IRV Prime

IRV Prime is a voting method variation of instant-runoff voting (classic IRV) developed by Marcos Boyington that selects a single winner using votes that express each voter's order of preference, like IRV. It is a generalization of Condorcet IRV that works when there is no Condorcet winner.

Classic IRV eliminates candidates which had a chance of winning the election while preserving candidates that will lose. IRV Prime finds every candidate which has a chance of winning (candidates in the Schwartz set) & eliminates candidates we know won't win (those losing in the final round).

Procedure

1. Find the instant-runoff voting (IRV) winners using the classic/standard rounds, where the candidate with least votes is eliminated & their votes redistributed (call this set Winners)
2. Find the candidates which satisfy the Schwartz set
3. Re-run the IRV election once for each candidate in the Schwartz set, preserving that candidate as well as all candidates in Winners set; each candidate in the Schwartz set which defeats one of the Winners is added to the Winners-Prime set.
4. Re-run the IRV election preserving all Winners and all Winners-Prime, proceeding with normal elimination when they are all that remain

Note that using the Schwartz set is simply to reduce the number of rounds; the results are the same if the set of all candidates or the Smith set is used instead.

Algorithm

 GetWinner(NumWinners, PrefSchedule)
 {
   // Step 1: Classic IRV, find classic winners
   RemainingCandidates = PrefSchedule
   while (RemainingCandidates.Length > NumWinners) {
     RemainingCandidates = RemainingCandidates.EliminateLeast()
   }
   Winners = RemainingCandidates.TopCandidates(NumWinners)
 
   // Step 2: Build round where only candidates remaining are
   // classic winners & schwartz set
   PrefMatrix = getPreferenceMatrix(PrefSchedule)
   SchwartzSet = getSchwartz(PrefMatrix)
   Keepers = union(Winners, SchwartzSet)
   PrimeRound = PrefSchedule.EliminateAllExcept(Keepers)
 
   // Step 3: Find candidates who can win against classic winners
   WinnersPrime = {}
   for (S in SchwartzSet) {
     Keepers = union(S, Winners)
     PossibleWinner = PrimeRound.EliminateAllExcept(Keepers)
     WinnersPrime = union(WinnersPrime, PossibleWinner.TopCandidates(1))
   }
 
   // Step 4: Run final round from prime round eliminating all who
   // lose against classic winners
   RemainingCandidates = PrimeRound.EliminateAllExcept(union(Winners, WinnersPrime))
   while (RemainingCandidates.Length > NumWinners) {
     RemainingCandidates = RemainingCandidates.EliminateLeast()
   }
 
   return RemainingCandidates.TopCandidates(NumWinners)
 }

Proof of failing later-no-harm

Consider the following votes:

102 A
101 B
100 C
5 A>B>C
5 B>C>A
5 C>A>B

A wins IRV (after C is eliminated) The Schwarz set is {ABC} A wins AB. C wins AC. So, WinnersPrime is {AC}. C wins {AC} Thus, C is overall winner.

....

Now, some C voters add a later preference:

102 A
101 B
100 C>B
5 A>B>C
5 B>C>A
5 C>A>B

B wins IRV (after C is eliminated). The Schwarz set is B. So B wins overall. This violates later-no-harm.

An example

Imagine the following vote preferences:

Vote Preference

Preference	Number of Votes
L, CR	47
R, CR	39
R	5
CR, R	5

Resulting in the following preference matrix:

Preference Matrix

Candidate Preference	R	CR	L
R over		39	45
CR over	52		44
L over	47	47

In instant-runoff voting, R is elected as CR is eliminated:

Last Round IRV

Candidate	Votes
R	49
L	47

In RCV Prime, because L does not end up in Winners-Prime but CR does, the final round instead looks like:

Last Round IRV

Candidate	Votes
CR	52
R	49

And CR wins with a larger majority than the candidate that would win with IRV

Strategy

Voters are left with only a single honest rank strategy to best serve their favorite candidate while minimizing the possibility of getting a less preferred candidate:

 1-3) Favorite candidates
 4) Favorite frontrunner (if not already covered in 1-3)
 5) Least-hated who is a frontrunner with opposing views

This voting strategy guarantees that a voter's favorite candidate will win if they are preferred by the largest number of voters; it also guarantees that if those candidates cannot win (do not have the largest number of voters), that a preferred candidate will win as long as they have the largest number of voters.

Comparison to cardinal systems

See also: Left, Center, Right

Cardinal methods always pick the utilitarian winner (B in the example below) even in theoretical scenarios where that candidate is never preferred:

Vote Preference

Preference	Number of Votes
Group 1 (C > B > A)	N
Group 2 (A > B > C)	N + 1

In this theoretical example, IRV Prime would pick A. In practice, however, a non-0 percentage of voters will rank B first. As long as C and B voters combined have more voters, the method will pick the utilitarian winner since it is a Condorcet method.

External Resources

• IRV-Prime Voting Method