Below, the term "approval" always refers to "implicit" approval, meaning a voter ranked some candidate
above bottom, above some other candidate.

Methods named with a double asterisk (**) are *two-round* methods, which in the second round go
back to the sincere preferences and assume all voters participate in the second round and vote sincerely.
For top-two runoffs this is probably fair, but otherwise it is probably cheating, since strategies are possible
but not explored. Interpret with caution.

**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. See __/lnharm__ for a calculator.
**AEC** or **Approval Elimination Condorcet**: 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."
**AER** or **Approval Elimination Runoff**: 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).
**Approval (implicit)**: We elect the candidate with the most approval.
**Approval TTR**** advances the two approval winners from a first round into a second round where the two go head to head.
**ATAR**, by analogy with STAR voting, means "Approval then automatic runoff." Elect the pairwise winner between the top two candidates on approval.
**AWMajCheck****: This means "Approval winner majority check," a non-eliminative two-round method. The approval winner of the first round becomes the tentative winner. In the second round, voters approve every candidate that they prefer to the tentative winner (which may be no one). Second-round voters use no other strategy, and this is definitely cheating (see note at top). If any candidate gets majority approval in the second round, the one of these with the most approval wins. Otherwise the tentative winner wins.
**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.
**BTR-IRV** is "bottom two runoff" IRV. In each round instead of eliminating the candidate with the fewest votes, eliminate the pairwise loser between the two candidates with the fewest votes. This is a Condorcet method.
**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.
**Borda**: I show all three interpretations of Borda that have appeared on this site. "BordaLNH" is
the Later-no-harm-compliant treatment that awards a full point per ballot for being ranked above *or equal to*
each other candidate. "BordaSC" handles equal ranking and truncation by awarding half points for candidates being
ranked at the same level. "BordaZFT" is the same, but there are zero points awarded to truncated candidates (those
ranked bottom or equal bottom).
**BPW (max)** and **BPW(chain)**: 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 attempt to extend this to 4+ candidates in a couple of ways.

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.)
**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.
**BTAR**, by analogy with STAR voting, means "Borda then automatic runoff." Elect the pairwise winner between the top two candidates on Borda score. This uses the third interpretation of Borda given above, where
truncated candidates receive no points.
**C//FPP** or **Condorcet//FPP**: Elect the Condorcet winner if there is one, otherwise elect the first preference winner.
**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**: This option paints regions a color if there is a Condorcet winner there, and otherwise
paints it black. This can help you understand if it's worth comparing different Condorcet methods with a given
scenario and settings.
**C//A** or **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.
**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.
**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.
**FPWMajCheck****: This means "First preference winner majority check," a non-eliminative two-round method. In the first round everyone votes for their favorite candidate. If there's a majority, we can stop there. The first preference winner becomes the tentative winner. In the second round, voters *use approval* and vote for every candidate that they prefer to the tentative winner (which may be no one). Second-round voters use no other strategy, and this is definitely cheating (see note at top). If any candidate gets majority approval in the second round, the one of
these with the most approval wins. Otherwise the tentative winner wins.
**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.
Craig Carey only intended his method 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.
**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.
**Majority favorite**: This option paints regions a color if there is a majority favorite there,
and otherwise
paints it black. This can in particular help you understand what kind of "spread" the standard deviation is actually
achieving.
**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).
**Margins Sorted Approval**: A Condorcet method proposed by Forest Simmons. Sort the candidates into
a list in
descending order of approval. Then while any candidates are pairwise defeated by the candidate following them in the
list, swap the order within the list of the two such candidates which have the closest approval scores. Finally, elect
the candidate
who is now first in the list.
**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.
**MinGS**: A Woodall method under which one elects the candidate X whose
fewest votes (pairwise) against some other candidate Y is the greatest. At __/calc__ this is called "MaxMin(Pairwise Support)," but I use the older name here as there is no equal (above-bottom) ranking in this simulation.
**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 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.
**Raynaud(WV)**: This is a Condorcet method in which, while there is no Condorcet winner, you
eliminate the candidate on the losing end of the strongest remaining defeat (per winning votes). It can give some
strange results as frontrunners can be eliminated before noise candidates.
**RCIPE**: Richard Fobes' method, "Ranked Choice Including Pairwise Elimination," which is broadly like IRV, except that for each elimination, if any remaining candidate loses pairwise to all other remaining candidates, then we eliminate this candidate instead of the candidate with the lowest vote count. The proposal includes essentially a fractional treatment of equal ranking, but there is no equal above-bottom ranking in this simulation.
**River(AWP)**: This is Jobst Heitzig's River method using "Approval-Weighted Pairwise," which is a form of James Green-Armytage's "Cardinal-Weighted Pairwise," which you can read about in __this interesting paper__ from *Voting matters*, 2004. Specifically, the defeat strength for some A over some B, which A pairwise defeats, is equal to the number of voters who approved A and didn't approve B.
**River(CCE)**: This is River(WV) 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 is very similar to
MDDA, but fixes its Plurality problem while losing Favorite Betrayal compliance.

RMPA resolution details have been added to the __/misc__
calculator.

These methods are also formerly called on the __/calc__ page "Majority Rule Approval" and "Majority Rule FPP."
**Single Contest** is an idea of mine from 2011. If there's a majority favorite, that candidate wins. Otherwise elect the pairwise winner between the pair of candidates who together minimize the number of voters who approve neither. Break ties in identifying the pair of candidates in favor of the (eligible) candidate with the most approval, and the most-approved candidate who can be in a pair with them.
**Smith//IRV**: First delete every candidate who isn't in the Smith set, and then perform IRV among the remaining candidates.
**STV3//CIRV**** or "fill three seats using STV and then conduct Condorcet//IRV with the three STV winners as the candidates, and don't allow truncation in round 2." I know it's oddly specific. Second-round voters can't truncate, but they could conceivably use compromise or burial. They will not, however, so we can call this cheating.
**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.
**TACC**: "Total Approval Chain Climbing," a 2005 method by Jobst Heitzig. Start with an empty set. Starting 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.
**Top-two Runoff****: Hold a one-on-one election between the two candidates who received the most first preferences in a first round. This is an actual two-round method where the second round makes use of the sincere preferences, not the truncated ballots.
**Voice of Reason**: Hopefully faithful to a proposal by Burt Monroe. Each voter receives one point for every pairwise preference of theirs which agrees with the overall preference among the voters. (Half point for expressing indifference, and no points for anyone if the overall preference were to be a tie.) Among the voters with the highest score, elect the first preference winner. (This last step is originally a lottery.) I don't check for a majority favorite, or do anything to account for the clone dependence.

*** - indicates a two-round method; see note at top.*
See

__here__ for examples of what to enter below. Note ⚠️ you could hang or crash your browser here!