lets assume that there are n=30 voters (total valid poll) and k=5 films to nominate (seats)
Quota = 6
9-6=3 excess votes are distributed to C and A
Films G, C, B, A and D:
A: MilkB: Slumdog MillionaireC: Curious Case of Benjamin ButtonD: The ReaderG: Frost/Nixon
Note that E, Quantum of Solace, was the Condorcet winner.
In previous Oscars- the nomination processes narrowed down the film to five nominees
Each fails to satisfy one desirable property
“The only voting method that isn't flawed is a dictatorship“
With three of more candidates an any number of voters, there does not exist a voting system that always produces a winner that satisfies the following criteria:
(Hodge and Klima)
Theorem: With three or more candidates and an odd number of votes, there does not exist- and there will never exist a voting system that satisfies both the Condorcet winner criterion and the independence of irrelevant alternatives and that always produces at least one winner in every election (COMAP).
In head to head:
A > B
B > C
12th Amendment- requires 270 votes in the electoral college to win a presidential election.
Is 269 – 269 tie possible?
Rank the following:
2) Research a ranking/decision making method (such as sports, Olympic games, election method in a foreign country). What method is used? Pick a particular occurrence and describe a surprising outcome.
William and Mary Links
Voting and Social Choice, Princeton University. http://www.math.princeton.edu/math_alive/6/index.shtml