A calculator to compare a lot of methods at once, without resolution details.
Methods described elsewhere on the site
/cce - CCE suggested winners (i.e. candidates who could win under any of the several methods suggested) /cond - the WV Condorcet methods /lnharm - Adjusted Condorcet Plurality (ACP), Chain Runoff, DSC, FPP, IFPP, IRV, MMPO, RUE FPP, Top-Two Runoff /misc - Approval Elimination Runoff (AER), Bucklin, CdlA, DAC, IBIFA, Iterated Bucklin, King of the Hill, MAMPO, no elimination IRV (type 1), QLTD
ACP variant: This is a variation of my Adjusted Condorcet Plurality (ACP) (see /lnharm).
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: 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."
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. 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
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 Smith is satisfied. The "max" version, in the absence of a
winner, elects the candidate with the most first preferences who beats or ties the plurality winner. 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
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).
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.
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.
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.
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.
MR FPP and MR Approval: Ideas of mine. "MR" stands for "majority rule"; this is a little presumptuous, so let's call the name
tentative. We choose a metric (first preferences or implicit approval), 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 attempting to confirm each of their majorities. In River a candidate is always "under" another candidate, initially themselves.
When you consider a proposition of X beating some Y, check whether Y is under themselves. If so,
then all candidates currently under Y are changed (if necessary) to be under whatever candidate X is under (possibly X
themselves, or indeed even Y, if X had been under Y). In these "MR" methods, we elect whichever candidate the metric leader
(first preference winner, etc.) ends up under.
MR FPP and MR Approval both satisfy Plurality and minimal defense and so could be included in the /misc
calculator. MR FPP is sometimes similar to King of the Hill, but doesn't satisfy Later-no-help. MR Approval is very similar to
MDDA, fixing the Plurality problem but losing Favorite Betrayal compliance.
No elimination IRV type 2: This is the same as type 1, a method presented on the "misc" calculator, 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 still satisfies minimal defense and Plurality.
It appears this particular rule was proposed by Bjarke Ebert in 2021.
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.
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.