If not Condorcet, then what?
Here is a calculator for a selection of rank ballot methods which satisfy neither Condorcet nor Later-no-harm.
All the methods satisfy Woodall's Plurality criterion (or a weaker form) and minimal defense, which means I would offer some endorsement of them.
Douglas Woodall's Descending Acquiescing Coalitions (DAC) and Quota-Limited Trickle-Down (QLTD)
Approval Elimination Runoff (AER)
No elimination IRV (one possible interpretation of this concept)
Chris Benham's IBIFA (Irrelevant Ballot-Independent Fallback Approval)
Etjon Basha's Iterated Bucklin
Three methods of mine: King of the Hill, MAMPO, and CdlA ("conditional approval").
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.
DAC: Proposed by Woodall in 1996 as his preferred alternative to IRV. This method works the same as DSC (from the Later-no-harm calculator) 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.
QLTD: 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.
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).
No elimination IRV: 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.
(At /calc I offer a "type 2" of this method which drops the "if the only candidate left to
be eliminated..." rule, and this version appears to have been suggested by Bjarke Ebert in 2021.)
IBIFA: Proposed by Chris Benham in 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.
Iterated Bucklin: Proposed by Etjon Basha in 2020. 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 (in contrast to several other methods here, in which preferences can only ever be added, not
withdrawn). 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 this calculator does not 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.
King of the Hill: Proposed by me in 2011. This (as well as Chain Runoff) was discovered by a random method generator seeking certain property combinations.
In "KOTH," 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 three Mike Ossipoff criteria (as my 2005 proposal MDDA had already done) 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).
The name stands for "Majority approval, minimum pairwise opposition."
CdlA: Proposed by me in 2003. The name is short for "conditional approval" which I don't 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.
Some other concepts and methods are also included in the table to facilitate comparison.
FPP, IRV, and DSC are Later-no-harm methods covered elsewhere. Schwartz and Condorcet are also described elsewhere.
"Implicit approval" is the notion that a candidate ranked above any other candidate has "approval" from the voter in
some sense. (This concept is overtly used by AER and MAMPO, and less overtly by some others.)
When an election is tied, I show (e.g. with color) all the possible winners in the main table, but there may still be some
potential for confusion in the presentation of other details. Below the table, some more in-depth information for certain methods can be displayed.
A few notes. I possess few proofs, so all comments should be taken as seeming to be true according to simulations done so far.
Minimal Defense: All the featured methods satisfy it. If a majority of voters rank A above B and rank B above no one, then B will lose. The "minimal defense"
required of this majority is to not rank B: They don't need to go any further by (for example) insincerely ranking A as their first choice. This helps prevent
the result from being affected by the presence of weak candidates. The concept comes
ultimately from Mike Ossipoff, who called it "SDSC," or "strong defensive strategy criterion."
Plurality criterion: All satisfy it except for IBIFA, which satisfies the weaker version I discuss on my CCE page. In other words, IBIFA would satisfy
Plurality if we say that Plurality doesn't have a voter's ballot count towards disqualifying their own compromise choices.
Condorcet Loser: satisfied by no elim IRV, AER, and seemingly IBIFA.
Mutual Majority: Failed at least by CdlA and King of the Hill. I haven't found failures in the other methods.
Monotonicity (Mono-raise): satisfied by Bucklin, QLTD, DAC, and MAMPO.
Mono-add-top: satisfied by DAC only.
Later-no-help: satisfied by Bucklin, QLTD, DAC, and King of the Hill. This property is noteworthy in that no Condorcet method can satisfy it.
Favorite Betrayal criterion (weak FBC): If equal rankings are allowed, weak FBC is satisfied by versions of Bucklin, QLTD, IBIFA, and MAMPO. This property
is also incompatible with Condorcet.
Ossipoff's "Strategy-Free" criterion: satisfied by MAMPO only. This property means that if there is a full majority for A over B, but no full majority over A, then B will not win.
(The concept is that B could not be the sincere Condorcet winner, while A might be, and it should be "strategy-free" to defeat B in this case.)
MAMPO was designed with the goal of simultaneously satisfying this property, minimal defense, Plurality, and the weak FBC.
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 doesn't allow equal ranking above the bottom. This is because not all of these methods
handle that kind of ballot.
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.