A calculator to compare a lot of methods at once, without resolution details.
Method list
ACP or Adjusted Condorcet Plurality: A method I discovered in Jan 2023. Let X be the candidate with the most first preferences. Edit all the ballots so that any candidates
ranked below X are truncated instead. On the revised ballot set, elect a Condorcet winner if there is one. Otherwise elect X. This method
is interesting because, as with IRV, one can neither harm nor help a candidate by ranking additional preferences below them.
ACP variant: This is a variation of the above.
It is the same, except that during the check for a
Condorcet winner, a candidate X still can only use their preferences from the adjusted matrix when trying to pairwise defeat another
candidate Y, but X contends with the Y's vote count from the original matrix. This makes it less likely that anyone
can win as Condorcet winner, and thus more likely that the FPP winner will be elected. The method also violates Later-no-help,
since preferences below the first preference winner can now aid that candidate in winning.
Approval Elimination Condorcet or AEC: While there is no Condorcet winner, disregard the remaining
candidate with the least implicit approval. Don't reassess approval after each step. Basically identical approaches are
MinMax(winner's approval) or the method "Definite Majority Choice."
Approval Elimination Runoff or AER: Like IRV, but successively eliminate the candidate with the least implicit approval (meaning the fewest number of the ballots ranking a candidate
above at least one other candidate). End once a candidate has votes from a majority of remaining voters (i.e. excluding exhausted ballots).
Benham: So named by James Green-Armytage after Chris Benham. In the absence of a Condorcet winner,
eliminate the plurality loser (and transfer
votes) repeatedly until you do have a Condorcet winner. Similar in effect to Smith//IRV.
Borda: On this page Borda uses symmetric completion for incomplete rankings, as that seems like the
most intuitive and least controversial way to do it.
But this means that the method doesn't satisfy Later-no-harm, in contrast to the treatment on the Later-no-harm calculator.
BPW: Eivind Stensholt's "Beats the plurality winner" is a Condorcet cycle resolution principle defined for
three candidates, in which in the absence of a Condorcet winner one elects the candidate who defeats the plurality winner.
It's an attempt to minimize burial opportunity.
I find that, in the three-candidate case, it does easily outperform Smith//IRV in terms of raw opportunities to use burial
strategy. My
efforts presented here to generalize to 4+ candidates, unfortunately, don't fare as well. I want to present BPW anyway because,
although it's probably too non-monotone to advocate, I find its results interesting and not exactly displeasing.
The "chain" version uses the technique of "chain climbing" popularized by Jobst Heitzig and Forest Simmons. Taking the candidates
from most to least
first preferences, add them to a set (which starts empty) if they don't lose pairwise to any candidate in the set. Elect the last
candidate who can be added to the set. This process ensures that the winner is in the Smith and uncovered sets. The "max" version, in the absence of a
Condorcet
winner, elects the candidate with the most first preferences who beats or ties the plurality winner.
(We don't verify whether the latter was even in the Smith set.)
Experimentally this is
a worse way of doing it, but it's simpler to understand than the "chain" version.
BRBO: An idea of mine. "Best response to best opposition." Elect the candidate X with the most pairwise support
in a pairwise contest with the candidate (or one of the candidates) with the greatest pairwise support against X. It doesn't satisfy
minimal defense.
Bucklin: A well-known method where voters successively reveal more compromise choices, at the same pace, until some candidate achieves a majority, or
all votes for compromise choices have been cast. Once a majority is achieved by one or more candidates, the one with the largest of those majorities wins.
CdlA: Proposed by me in 2003. The name is short for "conditional approval" which I don't want to lay claim to. Voters start by approving their top choice. In
subsequent rounds, they approve additionally every rank strictly better than the worst candidate (in the voter's view) that has ever been a round leader. Once a round leader is
a candidate who has already been a round leader, they are elected.
Chain Runoff:
Arrange all the candidates in a list from greatest to least first preference count.
Say that a candidate X
"has a majority" over another candidate Y if more than half of all the voters prefer X to Y.
Elect the first candidate in the list, where the very next candidate (if there is one) does not have a majority over
them. A Later-no-harm method proposed by myself in 2010.
Condorcet//Approval: The simple method where in the absence of a Condorcet winner, we elect the candidate with
the most implicit approval (i.e. ballots ranking a candidate above at least one other candidate). I
like this method, as it satisfies minimal defense among other properties, and makes a burial strategy look risky, as in the
Condorcet "cycle resolution" any insincere preferences count just as much as the sincere ones. This factor is why I don't
recommend that this method be used with explicit approval (i.e. with a ballot mechanism
to mark an approval cutoff at an arbitrary position in one's ranking). For an FBC-satisfying variant consider ICA below.
Cross Max: An idea of mine. Elect the candidate with the greatest pairwise support in any contest involving
the first preference winner (who may themselves win). The term "cross" refers to the typical arrangement of cells in the pairwise
matrix pertaining to one candidate. The method gives rather unusual results. It doesn't satisfy minimal defense.
DAC: Proposed by Woodall in 1996 as his preferred alternative to IRV. This method works the same as DSC (below) except that
a voter is counted as voting for each set of candidates
where for each candidate in the set, the voter prefers or ranks equal this candidate to each candidate outside the set. This means the voter may vote for sets that
include candidates that the voter ranked above no one. The sets are considered from most votes to least votes, attempting with each one to disqualify from winning every
candidate who isn't part of the given set.
DSC: A Later-no-harm method proposed by Douglas Woodall in 2003. Given n candidates, there are 2^n unique sets of candidates. Each such set receives a score,
equal to the number of voters who rank each candidate within that set strictly higher than every candidate outside that
set. Assess each set starting with the highest score proceeding to the lowest. Every candidate not in the current set
is disqualified from winning, unless this disqualifies all remaining candidates, in which case the set has no effect.
The last candidate remaining (i.e. never disqualified) is elected.
fpA minus max(fpC): A generalization of "fpA-fpC," a three-candidate method by Kristofer Munsterhjelm. In
the absence of a
Condorcet winner, each candidate is scored as their first preference count minus the greatest first preference count of any other
candidate who pairwise beats or ties them. Elect the candidate with the greatest score. In the three-candidate case this method
happens to be equivalent to Condorcet//IFPP (i.e. in the absence of a Condorcet winner, elect the IFPP winner).
It's noteworthy as a monotone Condorcet method with rather low burial incentive.
FPP or Plurality voting: The candidate with the most first preferences wins. All other preferences are ignored.
IBIFA: Chris Benham's method from 2010. This attempts to patch Bucklin so that the majority threshold requirement is replaced with a dynamic check for the
ability of a candidate's supporters to ensure victory (i.e. via truncation). The name stands for "Irrelevant Ballot-Independent Fallback Approval," so called because the outcome
should not change by adding, for example, some
write-in voters who under Bucklin could cause a majority threshold to no longer be reached.
Specifically one elects the candidate X who at the earliest round (and with the highest tally in that round, in case of a tie) can prevail in a hypothetical vote in which each
voter who would approve X in this round (i.e. a round in the Bucklin sense) approves only X, and every other voter approves all candidates they approved at any above-bottom rank
except for X.
IFPP or Improved First Past the Post: Elect the FPP winner if fewer than two candidates have over a third of the first preferences each. Otherwise,
elect the pairwise winner between the top two candidates as under an IRV final two. This method is
monotone, in contrast to IRV, but usually gives the same results as FPP.
IFPP was devised
by Craig Carey no later than 1998. He only intended it for three-candidate elections. For higher candidate counts I extend it such that
monotonicity is preserved, which means that the one-third rule is invariable.
Improved Condorcet Approval (ICA): Devised by me in 2005. A modification to Condorcet//Approval above,
which allows it to satisfy Ossipoff's Favorite
Betrayal criterion. It's discussed on the old 2005 webpage. When checking for a Condorcet winner, ballots tying two candidates in
the top rank ("tied at the top") count a vote to each side of the contest, but only to the advantage of each, such that if the
votes tying the two at the top are sufficient to determine the winner of the contest, neither is considered to defeat the other.
In this way, multiple
candidates can be considered pairwise undefeated. The implicit approval winner is elected from such candidates if possible. If
there are none, then the approval winner
among all candidates is elected. Note that this can only differ from ordinary C//A when there is equal ranking used at the top of
some ballots. For that reason, ICA results are not usually separately displayed below.
IRV: Initially count only first preferences. Repeatedly eliminate the candidate with the fewest votes, such that
always the highest-ranked non-eliminated preference receives a vote from the ballot. The last remaining candidate wins.
(One can also always stop if a candidate obtains votes from a majority of the ballots.)
Iterated Bucklin: Etjon Basha's 2020 proposal. Voters start by approving their top rank only. Voters gradually move their approval threshold
towards the latest round leader (but as in Bucklin, cannot go to the very bottom/unranked candidates). If the threshold has reached the rank of the current leader, it does not move.
Otherwise, it can move in either direction towards the current leader. The method ends when thresholds cease moving. This implies that when a candidate wins, they are being approved by every voter who could ever approve them.
Note that as a rule I don't use the rule regarding the situation where candidates tie
for the lead in some round. Instead, a tiebreaking score is generated for each
candidate at the very start of the method, and this is consulted to produce a decisive winner each round.
KOTH or King of the Hill: A method of mine from 2011. One elects the pairwise winner between the first preference winner (FPW) and the candidate with the most first preferences who has either a full majority pairwise win
over or loss to the FPW. The FPW wins if there is no such other candidate.
MAMPO: Proposed by me in 2007 in order to satisfy the same criteria as my 2005 proposal MDDA, but without violating the Plurality
criterion. If no candidate has majority implicit approval, then the candidate with the most implicit approval wins. Otherwise, of the candidates with majority approval,
the one with the lowest MMPO score wins. The MMPO score is the greatest number of voters that voted against the candidate in any
pairwise contest (even those involving candidates without majority approval, and regardless of which candidate won any given contest).
MaxMin(Pairwise Support): Woodall defines "MinGS" as the method under which one elects the candidate X whose
fewest votes (pairwise) against some other candidate Y is the greatest. This satisfies Plurality, Later-no-help, Mono-raise, and Mono-add-top,
but not mutual majority, even in very basic situations. Woodall further defines "DminGS" as a descending coalitions method where
each set's score is equal to the minimum vote count of any candidate in the set over any candidate outside the set. This adds
clone independence while sacrificing Later-no-help and Mono-add-top. Forest Simmons proposes to allow some candidate X to get a
vote against some candidate Y even when they are both ranked equal, but above bottom. Such a variation is called "MMPS" and
satisfies the weak Favorite Betrayal criterion (assuming equal ranking is allowed). For that reason I am using this rule and
this name for both the basic method and the descending coalitions version (marked "DC"; note this does not mean that the DC version satisfies Favorite Betrayal). The results of these methods are somewhat unusual.
MDDA: "Majority Defeat Disqualification Approval," devised by me in 2005. A method with the same properites as MAMPO (devised two years later) except that MDDA violates Plurality. Elect the most approved
candidate among those without a majority strength pairwise defeat, if possible. Otherwise just elect the most approved.
This method is also discussed in an interesting paper by Alex Small which
examines the possibilities for eliminating favorite betrayal incentives.
MinMax: A type of method in which one elects the candidate whose worst pairwise defeat is the mildest. It is here provided in WV, margins, and pairwise opposition forms. See the next item regarding the latter.
MinMax(Pairwise Opposition) or MMPO: Elect the candidate X whose selection minimizes the greatest number of voters who prefer some other candidate
Y to X. (It doesn't matter whether X or Y would prevail head-to-head.) MMPO may often produce ties, and in that case they are here resolved by electing the MMPO finalist with the most first preferences. Using Woodall's conventions the combined method can be called "MMPO,FPP".
No elimination IRV type 1: I don't know whether I am the first to formulate this method in this way: The "eliminations" of IRV become a mere status flag rather
than a real elimination. In each round voters cast votes going down their ballot until they have voted for a candidate who has not yet been eliminated. A round leader who
attains a majority (i.e. has votes from a majority of the voters) is elected. Otherwise the vote loser among non-eliminated candidates is marked as eliminated, and the
next round begins. There are two exceptions:
If no candidates remain to be eliminated, or if the only candidate left to be eliminated is the round leader, then the round leader is elected despite lacking a majority.
No elimination IRV type 2: This is the same as type 1, with one change.
If the round leader at some point is the last non-eliminated candidate, but they have not attained a majority, type 2 "eliminates" him
(in the same sense that candidates can be said to be "eliminated" under this method) instead of electing him, so that there is an additional round to count. This makes the method
more likely to elect the implicit approval winner.
It appears this particular rule was proposed by Bjarke Ebert in 2021.
QLTD or Quota-Limited Trickle-Down: Proposed by Woodall in 1996. This is the same as Bucklin except that if multiple candidates achieve a majority simultaneously, the win goes to
the candidate who required the lowest percentage of the votes they obtained, in the most recent round, to reach majority.
Ranked Pairs: Nicolaus Tideman's proposal (but in WV form) where, proceeding
from strongest defeats to weakest defeats, we gradually "lock" a final ranking of all
the candidates, ignoring any defeats that would introduce a cycle.
River: Jobst Heitzig's WV-based method which resembles Ranked Pairs except that defeats against candidates already defeated are not counted. A simpler procedure is available as well, which makes it easier to track whether a given defeat would create a cycle.
River(CCE): This is like the above, but applying the initial step proposed on the Condorcet Compromise Extension page, which is to remove candidates either barred by "weak Plurality" or not included in the "CCE top tier."
RMPA and RMPFPP, meaning River Majority Pass Approval or FPP: Ideas of mine.
We choose a metric (approval or FPP respectively), and apply Heitzig's River algorithm using only the full
pairwise majorities, using the chosen metric as the defeat strength. In practice this means going down the list of candidates in metric order and not worrying about defeat strength as such.
We just act on each of the candidate's pairwise majorities.
Start out each candidate in their own "bin." For each pairwise majority you encounter, take every candidate
from the pairwise loser's original bin (which may be no one, or may even include the pairwise winner)
and move them (if anyone) to the
bin that the pairwise winner is currently in. Once we've gone through all pairwise majorities, elect
the candidate in whose bin the metric leader ultimately ended up (possibly their own bin, of course).
RMPA and RMPFPP both satisfy Plurality and minimal defense and so could be included in the /misc calculator. RMPA is very similar to
MDDA, but fixes its Plurality problem while losing Favorite Betrayal compliance.
These methods were formerly called on this page "Majority Rule Approval" and "Majority Rule FPP."
Top-Two Runoff: Hold a one-on-one election between the two candidates who received the most first preferences in a first round. When conducted as a single round of voting (an instant runoff) this is better called the "contingent vote."
Schulze: Markus Schulze's method in which one elects the candidate whose strongest beatpath to each other candidate is at least as strong as each respective path back.
It's provided with WV as the measure of defeat strength.
SV: Eivind Stensholt's "Smallest Volume" method pursues the same goal as BPW, and has similar monotonicity
concerns. And again it's a three-candidate
method that I attempt to extend. In the absence of a Condorcet winner, we elect a candidate with a certain lowest score.
If for some candidate A, B is the candidate with the most first preferences who beats or ties A, and C is the candidate
with the most first preferences whom A beats or ties pairwise, then A's score is (50% - C's first preference vote share)
divided by (50% - B's first preference vote share). Any Condorcet Loser (who would have no candidate C) is precluded from
winning. Experimentally I don't find SV (or my generalization) to be better than BPW with respect to strategy incentives.
In some ways it seems worse. SV is just more monotone. But I should note that SV is motivated by a geometric argument that
I have not considered when evaluating it.
TACC: "Total Approval Chain Climbing," a 2005 method by Jobst Heitzig. "Chain climbing" is a mechanism popularized by him and also Forest Simmons. Start with an empty set. Starting (in this case) with the least approved candidate, evaluate
whether the candidate pairwise beats all candidates currently in the set. If so, add them to the set. Elect the last
candidate who can be added. In this way the TACC winner should never be "covered" by another candidate.
Use your own method(s)
See here for examples of what to enter below. Note ⚠️ you could hang or crash your browser here!
Instructions
Enter your ballots like this, one per line: 456: Alice>Bob>Carl=Debra
or equivalently to the above: 456: Alice>Bob
The number represents the size of the voting bloc. Decimals are OK. The size can also be left off and
it will then be randomized.
Note that this calculator does allow equal ranking above the bottom, but not every method is supported
in this case. With some methods, equal ranking is allowed only below the top rank.
Candidate names can contain spaces. Each candidate in the list should be separated by > or =. Pipes (i.e. |) and
any series of > will be interpreted
as single >s. Not every candidate needs to be listed; candidates present on the ballots but missing from one
faction's ranking will be interpreted
as ranked tied for last, below any explicitly ranked candidates.