Participation criterion

The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election as a result of receiving too many votes in support. More formally, it says that adding more voters who prefer Alice to Bob should not cause Alice to lose the election to Bob.

Voting systems that fail the participation criterion exhibit the no-show paradox, 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 outcome.

Positional methods and score voting satisfy the participation criterion. All methods satisfying paired majority-rule can fail in situations involving four-way cyclic ties. Most notably, instant-runoff voting and the two-round system often fail the participation criterion in competitive elections.

Ranked-Choice Voting
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.

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 moderate candidate to lose to the Bottom party: Here, the increase in support for the Top party allowed it to defeat the Center party in the first round. This makes the election an example of a center-squeeze, a class of elections where instant-runoff and plurality have difficulty electing the majority-preferred candidate because of vote-splitting in the first round.

Condorcet methods
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. However, with more than 3 candidates, every resolute and deterministic Condorcet method can sometimes fail participation. Similar incompatibilities have also been proven for set-valued voting rules.

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.

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.

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.

Quota rules
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. 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.

Quorum requirements
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 make the measure more likely to pass. A referendum that required a minimum number of yes votes (not counting no votes), by contrast, would pass the participation criterion. Many representative bodies have quorum requirements where the same dynamic can be at play.

Relationship to vote positivity
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).

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: The two voters rating A "Excellent" are unsure whether to participate in the election.

Voters not participating
Assume the 2 voters would not show up at the polling place.

The ratings of the remaining 3 voters would be: The sorted ratings would be as follows: Result: A has the median rating of "Fair" and B has the median rating of "Poor". Thus, A is elected majority judgment winner.

Voters participating
Now, consider the 2 voters decide to participate: The sorted ratings would be as follows: Result: A has the median rating of "Fair" and B has the median rating of "Good". Thus, B is the majority judgment winner.

Condorcet methods
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: This gives the pairwise counting method: The sorted list of victories would be: 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.

Voters participating
Now, assume an extra 4 voters, in the top row, decide to participate: The results would be tabulated as follows: The sorted list of victories would be: 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.