Defects in Voting MethodsChapter 12 sec 2.

4 Fairness criteria

1. The majority criterion 2. Condorcet’s criterion 3. independence of

irrelevant alternatives criterion

4. Monotonicity criterion

The Majority 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.

Majority Criterion

Which 4 methods satisfy the majority criterion?› Plurality, Borda, Plurality w/ Elmin.,

or Pairwise

› Plurality and Pairwise

Borda violates majority

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.

Use the Borda and the following preference table to determine whether the election satisfies the majority criterion.

Preference 8 10 5 21st C D C A2nd D A B B3rd B B D D4th A C A C

Who is the winner?

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%

Condorcet’s Criterion

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.

Condorcet

Note: The plurality method may violate Condorcet’s criterion.

Who was Condorcet? 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.

Quote: 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)

Independence of Irrelevant Alternatives.

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.

Means what!?

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.

Example

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.

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 8 6 6

1st A H G

2nd G G A

3rd H A H

If, 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.

Preference 8 6 6

1st A G G

2nd G A A

Therefore:

The plurality method does not satisfy the independence of irrelevant alternatives criterion.

Try this

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?

Solution

Using the pairwise the winner is W. However, if X and Y are removed, then Z wins the election. The criterion is not satisfied.

Monotonicity Criterion

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