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Theparticipation criterion is avoting 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 preferAlice toBob should not causeAlice to lose the election toBob.[3]
Voting systems that fail the participation criterion exhibit theno-show paradox,[4] where a voter is effectively disenfranchised by the electoral system because turning out to vote could make the result worse for them; such voters are sometimes referred to as havingnegative vote weights, particularly in the context ofGerman constitutional law, where courts have ruled such a possibility violates the principle ofone man, one vote.[5][6][7]
Positional methods andscore voting satisfy the participation criterion. All deterministic voting rules that satisfypairwise majority-rule[4][8] can fail in situations involving four-waycyclic ties, though such scenarios are empirically rare, and therandomized Condorcet rule is not affected by the pathology. Themajority judgment rule fails as well.[9]Instant-runoff voting and thetwo-round system both fail the participation criterion with high frequency in competitive elections, typically as a result of acenter squeeze.[1][2][10]
The no-show paradox is similar to, but not the same as, theperverse response paradox. Perverse response happens when anexisting voter can make a candidate win bydecreasing their rating of that candidate (or vice-versa). For example, underinstant-runoff voting, moving a candidate from first-place to last-place on a ballot can cause them to win.[11]
The most common cause of no-show paradoxes is the use ofinstant-runoff (often calledranked-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.[10][2]
A notable example is given in the2009 Burlington mayoral election, the United States' secondinstant-runoff election in the modern era, whereBob Kiss won the election as a result of 750 ballots ranking him in last place.[12]
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. Thisincrease in the number of voters who rank Bottomlast causes the Center candidate to lose to the Bottom party:
| More-popular Bottom | Less-popular Bottom | |||||
|---|---|---|---|---|---|---|
| Round 1 | Round 2 | Round 1 | Round 2 | |||
| Top |  N | +6 | Top | 31 | 46 | |
| Center | 30 | 55 Y | Center | N | ||
| Bottom | 39 | 39 | Bottom | 39 | 54Y | |
Thus the increase in support for the Top party allows it to defeat the Center party in the first round. This makes the election an example of acenter-squeeze, a class of elections where instant-runoff andplurality have difficulty electing the majority-preferred candidate.[13]
When there are at most 3 major candidates,Minimax Condorcet and its variants (such asranked pairs andSchulze's method) satisfy the participation criterion.[4] However, with more than 3 candidates,Hervé Moulin proved that every deterministic Condorcet method can sometimes fail participation.[4][8] Similar incompatibilities have also been shown forset-valued voting rules.[8][14][15] Therandomized Condorcet rule satisfies the criterion, but fails the closely-relatedmonotonicity criterion in situations with Condorcet cycles.[16]
Studies suggest such failures may be empirically rare, however. One study surveying 306 publicly-available election datasets found no participation failures for methods in theranked pairs-minimax family.[17]
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'smost-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.[18]
In fact, an even weaker property can be shown to be incompatible with the Condorcet criterion: it may be better for a voter to submit a completely reversed ballot than to submit a ballot that ranks all candidates honestly.[19]
Proportional representation systems usinglargest remainders forapportionment (such asSTV orHamilton's method) allow for no-show paradoxes.[5][20]
In Germany, situations where a voter's ballot has the opposite of its intended effect (e.g. a votefor a party or candidate causes them tolose) are callednegatives Stimmgewicht (lit. 'negative voting weights'). An infamous example occurred in the2005 German federal election, when an article inDer Spiegel laid out howCDU voters inDresden I would have to voteagainst their own party if they wished to avoid losing a seat in theBundestag.[5] This led to a lawsuit byelectoral reform organizationMehr Demokratie [de] andAlliance 90/The Greens, joined by the neo-NaziNPD of Germany, who argued the election law was undemocratic.[21]
TheFederal Constitutional Court agreed with the plaintiffs, ruling that negative vote weights violate theGerman constitution's guarantee ofequal and direct suffrage. The majority wrote that:[6][22]
A seat allocation procedure that allows an increase in votes to lead to a loss of seats, or results in more seats being won if [proportionally] fewer votes are cast for it, contradicts the meaning and purpose of a democratic election[...]
Such nonsensical relationships between voting and electoral success not only impair the equality of the right to vote and the equal opportunities of the parties, but also the principle of a popular election, as it is no longer apparent to the voter how their vote results in the success or failure of a candidate.[...]
Negative vote weights cannot be accepted as constitutional on the premise that they cannot be predicted or planned, and thus can hardly be influenced by the individual voter. To what extent this is true can be set aside, as such arbitrary results make a mockery of the democratic competition for support from the electorate.
The ruling forced theBundestag to abandon its old practice of ignoringoverhang seats, and instead adopt a new system of compensation involvingleveling seats.[21]
A common cause of no-show paradoxes is the use of aquorum. For example, if a publicreferendum requires 50% turnout to be binding, additional "no" votes may push turnout above 50%, causing the measure to pass. A referendum that instead required a minimum number of yes votes (e.g. >25% of the population voting "yes") would pass the participation criterion.[23]
Many representative bodies have quorum requirements where the same dynamic can be at play. For example, the requirement for a two-thirdsquorum in theOregon Legislative Assembly effectively creates an unofficial two-thirdssupermajority requirement for passing bills, and can result in a law passing if too many senators turn out to oppose it.[24] Deliberate ballot-spoiling strategies have been successful in ensuring referendums remain non-binding, as in the2023 Polish referendum.
The participation criterion can also be justified as a weaker form ofstrategyproofness: while it is impossible for honesty to always be thebest strategy (byGibbard's theorem), the participation criterion guarantees honesty will always be an effective, rather than actively counterproductive, strategy (i.e. a voter can always safely cast a sincere vote).[4] This can be particularly effective for encouraging honest voting if voters exhibitloss aversion. Rules with no-show paradoxes do not always allow voters to cast a sincere vote; for example, a sincere Palin > Begich > Peltola voter in the2022 Alaska special election would have been better off if they had not shown up at all, rather than casting an honest vote.
While no-show paradoxes can be deliberately exploited as a kind ofstrategic voting, systems that fail the participation criterion are typically considered to be undesirable because they expose the underlying system as logically incoherent or "spiteful" (actively seeking to violate the preferences of some voters).[25]
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 | # of voters | |
|---|---|---|
| A | B | |
| Excellent | Good | 2 |
| Fair | Poor | 2 |
| Poor | Good | 1 |
The two voters rating A "Excellent" are unsure whether to participate in the election.
Assume the 2 voters would not show up at the polling place.
The ratings of the remaining 3 voters would be:
| Candidates | # of voters | |
|---|---|---|
| A | B | |
| Fair | Poor | 2 |
| Poor | Good | 1 |
The sorted ratings would be as follows:
| Candidate |
| |||||||||
| A | ||||||||||
| B | ||||||||||
|
Result: A has the median rating of "Fair" and B has the median rating of "Poor". Thus,A is elected majority judgment winner.
Now, consider the 2 voters decide to participate:
| Candidates | # of voters | |
|---|---|---|
| A | B | |
| Excellent | Good | 2 |
| Fair | Poor | 2 |
| Poor | Good | 1 |
The sorted ratings would be as follows:
| Candidate |
| |||||||||
| A | ||||||||||
| B | ||||||||||
|
Result: A has the median rating of "Fair" and B has the median rating of "Good". Thus,B is the majority judgment winner.
This example shows how Condorcet methods can violate the participation criterion when there is apreference paradox. Assume four candidates A, B, C and D with 26 potential voters and the following preferences:
| Preferences | # of voters |
|---|---|
| A > D > B > C | 8 |
| B > C > A > D | 7 |
| C > D > B > A | 7 |
This gives thepairwise counting method:
X Y | A | B | C | D |
|---|---|---|---|---|
| A | 14 8 | 14 8 | 7 15 | |
| B | 8 14 | 7 15 | 15 7 | |
| C | 8 14 | 15 7 | 8 14 | |
| D | 15 7 | 7 15 | 14 8 | |
| Pairwise results for X, won-tied-lost | 1-0-2 | 2-0-1 | 2-0-1 | 1-0-2 |
The sorted list of victories would be:
| 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 |
Result: A > D, B > C and D > B are locked in (and the other three can't be locked in after that), so the full ranking is A > D > B > C. Thus,A is elected ranked pairs winner.
Now, assume an extra 4 voters, in the top row, decide to participate:
| Preferences | # of voters |
|---|---|
| A > B > C > D | 4 |
| A > D > B > C | 8 |
| B > C > A > D | 7 |
| C > D > B > A | 7 |
The results would be tabulated as follows:
X Y | A | B | C | D |
|---|---|---|---|---|
| A | Does not appear | 14 12 | 14 12 | 7 19 |
| B | 12 14 | Does not appear | 7 19 | 15 11 |
| C | 12 14 | 19 7 | Does not appear | 8 18 |
| D | 19 7 | 11 15 | 18 8 | Does not appear |
| Pairwise results for X, won-tied-lost | 1-0-2 | 2-0-1 | 2-0-1 | 1-0-2 |
The sorted list of victories would be:
| 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 |
Result: A > D, B > C and C > D are locked in first. Now, D > B can't be locked in since it would create a cycle B > C > D > B. Finally, B > A and C > A are locked in. Hence, the full ranking is B > C > A > D. Thus,B is elected ranked pairs winner by adding a set of voters who prefer A to B.
By eliminating the squeezing effect, Approval Voting would encourage the election of consensual candidates. The squeezing effect is typically observed in multiparty elections with a runoff. The runoff tends to prevent extremist candidates from winning, but a centrist candidate who would win any pairwise runoff (the "Condorcet winner") is also often "squeezed" between the left-wing and the right-wing candidates and so eliminated in the first round.
Of course, a method not satisfying participation will incentivize some strategic non-voting, as the voters in question will have an incentive not to vote (sincerely). But again, all voting methods incentivize strategic behavior[...] By contrast, weare troubled by failures of positive or negative involvement, as this shows that the method responds in the wrong way to unequivocal support for (resp. rejection of) a candidate.
