My guy lost? What’s up with that…

 In the 1950’s, Kenneth Arrow, a mathematical economist, proved that a method for determining election results that is democratic and always fair is mathematically impossible.  Basically, any system that we could ever create will have inherent flaws.

 What does “democratic” and “fair” mean?  There are four criteria that mathematicians and political scientists have agreed a fair voting system should meet.

 A majority of first place votes (over 50%) is different from a plurality, which is just the largest quantity of first place votes.

Number of Votes 632 First ChoiceEGF Second ChoiceFHG Third ChoiceGFH Fourth ChoiceHEE Clearly, Candidate E should win since they have more than half of the 1 st place votes. But under the Borda count, the results are much different. E = 29 pts, F = 32 pts, G = 30 pts, and H = 19 pts This puts candidate E in third place, with candidate F winning the election – A clear violation of the Majority Criterion.

“Daffy the duck and Jock the West Highland Terrier.” Virginmedia.com

 If candidate is favored when compared separately – that is, head-to-head – with every other candidate, then that candidate should win the election.  In a pairwise comparison, it is possible to have a three-way tie, where A beats B, B beats C, and C beats A. This was shown by the Marquis de Condorcet whose name is sometimes associated with the head-to-head criterion (the Condorcet Criterion).

Number of Votes8644 First ChoiceACCB Second ChoiceBBAA Third ChoiceCABC In a Pairwise Comparison, A beats B (12 to 10), and A beats C (12 to 10). This makes A the winner in this method. But if the Plurality method were used, C would win the election with 10 votes, more than A’s 8 or B’s 4.

 Who knew math could look this impressive… Ross, Chip, Prof. “Julia Set.” Abacus.bates.edu,

 If a candidate wins an election and, in a reelection, the only changes are changes that favor the candidate, then that candidate should win the reelection.

 An initial poll is taken to see where people stand (before the actual election):  If this initial poll was run under plurality with elimination, then V would be eliminated and W would beat G with 36 votes (to 22). Number of Votes st placeWVGG 2 nd placeGWVW 3 rd placeVGWV

 Because of the results of the initial “election”, the 8 people in the last column change their votes to match those in the 1 st column: Number of Votes st placeWVGG 2 nd placeGWVW 3 rd placeVGWV Number of Votes st placeWVG 2 nd placeGWV 3 rd placeVGW

 When this new configuration is run (for the actual election), the results are now different  Now, candidate G will be eliminated (instead of V), and V will win the election with 30 votes (to W’s 28). Number of Votes st placeWVG 2 nd placeGWV 3 rd placeVGW

 If a candidate wins an election and, in a recount, the only changes are that one or more of the other candidates are removed from the ballot, then that candidate should still win the election.

 Here is an election run with pairwise comparison  After running the comparisons, E has 2 pts, F and G tie with 1.5 each, and H has 1 pt. Number of Votes st placeEGHH 2 nd placeFFEE 3 rd placeGHGF 4 th placeHEFG

 Because they lost the initial voting, candidates F and G pull out of the election. This is what remains of the preference table…  Now, because of the removed candidates, H now defeats E in the reelection. Number of votes st placeEHHH 2 nd placeHEEE

Fairness Criteria Plurality Method Borda CountPlurality with Elimination Pairwise Comparison Majority Criterion Always satisfies May not satisfy Always satisfies Always satisfies Head-to- Head May not satisfy Always satisfies MonotonicityAlways satisfies Always satisfies May not satisfy Always satisfies Irrelevant Alternatives May not satisfy

 P ; #1, 5, 7, 9