Learning Outcomes
- Apply the fairness criteria to evaluate the fairness of contests whose outcome is determined using the voting methods of the previous two sections.
Fairness Criteria
The fairness criteria are statements that seem like they should be true in a fair election.
- Condorcet Criterion: If a choice is favored over every other candidate in one on one comparisons, that choice should be the winner.
- Majority Criterion: If a choice has a majority of first-place votes, that choice should be the winner.
- Monotonicity Criterion: If voters change their votes to increase the preference for a choice, it should not harm that choice’s chances of winning.
- Independence of Irrelevant Alternatives (IIA) Criterion: If a non-winning choice is removed from the ballot, it should not change the winner of the election.
In this section, we will consider examples that exploit some of the ways in which our voting methods fail to meet various fairness criteria. We have already hinted at some of these issues in previous examples.
Important Note
Even though a criterion may be violated by a voting method in one particular contest scenario, it does not mean that voting method always violates the criterion. It simply demonstrates that the voting method has the potential to violate the criterion in certain election scenarios.
What’s Wrong with the Plurality Method?
The following video includes a a review of the plurality method including an example that illustrates this method’s ability to violate the Condorcet fairness criterion.
The next example gives another illustration of an election in which the plurality winner is not the Condorcet Winner.
EXAMPLE
Consider a city council election in a district that is historically 60% Democratic voters and 40% Republican voters. Even though city council is technically a nonpartisan office, people generally know the affiliations of the candidates. In this election there are three candidates: Don and Key, both Democrats, and Elle, a Republican. A preference schedule for the votes looks as follows:
| 342 | 214 | 298 | |
| 1st choice | Elle | Don | Key |
| 2nd choice | Don | Key | Don |
| 3rd choice | Key | Elle | Elle |
Notice that a total of [latex]342+214+298=854[/latex] voters participated in this election. Computing percentage of first place votes:
- Don: 214/854 = 25.1%
- Key: 298/854 = 34.9%
- Elle: 342/854 = 40.0%
So in this election, there is no majority winner. It appears that the Democratic voters split their vote over the two Democratic candidates, allowing the Republican candidate Elle to win under the plurality method with only 40% of the vote.
Analyzing this election further, we calculate the one on one comparisons:
- Elle vs Don: 342 prefer Elle; 512 prefer Don: Don is preferred
- Elle vs Key: 342 prefer Elle; 512 prefer Key: Key is preferred
- Don vs Key: 556 prefer Don; 298 prefer Key: Don is preferred
So even though Don had the smallest number of first-place votes in the election, he is the Condorcet Winner, being preferred in every one-to-one comparison with the other candidates. Thus, the plurality method, in this scenario violates the Condorcet Criterion.
This example is worked out in the following video.
What’s Wrong with the Borda Count Method?
Let’s consider the example from the last section in which a group of mathematicians are meeting for a conference. Perhaps you already noticed a potential flaw of the Borda Count from that example. In particular, Seattle had a majority of first-choice votes, yet lost the election! This example demonstrates that the Borda Count method has the potential to violate the Majority Criterion. We will also note in this example the Borda Count method’s potential to violate the Condorcet Criterion.
Recall: Example
In this example, our group of mathematicians are getting together for a conference. The members are coming from four cities: Seattle, Tacoma, Puyallup, and Olympia.
Their approximate locations on a map are shown to the right.
The votes for where to hold the conference were:
| 51 | 25 | 10 | 14 | |
| 1st choice | Seattle | Tacoma | Puyallup | Olympia |
| 2nd choice | Tacoma | Puyallup | Tacoma | Tacoma |
| 3rd choice | Olympia | Olympia | Olympia | Puyallup |
| 4th choice | Puyallup | Seattle | Seattle | Seattle |
Verify that the Borda Count method violates the Majority Criterion in this scenario.
While we have demonstrated the Borda Count method’s ability to violate at least two of the fairness criteria, there are some important benefits to this method. In particular, Borda can sometimes choose a more broadly acceptable option over the one with majority support and for this reason is sometimes described as a consensus-based voting system. In the previous example, Tacoma is probably the best compromise location. This is a different approach than plurality and instant run-off voting that focus on first-choice votes; Borda Count considers every voter’s entire ranking to determine the outcome.
Because of this consensus behavior, Borda Count, or some variation of it, is commonly used in awarding sports awards. Variations are used to determine the Most Valuable Player in baseball, to rank teams in NCAA sports, and to award the Heisman trophy.
What’s Wrong with Run-off Methods?
Next, we consider election scenarios in which run-off methods violate, in the first example, the Condorcet Criterion and in the second example, the Monotonicity Criterion.
Examples
Example 1: Returning to the first example on this page, we consider again our City Council Election. Use a run-off method to find the winner of this election. (Recall that IRV and single run-off are the same when we have only three choices.) Then, verify that a run-off method used on this election scenario would violate the Condorcet Criterion.
| 342 | 214 | 298 | |
| 1st choice | Elle | Don | Key |
| 2nd choice | Don | Key | Don |
| 3rd choice | Key | Elle | Elle |
The following video works out this example in which we find that the IRV method violates the Condorcet Criterion in an election for a city council seat.
Example 2: Suppose a student club at Colorado Mesa University has three candidates for Club President: Adams, Brown, and Carter. A preference ballot is used and the preference schedule is shown below. First, determine a winner using a run-off method. Note that in this election scenario, the Monotonicity Criterion will be violated.
| 37 | 22 | 12 | 29 | |
| 1st choice | Adams | Brown | Brown | Carter |
| 2nd choice | Brown | Carter | Adams | Adams |
| 3rd choice | Carter | Adams | Carter | Brown |
Example 2 is explained in the following video.
What’s Wrong with the Pairwise Comparisons Method?
The idea behind the Independence of Irrelevant Altenartive Criterion is this: suppose a voting method yields a winner and one of the losing choices is no longer available (disqualified, say) then, applying the same preferences and the same voting method should still produce the same winner.
Example
Consider the preference schedule shown below. Show that the Independence of Irrelevant Alternatives Criterion is violated by the pairwise comparisons method used on this preference schedule.
| 11 | 7 | 6 | 9 | 3 | |
| 1st choice | B | D | C | A | D |
| 2nd choice | A | B | A | C | C |
| 3rd choice | C | C | D | D | B |
| 4th choice | D | A | B | B | A |
Important Note (Worth repeating!)
Even though a criterion may be violated by a voting method in one particular contest scenario, it does not mean that voting method always violates the criterion. It simply demonstrates that the voting method has the potential to violate the criterion in certain election scenarios.
