Section 7.7: Monotoncity Criterion
- Page ID
- 219584
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Imagine this scenario: Election Day arrives. The votes are counted using the plurality with elimination method or Instant Runoff Voting (IRV), and Candidate A wins. Now imagine a parallel universe where everything is identical except:
- A small group of voters who originally ranked Candidate A in 3rd place decide they like A more
- They move A up to 1st place on their ballots
- Nothing else changes and all other rankings remain the same
Question: Should Candidate A still win? Obvious answer: Of course! If Candidate A won before, and then gained additional support, A should definitely still win. Getting more support should never hurt a candidate.
The shocking reality: Under IRV (and some other methods), Candidate A might now lose. This paradox, where gaining additional support causes a candidate to lose, is where we investigate our third fairness criterion called the Monotonicity criterion.
Monotonicity Criterion (also called the Monotonicity Fairness Criterion or Participation Criterion):
If a candidate wins an election, and then in a re-election (or hypothetical scenario) some voters increase their support for that candidate (by ranking them higher) while everything else remains unchanged, that candidate should still win.
Receiving additional support or moving up on voters' preference rankings should never hurt a candidate's chances of winning.
Unpacking the Definition
Consider an election in which Candidate X wins under a given voting method. Now suppose that some voters modify their ballots so that Candidate X is ranked higher than before. These changes are strictly favorable to Candidate X: the candidate may be moved from third place to first, from second to first, or from a lower position to a higher one. Aside from the necessary adjustments required to move Candidate X upward, the relative ordering of all other candidates on those ballots remains unchanged. All other voters’ ballots, as well as all other conditions of the election, remain exactly the same.
Under a voting system satisfying monotonicity, Candidate X should still win the election after these changes. A violation occurs if, despite receiving greater support in the form of higher rankings, Candidate X no longer wins.
The Intuitive Principle
The logic behind this requirement is straightforward. In a reasonable voting system, increased support for a candidate should never work to that candidate’s disadvantage. If higher rankings represent stronger approval, then granting Candidate X more favorable rankings should improve, or at least not worsen, Candidate X’s chances of winning.
This principle aligns with common ideas of causality and voter empowerment. Voters should be able to support their preferred candidate without fear that doing so might backfire. If Candidate X defeats another candidate with a certain level of support, it would be logically inconsistent for Candidate X to lose after receiving even more support. A voting rule that violates this expectation creates perverse incentives by forcing voters to worry that honest and enthusiastic support could harm the very candidate they wish to help. In everyday terms, ranking Candidate X higher should help Candidate X, not hurt them.
Why This Matters: The Participation Paradox
Monotonicity violations have a troubling implication known as the participation paradox. If ranking Candidate X higher can cause Candidate X to lose, then it may be possible for a voter to better support Candidate X by abstaining from the election altogether or by ranking Candidate X lower. In extreme cases, staying home may be the most effective way to help Candidate X win.
For example, consider an election conducted under instant‑runoff voting. Suppose that Candidate X would win if a particular voter did not participate. If that voter instead casts a ballot ranking Candidate X first, the added support may alter the elimination order in early rounds, ultimately causing Candidate X to lose. The voter’s intention is to help Candidate X, but the effect of voting is the opposite. The rational strategy, paradoxically, is not to vote at all. This outcome is deeply counterintuitive and undermines the democratic ideal that participation and honest expression of preferences should be encouraged rather than punished.
Mathematical Explanation: How Violations Occur
Monotonicity violations arise from the internal mechanics of certain voting systems, particularly instant‑runoff voting. These systems eliminate candidates sequentially, and the order of elimination plays a crucial role in determining the final outcome. Because the elimination process is path‑dependent, small changes in the distribution of first‑place votes can trigger entirely different elimination sequences, leading to different winners.
When a candidate is eliminated, their votes are transferred to other candidates according to the next preferences listed on those ballots. The destination of these transferred votes influences which candidate is eliminated next, and this process continues until a winner emerges. Changing how some voters rank Candidate X can redirect vote transfers in subtle but consequential ways, producing cascading effects throughout the successive rounds.
As a result, Candidate X may benefit from certain opponents being eliminated early and from receiving transferred votes later in the process. By increasing support for Candidate X early on, voters may inadvertently prevent a favorable elimination sequence from occurring. A different candidate may be eliminated instead, leading to a transfer pattern that ultimately deprives Candidate X of the support needed to win. Thus, although Candidate X gains direct support, the altered elimination order changes the flow of transferred votes in a way that harms Candidate X’s overall chances.
In short, the additional support changes the path of the election, and the new path leads to a worse outcome. This counterintuitive result explains how a voting system can violate monotonicity even when voters act sincerely and consistently.
Consider the following preference table with Candidates A, B, & C:
| Number of Voters | 5 | 4 | 3 | 2 | 1 |
|---|---|---|---|---|---|
| First Choice | A | B | C | C | A |
| Second Choice | C | A | B | A | B |
| Third Choice | B | C | A | B | C |
- Use the plurality-with-elimination method to determine a winner.
- Suppose that all 3 voters that voted C > B > A in the original election are persuaded to change their ballots to match the 5 voters that voted A > C > B and a re-election happens. Does this violate the monotonicity criterion?
✅ Solution:
- Plurality-with-elimination method:
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Candidate A received 6 out of 15 first place votes or 40%
- Candidate B received 4 out of 15 first place votes or 26.7%
- Candidate C received 5 out of 15 first place votes or 33.3%
- So, no majority winner.
- Step 2: Eliminate the weakest candidate.
Candidate B received the fewest first place votes, so eliminate Candidate B by removing them from the table.
| Number of Voters | 5 | 4 | 3 | 2 | 1 |
|---|---|---|---|---|---|
| First Choice | A | C | C | A | |
| Second Choice | C | A | A | ||
| Third Choice | C | A | C |
Then move up all other candidates in the next available slots and remove the last row.
| Number of Voters | 5 | 4 | 3 | 2 | 1 |
|---|---|---|---|---|---|
| First Choice | A | A | C | C | A |
| Second Choice | C | C | A | A | C |
- Step 3: Count first-place votes. (Round 2)
- Candidate A received 10 out of 15 first place votes or 66.7%
- Candidate C received 5 out of 15 first place votes or 33.3%
- So, a majority exists.
Thus, Candidate A is the winner by the plurality-with-elimination method.
- So, all 3 voters that voted C > B > A in the election are persuaded to change their ballots to match the 5 voters that voted A > C > B. In simplest terms, this means we eliminate the column of 3 voters that voted C > B > A and change the other column's number of voters that voted A > C > B from 3 to 8.
| Number of Voters | 8 | 4 | 2 | 1 |
|---|---|---|---|---|
| First Choice | A | B | C | A |
| Second Choice | C | A | A | B |
| Third Choice | B | C | B | C |
Now, we use the same method as we did in the original election to see if the same candidate would win. So, we will use the plurality-with-elimination method once again on the re-election.
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Candidate A received 9 out of 15 first place votes or 60%
- Candidate B received 4 out of 15 first place votes or 26.7%
- Candidate C received 2 out of 15 first place votes or 13.3%
- So, a majority exists.
Candidate A wins the re-election.
Thus, Candidate A is the winner under the plurality‑with‑elimination method in the original election. Because Candidate A also wins the re‑election after receiving increased support, the outcome does not violate the monotonicity criterion. Therefore, there is NO monotonicity violation.
Consider the following preference table with Candidates A, B, & C:
| Number of Voters | 34 | 19 | 10 | 27 |
|---|---|---|---|---|
| First Choice | A | B | B | C |
| Second Choice | B | C | A | A |
| Third Choice | C | A | C | B |
- Use the plurality-with-elimination method to determine a winner.
- Suppose that 6 of the 10 voters that voted B > A > C in the election are persuaded to change their ballots to A > C > B and a re-election happens. Does this violate the monotonicity criterion?
✅ Solution:
- Plurality-with-elimination method:
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Candidate A received 34 out of 90 first place votes or 37.8%
- Candidate B received 29 out of 90 first place votes or 32.2%
- Candidate C received 27 out of 90 first place votes or 30%
- So, no majority winner.
- Step 2: Eliminate the weakest candidate.
Candidate C received the fewest first place votes, so eliminate Candidate C by removing them from the table.
| Number of Voters | 34 | 19 | 10 | 27 |
|---|---|---|---|---|
| First Choice | A | B | B | |
| Second Choice | B | A | A | |
| Third Choice | A | B |
Then move up all other candidates in the next available slots and remove the last row.
| Number of Voters | 34 | 19 | 10 | 27 |
|---|---|---|---|---|
| First Choice | A | B | B | A |
| Second Choice | B | A | A | B |
- Step 3: Count first-place votes. (Round 2)
- Candidate A received 61 out of 90 first place votes or 67.8%
- Candidate B received 29 out of 90 first place votes or 32.2%
- So, a majority exists.
Thus, Candidate A is the winner by the plurality-with-elimination method.
- So, 6 of the 10 voters that voted B > A > C in the election are persuaded to change their ballots to vote A > C > B. In simplest terms, this means we add an extra column of 6 voters that vote A > C > B and change the other column's number of voters that voted B > A > C from 10 to 4.
| Number of Voters | 34 | 19 | 4 | 27 | 6 |
|---|---|---|---|---|---|
| First Choice | A | B | B | C | A |
| Second Choice | B | C | A | A | C |
| Third Choice | C | A | C | B | B |
Now, we use the same method as we did in the original election to see if the same candidate would win. So, we will use the plurality-with-elimination method once again on the re-election.
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Candidate A received 40 out of 90 first place votes or 44.4%
- Candidate B received 23 out of 90 first place votes or 25.6%
- Candidate C received 27 out of 90 first place votes or 30%
- So, no majority winner.
- Step 2: Eliminate the weakest candidate.
Candidate B received the fewest first place votes, so eliminate Candidate B by removing them from the table.
| Number of Voters | 34 | 19 | 4 | 27 | 6 |
|---|---|---|---|---|---|
| First Choice | A | C | A | ||
| Second Choice | C | A | A | C | |
| Third Choice | C | A | C |
Then move up all other candidates in the next available slots and remove the last row.
| Number of Voters | 34 | 19 | 4 | 27 | 6 |
|---|---|---|---|---|---|
| First Choice | A | C | A | C | A |
| Second Choice | C | A | C | A | C |
- Step 3: Count first-place votes. (Round 2)
- Candidate A received 44 out of 90 first place votes or 48.9%
- Candidate B received 46 out of 90 first place votes or 51.1%
- So, a majority exists.
-
Thus, Candidate B is the winner by the plurality-with-elimination method.
Candidate B wins the re-election.
Thus, Candidate A was the winner under the plurality‑with‑elimination method in the original election. Because Candidate B had won the re‑election after Candidate A received increased support, the outcome does violate the monotonicity criterion. Therefore, YES there is a monotonicity violation.
A city council is choosing one feature to build in a new public park. Residents rank the options based on preference. The city uses a plurality‑with‑elimination (instant‑runoff) voting system.
| Number of Voters | 36 | 30 | 24 | 10 |
|---|---|---|---|---|
| First Choice | Playground | Pickle Ball Court | Dog Park | Dog Park |
| Second Choice | Dog Park | Garden | Pickle Ball Court | Playground |
| Third Choice | Pickle Ball Court | Playground | Garden | Pickle Ball Court |
| Fourth Choice | Garden | Dog Park | Playground | Garden |
- Use the plurality-with-elimination method to determine a winner.
- Suppose that only 5 of the 10 voters that voted Dog Park > Playground > Pickle Ball Court > Garden in the election are persuaded to change their ballots to match the 36 voters that voted Playground > Dog Park > Pickle Ball Court > Garden and a re-election happens. Does this violate the monotonicity criterion?
✅ Solution:
- Plurality-with-elimination method:
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Playground received 36 out of 100 first place votes or 36%
- Pickle Ball Court received 30 out of 100 first place votes or 30%
- Dog Park received 34 out of 100 first place votes or 34%
- Garden received 0 out of 100 first place votes or 0%
- So, no majority winner.
- Step 2: Eliminate the weakest candidate.
Garden received the fewest first place votes, so eliminate Garden by removing them from the table.
| Number of Voters | 36 | 30 | 24 | 10 |
|---|---|---|---|---|
| First Choice | Playground | Pickle Ball Court | Dog Park | Dog Park |
| Second Choice | Dog Park | Pickle Ball Court | Playground | |
| Third Choice | Pickle Ball Court | Playground | Pickle Ball Court | |
| Fourth Choice | Dog Park | Playground |
Then move up all other features in the next available slots and remove the last row.
| Number of Voters | 36 | 30 | 24 | 10 |
|---|---|---|---|---|
| First Choice | Playground | Pickle Ball Court | Dog Park | Dog Park |
| Second Choice | Dog Park | Playground | Pickle Ball Court | Playground |
| Third Choice | Pickle Ball Court | Dog Park | Playground | Pickle Ball Court |
- Step 3: Count first-place votes. (Round 2)
- Playground received 36 out of 100 first place votes or 36%
- Pickle Ball Court received 30 out of 100 first place votes or 30%
- Dog Park received 34 out of 100 first place votes or 34%
- So, no majority winner.
- Step 4: Repeat Until a Winner Emerges
- Eliminate the weakest candidate.
Pickle Ball Court received the fewest first place votes, so eliminate Pickle Ball Court by removing them from the table.
| Number of Voters | 36 | 30 | 24 | 10 |
|---|---|---|---|---|
| First Choice | Playground | Dog Park | Dog Park | |
| Second Choice | Dog Park | Playground | Playground | |
| Third Choice | Dog Park | Playground |
Then move up all other features in the next available slots and remove the last row.
| Number of Voters | 36 | 30 | 24 | 10 |
|---|---|---|---|---|
| First Choice | Playground | Playground | Dog Park | Dog Park |
| Second Choice | Dog Park | Dog Park | Playground | Playground |
- Step 3 (Repeated): Count first-place votes. (Round 3)
- Playground received 66 out of 100 first place votes or 66%
- Dog Park received 34 out of 100 first place votes or 34%
- So, a majority exists.
Thus, Playground is the winner by the plurality-with-elimination method.
- So, only 5 of the 10 voters that voted Dog Park > Playground > Pickle Ball Court > Garden in the election are persuaded to change their ballots to match the 36 voters that voted Playground > Dog Park > Pickle Ball Court > Garden. In simplest terms, this means we change the column of 10 voters that voted Dog Park > Playground > Pickle Ball Court > Garden from 10 to 5 and change the other column's of 36 voters that voted Playground > Dog Park > Pickle Ball Court > Garden from 36 to 41.
| Number of Voters | 41 | 30 | 24 | 5 |
|---|---|---|---|---|
| First Choice | Playground | Pickle Ball Court | Dog Park | Dog Park |
| Second Choice | Dog Park | Garden | Pickle Ball Court | Playground |
| Third Choice | Pickle Ball Court | Playground | Garden | Pickle Ball Court |
| Fourth Choice | Garden | Dog Park | Playground | Garden |
Now, we use the same method as we did in the original election to see if the same candidate would win. So, we will use the plurality-with-elimination method once again on the re-election.
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Playground received 41 out of 100 first place votes or 41%
- Pickle Ball Court received 30 out of 100 first place votes or 30%
- Dog Park received 29 out of 100 first place votes or 29%
- Garden received 0 out of 100 first place votes or 0%
- So, no majority winner.
- Step 2: Eliminate the weakest candidate.
Garden received the fewest first place votes, so eliminate Garden by removing them from the table.
| Number of Voters | 41 | 30 | 24 | 5 |
|---|---|---|---|---|
| First Choice | Playground | Pickle Ball Court | Dog Park | Dog Park |
| Second Choice | Dog Park | Pickle Ball Court | Playground | |
| Third Choice | Pickle Ball Court | Playground | Pickle Ball Court | |
| Fourth Choice | Dog Park | Playground |
Then move up all other features in the next available slots and remove the last row.
| Number of Voters | 41 | 30 | 24 | 5 |
|---|---|---|---|---|
| First Choice | Playground | Pickle Ball Court | Dog Park | Dog Park |
| Second Choice | Dog Park | Playground | Pickle Ball Court | Playground |
| Third Choice | Pickle Ball Court | Dog Park | Playground | Pickle Ball Court |
- Step 3: Count first-place votes. (Round 2)
- Playground received 41 out of 100 first place votes or 41%
- Pickle Ball Court received 30 out of 100 first place votes or 30%
- Dog Park received 29 out of 100 first place votes or 29%
- So, no majority winner.
- Step 4: Repeat Until a Winner Emerges
- Eliminate the weakest candidate.
Dog Park received the fewest first place votes, so eliminate Dog Park by removing them from the table.
| Number of Voters | 41 | 30 | 24 | 5 |
|---|---|---|---|---|
| First Choice | Playground | Pickle Ball Court | ||
| Second Choice | Playground | Pickle Ball Court | Playground | |
| Third Choice | Pickle Ball Court | Playground | Pickle Ball Court |
Then move up all other features in the next available slots and remove the last row.
| Number of Voters | 41 | 30 | 24 | 5 |
|---|---|---|---|---|
| First Choice | Playground | Pickle Ball Court | Pickle Ball Court | Playground |
| Second Choice | Pickle Ball Court | Playground | Playground | Pickle Ball Court |
- Step 3 (Repeated): Count first-place votes. (Round 3)
- Playground received 46 out of 100 first place votes or 46%
- Pickle Ball Court received 54 out of 100 first place votes or 54%
- So, a majority exists.
Thus, Pickle Ball Court is the winner by the plurality-with-elimination method.
Pickle Ball Court wins the re-election.
Thus, Playground was the winner under the plurality‑with‑elimination method in the original election. Because Pickle Ball Court had won the re‑election after Playground received increased support, the outcome does violate the monotonicity criterion. Therefore, YES there is a monotonicity violation.
So, let's summarize what we have learned from the examples above.
Does the plurality-with-elimination method violate the monotonicity criterion?
(As we saw in Examples #7.7.2 & 7.7.3). YES. The plurality‑with‑elimination method violates the monotonicity criterion, because increasing support for a winning candidate can change the elimination order and vote transfers, causing that candidate to lose despite gaining support.
As we update the fairness criteria table from Section 7.3, we now examine how the previously studied voting methods perform with respect to all of the fairness criteria.
Does the plurality method violate the monotonicity criterion?
The plurality method NEVER violates the monotonicity criterion, because increasing support for a candidate can only increase that candidate’s number of first‑place votes, and no other aspect of the election outcome is affected.
Does the Borda count method violate the monotonicity criterion?
The Borda count method NEVER violates the monotonicity criterion, because ranking a candidate higher always increases that candidate’s point total and cannot reduce their chances of winning.
Does the plurality-with-elimination method violate the Condorcet criterion?
YES. The plurality‑with‑elimination method violates the Condorcet criterion, because a candidate who wins every head‑to‑head comparison can be eliminated early due to having too few first‑place votes.
Does the plurality-with-elimination method violate the Majority criterion?
The plurality‑with‑elimination method NEVER violates the Majority criterion, because any candidate who receives a majority of first‑place votes is immediately declared the winner.
Thus, here is our updated fairness criteria table from Section 7.3:
| Fairness Criterion \(\Large\longrightarrow\) Major Voting Method \(\Large\downarrow\) |
Condorcet Criterion |
Majority Criterion |
Monotonicity Criterion |
Independence of Irrelevant Alternatives Criterion |
|---|---|---|---|---|
| Plurality | Violation Possible | \(\checkmark\) | \(\checkmark\) | ? |
| Borda Count | Violation Possible | Violation Possible | \(\checkmark\) | ? |
| Plurality-with-Elimination | Violation Possible | \(\checkmark\) | Violation Possible | ? |
| Pairwise Comparison | ? | ? | ? | ? |
So, we are still looking for a major voting method that satisfies all four fairness criterion's and unfortunately, we have determined that the plurality method and the Borda count method have at least one possible violation for each.
-
Consider the following preference table with Candidates A, B, & C:
Number of Voters 5 4 3 2 First Choice A B C C Second Choice B A A B Third Choice C C B A - Use the plurality-with-elimination method to determine a winner.
- Suppose that the 2 of the 3 voters that voted C > A > B in the original election are persuaded to change their ballots to A > C > B and a re-election happens. Does this violate the monotonicity criterion?
-
A survey of 36 college students were asked to rank their favorite series in order on some of the most watched series on Netflix. The results are summarized in a preference table:
Number of Voters 11 10 9 6 First Choice Wednesday Squid Games Stranger Things Stranger Things Second Choice Stranger Things Wednesday Squid Games Wednesday Third Choice Squid Games Stranger Things Wednesday Squid Games - Use the plurality-with-elimination method to determine a winner.
- Suppose that all 6 voters that voted Stranger Things > Wednesday > Squid Games in the original election are persuaded to change their ballots to match the 11 voters that voted Wednesday > Stranger Things > Squid Games and a re-election happens. Does this violate the monotonicity criterion?
-
A marine science department wants to vote on a department chair for the upcoming fiscal year. The results are summarized in a preference table:
Number of Voters 7 6 5 3 First Choice Dennis Rob Karen Karen Second Choice Karen Dennis Rob Rob Third Choice Rob Karen Mary Mary Fourth Choice Mary Mary Dennis Dennis - Use the plurality-with-elimination method to determine a winner.
- Suppose that only 3 of the 5 voters that voted Karen > Rob > Mary > Dennis in the election are persuaded to change their ballots to match the 7 voters that voted Dennis > Karen > Rob > Mary and a re-election happens. Does this violate the monotonicity criterion?
- Answers
-
- a) Candidate A; b) Candidate A wins re-election; no, the monotonicity criterion is not violated.
- a) Wednesday; b) Squid Games wins re-election; yes the monotonicity criterion is violated.
- a) Dennis; b) Rob wins re-election; yes the monotonicity criterion is violated.