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 6 3 2
Voters
1st 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 5
Voters
1st choice
C
2nd choice
S
3rd choice
P
4
3
S
C
P
P
S
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 5
Voters
1st choice
A
2nd choice
B
3rd choice
C
4
3
B
C
A
C
B
A
C drops out
 Does the Borda count method satisfy this criterion?
Number of 5
Voters
1st choice
A
2nd choice
C
3rd choice
B
3
1
C
B
A
B
C
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