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


  • 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 places candidate X on a higher (lower number) tier than candidate Y, X receives a vote in the pairwise X-Y contest, and visa-versa. For example, if Alice is given a ranking of 3rd-tier, while Bob is given a ranking of 5th-tier, Alice receives a vote in the Pairwise(Alice,Bob) contest. The number of tiers in between is not important, only the relative higher/lower position.
  • The T3 winner is the pairwise winner of the highest approved candidate ("A") versus the pairwise winner between the second ("B") and third ("C") most approved candidates wins the election. In other words, T3-winner = PW(A, PW(B,C)).
  • If the T3 winner is not the Approval winner ("A"), find the pairwise winner between A and the pairwise loser of B vs C in order to determine the T3 runner-up and third-place.



  • 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.
    • Repeat until no untested Smith set candidates remain.
  • 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.