The Majority Criterion
If a candidate X has a majority of the first-place votes in an election, then candidate X should be the winner of the election.
The Condorcet Criterion
If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison, then candidate X should be the winner of the election.
The Monotonicity Criterion
If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor X (and only X), then X should remain a winner of the election.
The Independence-of-Irrelevant-Alternatives Criterion (IIA)
If candidate X is a winner of an election and in a recount one of the non-winning candidates is removed from the ballots, then X should still be winner of the election.
The Majority Criterion
If a candidate X has a majority of the first-place votes in an election, then candidate X should be the winner of the election.
If a choice receives a majority of the first-place votes in an election but does not win the election, we have a violation of the majority criterion. When there is no majority choice in an election, the majority criterion does not apply.
Ex. A candidate that has a majority of the first-place votes is automatically the winner under the plurality method so the plurality method satisfies the majority criterion.
· Does the Borda count method satisfy the majority criterion?
Number of Voters / 6 / 3 / 21st choice / C / P / P
2nd choice / P / S / S
3rd choice / S / C / C
C = Coke P = Pepsi S = Seven up
· Does the plurality-with-elimination method satisfy the majority criterion?
Yes, if there is a candidate that is the first choice of a majority of the voters, then using this method, that candidate will be declared the winner of the election in the first round.
· Does the pairwise comparisons method satisfy the majority criterion?
Yes, if candidate X is the first choice of a majority of the voters, then every head-to-head comparison between X and any other candidate will result in a win for X. X will win every pairwise comparison.
The Condorcet Criterion
If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison, then candidate X should be the winner of the election.
A candidate who wins in every head-to-head comparison against each of the other candidates is called the Condorcet candidate. When there is a Condorcet candidate, then that candidate should be the winner. When there is no Condorcet candidate, the Condorcet criterion does not apply.
· Which method guarantees that if there is a condorcet candidate, then that candidate will win?
· Does the plurality method satisfy the Condorcet criterion?
Number of Voters / 5 / 4 / 31st choice / C / S / P
2nd choice / S / C / S
3rd choice / P / P / C
· Does the Borda count method satisfy the Condorcet criterion?
No, in the MAS election, Carmen (C) is a Condorcet candidate but Boris (B) is the winner using the Borda count method, (Since there is no candidate with a majority of first-place votes, the majority criterion is not violated.)
· Does the plurality-with-elimination method satisfy the Condorcet criterion? Ex #60 a, b
The Monotonicity Criterion
If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor X (and only X), then X should remain a winner of the election.
· Does the plurality method satisfy the monotonicity criterion?
Yes, if choice X is the winner of an election using the plurality method and , in a reelection , the only changes in the ballots are changes that only favor X, then no candidate other than X can increase his/her first place votes and so X is still the winner of the election.
· Does the Borda count method satisfy the monotonicity criterion?
Yes, if choice X is the winner of an election using this method and , in a reelection, the only changes in the ballots are changes that only favor X, then X will get more points while the other candidates’ points will stay the same or decrease, so X is still the winner of the election.
· Does the plurality-with-elimination method satisfy the monotonicity criterion?
Ex # 34
· Does the pairwise comparisons method satisfy the monotonicity criterion?
Yes, if X is the winner of an election using this method and, in a reelection, the only changes in the ballots are changes that only favor X, then candidate X will still win every pairwise comparison he/she won in the original election and possibly even some new ones – while no other candidate will win any new pairwise comparisons (since there were no changes favorable to any other candidate). Consequently, X is also the winner of the reelection.
The Independence-of-Irrelevant-Alternatives Criterion
If candidate X is a winner of an election and in a recount one of the non-winning candidates is removed from the ballots, then X should still be winner of the election.
· Does the plurality method satisfy this criterion?
Number of Voters / 5 / 4 / 31st choice / A / B / C
2nd choice / B / C / B
3rd choice / C / A / A
C drops out
· Does the Borda count method satisfy this criterion?
Number of Voters / 5 / 3 / 11st choice / A / C / B
2nd choice / C / B / C
3rd choice / B / A / A
B drops out
· Does the plurality-with-elimination method satisfy this criterion?
Ex #60 b, c [ B drops out - not D]
· Does the pairwise comparisons method satisfy this criterion?
Ex #35