Transcript Defects in Voting Methods - University of New Mexico

1.
The majority criterion
2. Condorcet’s criterion
3. independence of
irrelevant alternatives
criterion
4. Monotonicity criterion
 Def.
› If a majority* of the voters rank a
candidate as their first choice, then
that candidate should win the
election.
› * Majority means more than half.
 Which
4 methods satisfy the
majority criterion?
› Plurality, Borda, Plurality w/ Elmin., or
Pairwise
› Plurality and Pairwise

Car Magazine has narrowed down your
choice of car of the year to the following
four cars. (A)udi A8, (B)MW Z4, (D)odge
Viper, and (C)orvette Z06. A people
editorial panel, using the Borda method
will make the final decision.
Preference
1st
2nd
3rd
4th

8
C
D
B
A
10
D
A
B
C
5
C
B
D
A
2
A
B
D
C
Use the Borda and the following
preference table to determine whether
the election satisfies the majority
criterion.
 The
winner is D if 78 pts. But C
has the majority of first place
votes, the majority criterion is
not satisfied.
 C = 13/25 = 52%,
 A = 2/25 = 8%,
 D = 10/25 = 40%
 Def.
› If candidate X can defeat each
of the other candidates in a
head to head vote, then X is the
winner of the election.
 Note:
The plurality method
may violate Condorcet’s
criterion.

Marie-Jean-Antonie-Nicolas Caritat, the
Marquis de Condorcet, was an
eighteenth century French Nobleman.

Regarded as a philosopher and
mathematician.

In 1785 Wrote a paper, Essay on the
Applications of Mathematics to the
Theory of Decision Making.
 Believed
that scientists could employ
mathematics to free society from its
reliance on greedy capitalists and to
develop insurance programs to
protect the poor, and, once we
discover appropriate moral laws,
crime and war would disappear.
 Believed
in abolishing slavery,
opposed capital punishment,
advocated free speech, and
supported early feminists lobbying for
equal rights.

Women, having these same qualities,
must necessarily possess equal rights.
Either no individual of the human species
has any true rights, or all have the same.
And he who votes against the rights of
another, or whatever religion, color, or
sex, has thereby abjured his own.*

*1790 (On Admission of Women to the
Rights of Citizenship)
 Def.
› If candidate X wins an election,
some nonwinners are removed from
the ballot, and a recount is done,
then X still wins the election.

Suppose that after an election has been
conducted using preference ballots we
discover that one of the losing
candidates should never have been
listed on the ballot. Rather than have a
new election, we could strike that
candidate’s name from each ballot and
do a recount.

The county board of supervisors is voting
on methods to raise taxes to finance a
new sports stadium. The options are
levying a tax on (H)otel rooms,
increasing the tax on (A)lcohol, and
increasing the tax on (G)asoline.
Preference
1st
2nd
3rd
8
A
G
H
6
H
G
A
6
G
A
H

A has the most votes. Does this satisfy
the independence of irrelevant
alternatives criterion?

Before we answer this question, does the
removal of one of the losing options
changes the outcome of the election?
Preference
1st
2nd
 If,
8
A
G
6
G
A
6
G
A
due to lobbying pressure, the
board removes the tax on hotel
rooms as an option. We see now
that gasoline tax now wins by a
vote of 12 to 8.
The
plurality method does
not satisfy the
independence of irrelevant
alternatives criterion.

Preference
5
3
3
1
1st
W
Z
Y
Z
2nd
X
W
X
W
3rd
Y
Y
Z
X
4th
Z
X
W
Y
Use pairwise to determine the winner of
the election summarized in the table
above. Is the criterion satisfied?

Using the pairwise the winner is W.
However, if X and Y are removed, then Z
wins the election. The criterion is not
satisfied.
 Def.
› If X wins an election and in a
reelection all voters who change
their votes only change their votes
to favor X, then X also wins the
reelection.
 Which
of the 4 would most likely
violate this criterion?
 The
plurality with elimination