Download Plurality-with-elimination and more Exercises Political Systems in PDF only on Docsity!
Voting Methods discussed thus far: Plurality Borda Count Note: neither require a majority to select a winner What if we Need a Majority?
Plurality-with-elimination
F Also called Instant Runoff Voting F Guarentees winner has a majority of the votes F Eliminates low-vote candidates F Preference ballots- no need to run multiple elections
Round One
Count first place votes. Check for majority; if there is, declare the winner and you are finished. If no majority, eliminate the candidate with the least number of first place votes.
Round Two
Recount the first place votes. Check for majority and winner. Eliminate.
Rounds Three and so on
Repeat until a candidate has a majority.
Try plurality-with-elimination on the MAS Example: Preference Schedule: MAS Election Number of voters 14 10 8 4 1 First choice A C D B C Second choice B B C D D Third choice C D B C B Fourth choice D A A A A
Round One
Count first place votes:
A: 14, B: 4, C: 11, D: 8
Eliminate candidate B and rewrite the preference
schedule:
Preference Schedule: MAS Election(w/o B) Number of voters 14 10 8 4 1 First choice A C D D C Second choice C D C C D Third choice D A A A A
Round Two
Count first place votes:
A: 14, C: 11, D: 12
Eliminate candidate C and rewrite the preference
schedule:
As a result of the straw poll, 4 people change their first place vote from A to C, giving C more first place votes.
Preference Schedule, real election: Preference Schedule: Olympics Number of voters 7 8 14 First choice A B C Second choice B C A Third choice C A B
No majority, drop A.
Preference Schedule: Olympics Number of voters 7 8 14 First choice B B C Second choice C C B
B has majority and wins!
Monotonicity Criterion
If choice X is the winner of an election and, in a re- election, the only changes in the ballots are changes that favor X, then X should remain a winner of the election.
Method of Pairwise Comparisons
Compare each candidate one-on-one. A win is worth one point, a tie worth 12 point, and a loss worth nothing.
Make tables comparing each candidate: For example:
A B
Repeat for all combinations.
Math Appreciation Society Example:
Preference Schedule: MAS Election Number of voters 14 10 8 4 1 First choice A C D B C Second choice B B C D D Third choice C D B C B Fourth choice D A A A A
A B
A C
A D
B C
B D
C D
A B A C A D A E
B C B D B E
C D C E D E
Point totals: A gets 3 points. B gets 2^12 points. C gets 2 points. D gets 1^12 points. E gets 1 point. Allen wins.
What happens if Castillo accepts a different scholarship and no longer plays football?
Recount points, ignoring all comparisons with C.
A gets 2 points. B gets 2^12 points. D gets 1 21 points. E gets 0 points.
Byers wins.
Independence of Irrelevant
Alternatives Criterion
If candidate X is a winner of an election and in a recount one of the non-winning choices is removed from the ballots, then X should still be a winner of the election.
Method of Pairwise Comparisons violates the Independence of Irrelevant Alternatives Criterion.
What else can happen?
Icelandia State University Hockey Team votes on where to go to dinner- Andrechuk’s, Bure’s, or Chelio’s. Preference Schedule: Where to eat? Number of voters 4 2 6 3 First choice A B C B Second choice B C A A Third choice C A B C
Note that A beats B, B beats C, and C beats A. Three-way Tie!
Be sure to decide how to break ties before doing the election!
Summary G Plurality-with-Elimination- Remove the candidate with the least number of first place votes until one candidate has a majority. This candidate wins. G Guarentees the winner has a majority of votes G Instant runoff- no need to hold a separate election every time a candidate is dropped out of the race. G P-W-E Violates the Monotonicity Criterion G Used to choose Olympic Host City F Method of Pairwise Comparisons- compare candidates one-on-one, wins = 1 point, ties =^12 F Satisfies the Condorcet, Monotonicity and Majority Criteria F Violates the Independence of Irrelevant Alternatives Criterion F Possible for everyone to tie F n(n^2 − 1) pairs to compare
