# Cloaked Participation

The **Plurality-Cloaked Participation Criterion (PCPC)** states that *adding a ballot that bullet-votes for only X should neither decrease the winning probability of X, nor change the winning probabilities among the other candidates relative to each other. A method that does not allow bullet-voting for only one candidate is considered failing the criterion.*

The **Approval-Cloaked Participation Criterion (ACPC)** is a stronger requirement which states that *adding a ballot that votes a set S of candidates equal-top and all candidates outside that set equal-bottom should neither decrease the winning probability of any candidate in S, nor change the winning probabilites among the candidates in set S relative to each other, nor change the winning probabilities among the candidates outside of S relative to each other. A method that does not allow voting by putting one or several candidates in the first rank and all other candidates in the last rank is considered failing the criterion.* Satisfying ACPC implies satisfying PCPC.

Satisfying ACPC does not imply compliance with the Participation Criterion but protection for semi-honest voters (those who compress their true preferences) against the No-Show Paradox.

Methods that satisfy ACPC: Approval, Range.

Methods that fail both ACPC and PCPC: IRV, Runoff voting in general, Bucklin, the Schulze method.

Example for IRV failing PCPC:

49 Z 26 Y 25 X>Y

Y wins.

Now add 3 voters bullet-voting for X.

49 Z 26 Y 25 X>Y 3 X

Z wins. The added voters who bullet-voted for X changed the winner without changing it to X.

Example for Bucklin failing PCPC:

49 Z 51 Y>Z

Y wins.

Now add 10 voters plumbing for X.

49 Z 51 Y>Z 10 X

Z wins. The added voters who bullet-voted for X changed the winner without changing it to X.