The participation criterion, sometimes called voter monotonicity, is a voting system criterion that says candidates should never lose an election as a result of receiving too many votes in support.[1] [2] More formally, it says that adding more voters who prefer Alice to Bob should not cause Alice to lose the election to Bob.[3]
Voting systems that fail the participation criterion exhibit the no-show paradox,[4] where a voter is effectively disenfranchised by the electoral system because turning out to vote would make the outcome worse. In such a scenario, these voters' ballots are treated as less than worthless, actively harming their own interests by reversing an otherwise-favorable result.[5]
The criterion can also be described as a weaker form of strategyproofness: while it is impossible for honesty to always be the perfect strategy (by Gibbard's theorem), the participation criterion guarantees honesty will always be an effective, rather than counterproductive, strategy (i.e. an honest vote will make the outcome better, not worse). Strategy in non-participatory systems can become highly complex, as casting an honest vote is not a potential fallback option for honest voters.
Positional methods and score voting satisfy the participation criterion. All methods satisfying paired majority-rule[6] can fail in situations involving four-way cyclic ties, though such scenarios are empirically rare. Most notably, instant-runoff voting and the two-round system fail the participation criterion with high frequency in competitive elections, typically as a result of center squeeze.[7]
The most common cause of no-show paradoxes is the use of instant-runoff (often called ranked-choice voting in the United States). In instant-runoff voting, a no-show paradox can occur even in elections with only three candidates, and occur in 50%-60% of all 3-candidate elections where the results of IRV disagree with those of plurality.
A notable example is given in the 2009 Burlington mayoral election, the United States' second instant-runoff election in the modern era, where Bob Kiss won the election as a result of 750 ballots ranking him in last place.[8]
An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Bottom party initially loses. However, say that a group of pro-Top voters joins the election, making the electorate more supportive of the Top party, and more strongly opposed to the Bottom party. This increase in the number of voters who rank Bottom last causes the Center candidate to lose to the Bottom party:
| Less-popular Bottom | |||||||
| Round 1 | Round 2 | Round 1 | Round 2 | ||||
|---|---|---|---|---|---|---|---|
| Top | +6 | Top | 31 | 46 | |||
| Center | 30 | 55 | Center | ||||
| Bottom | 39 | 39 | Bottom | 39 | 54 |
When there are at most 3 major candidates, Minimax Condorcet and its variants (such as Ranked Pairs and Schulze's method) satisfy the participation criterion.[10] However, with more than 3 candidates, every resolute and deterministic Condorcet method can sometimes fail participation.[11] Similar incompatibilities have also been proven for set-valued voting rules.[12] [13]
Studies suggest such failures may be empirically rare, however. One study surveying 306 publicly-available election datasets found no participation failures for methods in the ranked pairs-minimax family.[14]
Certain conditions weaker than the participation criterion are also incompatible with the Condorcet criterion. For example, weak positive involvement requires that adding a ballot in which candidate A is one of the voter's most-preferred candidates does not change the winner away from A. Similarly, weak negative involvement requires that adding a ballot in which A is one of the voter's least-preferred does not make A the winner if it was not the winner before. Both conditions are incompatible with the Condorcet criterion.[15]
In fact, an even weaker property can be shown to be incompatible with the Condorcet criterion: it can be better for a voter to submit a completely reversed ballot than to submit a ballot that ranks all candidates honestly.[16]
Proportional representation systems using largest remainders for apportionment (such as STV or Hamilton's method) do not pass the participation criterion. This happened in the 2005 German federal election, when CDU voters in Dresden were instructed to vote for the FDP, a strategy that allowed the party an additional seat.[17] As a result, the Federal Constitutional Court ruled that negative voting weights violate the German constitution's guarantee of the one man, one vote principle.[18]
One common failure of the participation criterion in elections is not in the use of particular voting systems to elect candidates to office, but in simple yes or no measures that place quorum requirements. A public referendum, for example, if it required majority approval and a certain number of voters to participate in order to pass, would fail the participation criterion, as a minority of voters preferring the "no" option could cause the measure to fail by simply not voting rather than voting no. In other words, the addition of a "no" vote may cause the measure to pass. A referendum requiring a minimum number of yes votes (not counting no votes) would pass the participation criterion.[19]
Many representative bodies have quorum requirements where the same dynamic can be at play. For example, the requirement for a two-thirds quorum in the Oregon Legislative Assembly effectively creates an unofficial two-thirds supermajority requirement for passing bills, and can result in a law passing if too many senators turn out to oppose it.[20] Similar ballot-spoiling strategies have been used to ensure referendums remain non-binding, as in the 2023 Polish referendum.
Negative vote weight refers to an effect that occurs in certain elections where votes can have the opposite effect of what the voter intended. A vote for a party might result in the loss of seats in parliament, or the party might gain extra seats by not receiving votes. This runs counter to the intuition that an individual voter voting for an option in a democratic election should only increase the chances of that option winning the election overall, compared to not voting (a no-show pathology) or voting against it (a monotonicity or negative response pathology).
See main article: article and Majority judgment. This example shows that majority judgment violates the participation criterion. Assume two candidates A and B with 5 potential voters and the following ratings:
| Candidates |
voters | ||
|---|---|---|---|
| A | B | ||
| Excellent | Good | 2 | |
| Fair | Poor | 2 | |
| Poor | Good | 1 | |
Assume the 2 voters would not show up at the polling place.
The ratings of the remaining 3 voters would be:
| Candidates |
voters | ||
|---|---|---|---|
| A | B | ||
| Fair | Poor | 2 | |
| Poor | Good | 1 | |
| Candidate |
| |||||||||||
| A | ||||||||||||
| B | ||||||||||||
|
Now, consider the 2 voters decide to participate:
| Candidates |
voters | ||
|---|---|---|---|
| A | B | ||
| Excellent | Good | 2 | |
| Fair | Poor | 2 | |
| Poor | Good | 1 | |
| Candidate |
| |||||||||||
| A | ||||||||||||
| B | ||||||||||||
|
This example shows how Condorcet methods can violate the participation criterion when there is a preference paradox. Assume four candidates A, B, C and D with 26 potential voters and the following preferences:
| Preferences |
| |
|---|---|---|
| A > D > B > C | 8 | |
| B > C > A > D | 7 | |
| C > D > B > A | 7 |
| X | ||||||
| A | B | C | D | |||
|---|---|---|---|---|---|---|
| Y | A | [X] 14 [Y] 8 | [X] 14 [Y] 8 | [X] 7 [Y] 15 | ||
| B | [X] 8 [Y] 14 | [X] 7 [Y] 15 | [X] 15 [Y] 7 | |||
| C | [X] 8 [Y] 14 | [X] 15 [Y] 7 | [X] 8 [Y] 14 | |||
| D | [X] 15 [Y] 7 | [X] 7 [Y] 15 | [X] 14 [Y] 8 | |||
| Pairwise results for X, won-tied-lost | 1-0-2 | 2-0-1 | 2-0-1 | 1-0-2 | ||
| Pair | Winner | |
|---|---|---|
| A (15) vs. D (7) | A 15 | |
| B (15) vs. C (7) | B 15 | |
| B (7) vs. D (15) | D 15 | |
| A (8) vs. B (14) | B 14 | |
| A (8) vs. C (14) | C 14 | |
| C (14) vs. D (8) | C 14 |
Now, assume an extra 4 voters, in the top row, decide to participate:
| Preferences |
| |
|---|---|---|
| A > B > C > D | 4 | |
| A > D > B > C | 8 | |
| B > C > A > D | 7 | |
| C > D > B > A | 7 |
| X | ||||||
| A | B | C | D | |||
|---|---|---|---|---|---|---|
| Y | A | [X] 14 [Y] 12 | [X] 14 [Y] 12 | [X] 7 [Y] 19 | ||
| B | [X] 12 [Y] 14 | [X] 7 [Y] 19 | [X] 15 [Y] 11 | |||
| C | [X] 12 [Y] 14 | [X] 19 [Y] 7 | [X] 8 [Y] 18 | |||
| D | [X] 19 [Y] 7 | [X] 11 [Y] 15 | [X] 18 [Y] 8 | |||
| Pairwise results for X, won-tied-lost | 1-0-2 | 2-0-1 | 2-0-1 | 1-0-2 | ||
| Pair | Winner | |
|---|---|---|
| A (19) vs. D (7) | A 19 | |
| B (19) vs. C (7) | B 19 | |
| C (18) vs. D (8) | C 18 | |
| B (11) vs. D (15) | D 15 | |
| A (12) vs. B (14) | B 14 | |
| A (12) vs. C (14) | C 14 |