Participation criterion

The participation criterion is a voting system criterion applicable to both single and multiple winner ranked voting systems. A method that passes this criterion ensures a voter that it's always better to cast a full honest vote than to not show up for the election at all. It does this by guaranteeing that adding a ballot can never change the winner from someone who is ranked higher on that ballot to someone who is ranked lower.

While the criterion ensures that a voter can't benefit from staying home rather than voting honestly, a voter may do even better by engaging in tactical voting; participation does not imply that the method is strategy-proof.

Definition
For deterministic single-winner methods, the criterion is defined as follows:

For multi-winner methods and methods that involve an element of chance, the definition is:

Semi-honest participation criterion
This is a weaker form of the participation criterion. It states that for any set of ballots, an extra voter with a given preference set must be able to cast a ballot which is semi-honest and meaningfully expressive, without making the result worse. Meaningfully expressive means that if the voter prefers some set of candidates to the winner, the non-harmful ballot must be able to express that preference.

Complying methods
This criterion is important in the context of the Balinski–Young theorem. Failing the participation criterion is an an example of failing population monotonicity.

Every weighted positional method that gives higher ranked candidates higher scores passes the participation criterion. In particular, Plurality voting and the Borda count both pass. Furthermore, Approval voting, Cardinal Ratings, and Woodall's DAC and DSC methods all pass the participation criterion. All Condorcet methods, Bucklin voting, and IRV fail.

It's possible to pass both Condorcet and Participation for three candidates and any number of voters, or for four candidates up to 11 voters inclusive. This result also holds for certain probabilistic extensions of the Condorcet criterion.

All Monroe type multi-member systems fail participation.