Ranked preference approval voting

Ranked Preference Approval Voting (RPAV) is a general term used to describe voting methods with approval inferred from a ranked ballot (equal ranking and ranking-gaps allowed), but is used specifically for two different single-winner methods and one multiwinner proportional representation method.

Single Winner Ballot format

The goal of RPAV single-winner is to emulate a general ranking with an explicit approval cutoff through use of a fixed ranking format with constant approval cutoff level. If you want to emulate the effect of being able to put an explicit approval cutoff somewhere in an M-level ranking, then you need 2*M ranks, with the top M ranks approved. This lets you rank M candidates as approved, or up to M-1 candidates disapproved but not last.

To de-emphasize *rating*, even though it could be considered equivalent to a score ballot, the RPAV ballot is set up as N ranked *tiers*. The terminology *tier* is chosen because a rank level is not exclusive --- more than one candidate can be ranked on a tier level --- and it is not necessary to rank a candidate on each tier. The default number of tiers is 6, which lets voters put an explicit approval cutoff somewhere in 3 ranking levels, an adequate level of resolution for most public elections.

Tier	Approved	Description
1	Yes	Most/Strongly approved
2	Yes	Approved
3	Yes	Slightly/Barely approved
4	No	Slightly/Barely disapproved
5	No	Disapproved
6	No	Most/Strongly Disapproved; Rejected; Unknown

Single Winner RPAV methods

Top Three Tournament

RPAV-T3:

• Use a 6-tier approval ballot as above.
• Sort the candidates in descending order of approval, and take the top three approved candidates. (A = approval winner, B = approval runner-up, C = approval third place)
• Use rankings to form a pairwise matrix for those three candidates.
  • If a ballot rates candidate X higher than candidate Y, X receives a vote in the pairwise X-Y contest, and visa-versa.
• The winner is the pairwise winner of the highest approved candidate versus the pairwise winner between the second and third most approved candidates wins the election. In other words, T3-winner = PW(A, PW(B,C))

Smith//Approval

RPAV-Smith-Approval:

• Same ballot as RPAV-T3.
• Find Smith Set:
  • Compute Pairwise matrix
  • Initialize Smith Set as empty set
  • Find candidate(s) with the smallest number of pairwise losses, add them to Smith Set
  • For each untested Smith Set candidate, add in any candidates not already in Smith Set who defeat that candidate.
• Winner is the highest approved member of the Smith set. For three candidates, the winner of RPAV-T3 is the same as the winner of RPAV-Smith-Approval.