11.2: Defects in Voting Methods

Now that we have reviewed four different voting methods, how do you decide which method to use? One question to ask is which method is the fairest? Unfortunately, there is no completely fair method. What are the fairness criteria? They are guidelines that people use to help decide which voting method would be best to use under certain circumstances. They are the Majority Criterion, Condorcet Criterion, Monotonicity Criterion, and Independence of Irrelevant Alternatives Criterion.

The Majority Criterion

If a candidate receives a majority of the 1st-place votes in an election, then that candidate should be the winner of the election.

Example 1: Majority Criterion Violated

Suppose a group is planning to have a conference in one of four Arizona cities: Flagstaff, Phoenix, Tucson, or Yuma. The votes for where to hold the conference are summarized in the preference schedule shown below in Table \(\PageIndex{1}\).

Table \(\PageIndex{1}\): Preference Schedule for Conference City

Number of voters	51	25	10	14
1st choice	Flagstaff	Phoenix	Yuma	Tucson
2nd choice	Phoenix	Yuma	Phoenix	Phoenix
3rd choice	Tucson	Tucson	Tucson	Yuma
4th choice	Yuma	Flagstaff	Flagstaff	Flagstaff

Solution

If we use the Borda Count Method to determine the winner then the number of Borda points that each candidate receives are shown in Table \(\PageIndex{2}\).

Table \(\PageIndex{2}\): Preference Schedule for Conference City with Borda Points

Number of voters	51	25	10	14
1st choice 4 points	Flagstaff 204	Phoenix 100	Yuma 40	Tucson 56
2nd choice 3 points	Phoenix 153	Yuma 75	Phoenix 30	Phoenix 42
3rd choice 2 points	Tucson 102	Tucson 50	Tucson 20	Yuma 28
4th choice 1 point	Yuma 51	Flagstaff 25	Flagstaff 10	Flagstaff 14

The totals of all the Borda points for each city are:

Phoenix:

Yuma:

Tucson:

Phoenix wins using the Borda Count Method. However, notice that Flagstaff actually has the majority of first-place votes. There are 100 voters total and 51 voters voted for Flagstaff in first place (51/100 = 51% or a majority of the first-place votes). So, Flagstaff should have won based on the Majority Criterion. This shows how the Borda Count Method can violate the Majority Criterion.

The Condorcet Criterion

If there is a candidate that in a head-to-head comparison is preferred by the voters over every other candidate, then that candidate should be the winner of the election. This candidate is known as the Condorcet candidate.

Example 2: The Condorcet Criterion Violated

Suppose you have a vacation club trying to figure out where it wants to spend next year’s vacation. The choices are Hawaii (H), Anaheim (A), or Orlando (O). The preference schedule for this election is shown below in Table \(\PageIndex{3}\).

Table \(\PageIndex{3}\): Preference Schedule of Vacation Election

Number of voters	1	3	3	3
1st choice	A	A	O	H
2nd choice	O	H	H	A
3rd choice	H	O	A	O

Solution

Using the Plurality Method, A has four first-place votes, O has three first-place votes, and H has three first-place votes. So, Anaheim is the winner. However, if you use the Method of Pairwise Comparisons, A beats O (A has seven while O has three), H beats A (H has six while A has four), and H beats O (H has six while O has four). Thus, Hawaii wins all pairwise comparisons against the other candidates, and would win the election. In fact Hawaii is the Condorcet candidate. However, the Plurality Method declared Anaheim the winner, so the Plurality Method violated the Condorcet Criterion.

The Monotonicity Criterion

If candidate X is a 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.

Example 3: Monotonicity Criterion Violated

Suppose you have a voting system for a mayor. The resulting preference schedule for this election is shown below in Table \(\PageIndex{4}\).

Table \(\PageIndex{4}\): Preference Schedule of Mayoral Election

Number of voters	37	22	12	29
1st choice	Adams	Brown	Brown	Carter
2nd choice	Brown	Carter	Adams	Adams
3rd choice	Carter	Adams	Carter	Brown

Solution

Using the Plurality with Elimination Method, Adams has 37 first-place votes, Brown has 34, and Carter has 29, so Carter would be eliminated. Carter’s votes go to Adams, and Adams wins. Suppose that the results were announced, but then the election officials accidentally destroyed the ballots before they could be certified, so the election must be held again. Wanting to “jump on the bandwagon,” 10 of the voters who had originally voted in the order Brown, Adams, Carter; change their vote to the order of Adams, Brown, Carter. No other voting changes are made. Thus, the only voting changes are in favor of Adams. The new preference schedule is shown below in Table \(\PageIndex{5}\).

Table \(\PageIndex{5}\): Preference Schedule of Mayoral Re-election

Number of voters	47	22	2	29
1st choice	Adams	Brown	Brown	Carter
2nd choice	Brown	Carter	Adams	Adams
3rd choice	Carter	Adams	Carter	Brown

Now using the Plurality with Elimination Method, Adams has 47 first-place votes, Brown has 24, and Carter has 29. This time, Brown is eliminated first instead of Carter. Two of Brown’s votes go to Adams and 22 of Brown’s votes go to Carter. Now, Adams has 47 + 2 = 49 votes and Carter has 29 + 22 = 51 votes. Carter wins the election. This doesn’t make sense since Adams had won the election before, and the only changes that were made to the ballots were in favor of Adams. However, Adams doesn’t win the re-election. The reason that this happened is that there was a difference in who was eliminated first, and that caused a difference in how the votes are re-distributed. In this example, the Plurality with Elimination Method violates the Monotonicity Criterion.

The Independence of Irrelevant Alternatives Criterion

If candidate X is a winner of an election and one (or more) of the other candidates is removed and the ballots recounted, then X should still be a winner of the election.

Example 4: The Independence of Irrelevant Alternatives Criterion Violated

A committee is trying to award a scholarship to one of four students: Anna (A), Brian (B), Carlos (C), and Dmitri (D). The votes are shown below.

Table \(\PageIndex{6}\): Preference Schedule for Scholarship

Number of voters	5	5	6	4
1st choice	D	A	C	B
2nd choice	A	C	B	D
3rd choice	C	B	D	A
4th choice	B	D	A	C

Solution

Using the Method of Pairwise Comparisons:

A vs B: 10 votes to 10 votes, A gets ½ point and B gets ½ point

A vs C: 14 votes to 6 votes, A gets 1 point

A vs D: 5 votes to 15 votes, D gets 1 point

B vs C: 4 votes to 16 votes, C gets 1 point

B vs D: 15 votes to 5 votes, B gets 1 point

C vs D: 11 votes to 9 votes, C gets 1 point

So A has 1½ points, B has 1 point, C has 2 points, and D has 1 point. So Carlos is awarded the scholarship.

Now suppose it turns out that Dmitri didn’t qualify for the scholarship after all. Though it should make no difference, the committee decides to recount the vote. The preference schedule without Dmitri is below.

Table \(\PageIndex{7}\): Preference Schedule for Scholarship with Dmitri Removed

Number of voters	10	6	4
1st choice	A	C	B
2nd choice	C	B	A
3rd choice	B	A	C

Using the Method of Pairwise Comparisons:

A vs B: 10 votes to 10 votes, A gets ½ point and B gets ½ point

A vs C: 14 votes to 6 votes, A gets 1 point

B vs C: 4 votes to 16 votes, C gets 1 point

So A has 1½ points, B has ½ point, and C has 1 point. Now Anna is awarded the scholarship instead of Carlos. This is an example of The Method of Pairwise Comparisons violating the Independence of Irrelevant Alternatives Criterion.

Arrow's Impossibility Theorem

No voting system can satisfy all four fairness criteria in all cases.

In summary, every one of the fairness criteria can possibly be violated by at least one of the voting methods as shown in Table \(\PageIndex{8}\). However, keep in mind that this does not mean that the voting method in question will violate a criterion in every election. It is just important to know that these violations are possible.

Table \(\PageIndex{8}\): Summary of Violations of Fairness Criteria

	Plurality	Borda Count	Plurality with Elimination	Pairwise Comparisons
Majority Criterion	*	Violation Possible	*	*
Condorcet Criterion	Violation Possible	Violation Possible	Violation Possible	*
Monotonicity Criterion	*	*	Violation Possible	*
Independence of Irrelevant Alternatives Criterion	Violation Possible	Violation Possible	Violation Possible	Violation Possible

* The indicated voting method does not violate the indicated criterion in any election.