Condorcet's method is the method that genuinely lives up to the promise of rank-balloting. The choice between Approval & Condorcet is a difficult one. Both methods have exclusive criterion compliances, compliances that no other method has. I feel that Condorcet's exclusive compliances are more powerful under typical conditions, for getting rid of the lesser-of-2-evils problem. But both methods bring so much improvement that Approval's easier implementation & proposability could tip the balance in its favor where those 2 considerations predominate.
If candidate A is our favorite, but we probably need candidate B to beat the despised candidate C, then ideally we'd like to vote candidate B over candidate C, just to be safe, but we'd also like to vote A over B. And we'd like both of those preferences to be reliably & fully counted.
So what we want is really about pairwise preferences. So then, when we rank the candidates, doesn't it make sense to count the pairwise preferences directly? Count my preference for B over C, and my preference for A over B, and for A over C. And we mustn't count the A over C preference as stronger, because that creates a dilemma of which of our preferences we will give give a "full-strength" vote by giving them maximum separation in our ranking. So we count all pairwise preferences equally. Only then can we say that we're _fully_ counting all of our voted preferences.
Often this count will turn up a candidate who wins his pairwise comparisons with each of the other candidates. I'll call him a "BeatsAll" winner.
There are a number of proposals that search for a BeatsAll winner. They differ in what they do when there isn't one. There could, for example, be a situation where A beats B, B beats C, and C beats A, when everyone's pairwise preferences are tallied. That's called a "cycle", or a "circular tie".
Well, since our original goal was to get rid of the lesser-of- 2-evils problem, what can we do to at least reassure the voter who is dominated by that problem? That's a voter for whom the important thing is to cast an effective vote against someone he most dislikes, so much so that he'll give up voting for his favorite if necessary in order to do so. So let's reassure him that his vote against Mr. Worst will be fully & reliably counted, without his having to move some lesser-evil up to 1st place.
Let's count the candidates' votes against them, for the purpose of solving a "circular tie" , a situation with no BeatsAll winner.
So then, we could define this method:
1. Candidate A beats candidate B if more voters rank A over B than vice-versa. 2. If one candidate beats each one of the others, then he wins. 3. If not, then the winner is the candidate whose greatest defeat by another candidate is the least.
(B's defeat by A is measured by how many people ranked A over B)
We call that method "Plain Condorcet" ("PC").
It turns out that this method was proposed in the late 18th century, by the founder of voting theory.
The Marquis de Condorcet, in the period just after the French Revolution, participated in the discussion of how the new government should be set up. This included voting system proposals.
He proposed that if there's a candidate such as I've called BeatsAll winner, that candidate should win. And he proposed a natural & obvious solution for when there's no BeatsAll winner.
Condorcet made 2 suggestions for solving such "circular ties": a "drop-weakest" proposal, and a "keep-strongest" proposal. What follows is his drop-weakest proposal. What I've been calling "pairwise-defeats", Condorcet called "propositions":
Condorcet's drop-weakest proposal:
If the propositions can't all exist together [because there's a cycle instead of a BeatsAll winner], then, one at a time, drop the proposition with the smallest majority, until there's an unbeaten candidate.
In other words, drop the weakest defeat. Repeat till there's an unbeaten candidate.
(The strength of B's defeat by A is measured by how many people ranked A over B).
What could be more natural, when the pairwise defeats are inconsistent, than to sequentially drop the weakest ones?
Now, if Condorcet's words are taken literally, then what he proposed is equivalent to the method that I've called "PC". PC is the literal interpretation of Condorcet's "drop-weakest" proposal.
Here's a criterion met only by Condorcet versions such as PC:
The "Condorcet winner" (CW) is the candidate who, when compared separately to each one of the other, is preferred to him by more people than vice-versa. This is about sincere preferences, which may differ from actual voting with some methods.
If no one falsifies a preference, and if there's a CW, and if a majority prefer the CW to Y, and vote sincerely, then Y shouldn't win.
PC meets WDSC ( Weak Defensive Strategy Criterion, defined in the Approval section, and at the Strategy Criteria page). ***
Here's the accepted interpretation of Condorcet's excellent "keep-strongest" proposal:
In order of stronger defeats first, consider each defeat, in turn, as follows: Keep it if it doesn't contradict (form a cycle with) already-kept defeats.
When all defeats have been so considered, a candidate wins if s/he has no kept defeats.
That definition is intended to be taken literally. For example, when we start by considering the strongest defeat, it of course doesn't form a cycle with any already-kept defeats, because no defeats have been kept yet, since we haven't even considered any other defeats yet.
When RP's defining rule, as stated above, is followed, it turns out that the strongest and 2nd strongest defeats are always kept.
A cycle is a cyclic sequence of defeats, such as: A beats B beats C beats A.
RP has the great advantage of combining additional important criterion compliances with an extremely brief & simple definition, making RP a top choice, likely the best choice, for a public Condorcet proposal.
Some additional criteria met by RP: Strong Defensive Strategy Criterion, and Generalized Strategy-Free Criterion (GSFC). SFC only applies when there's a CW, but GSFC always applies.
Generalized Strategy-Free Criterion (GSFC):
The sincere Smith set is the smallest set of candidates such that every candidate in the set is preferred to every candidate outside the set by more voters than vice-versa. Again, note that we're talking about sincere preferences rather than votes.
If no one falsifies a preference, and if X is a member of the sincere Smith set, and Y is not, and if a majority of all the voters prefer X to Y, and vote sincerely, then Y shouldn't win.
A precise definition of voting sincerely, and some other preliminary definitions, may be found at the Strategy Criteria article, at this website, linked-to from the main table of contents. Comments regarding the criteria may be found there also.
Strong Defensive Strategy Criterion (SDSC):
If a majority of all the voters prefer A to B, then they should have a way of ensuring that B won't win, without any member of that majority voting a less-liked candidate equal to or over a more-liked candidate.
(Note that if you don't vote for A or B, you aren't voting them equal, since you're not voting for them at all).
SDSC differs from WDSC only in that "...over a more-liked candidate" is replaced by "...equal to or over a more-liked candidate".
Another interpretation of Condorcet's "drop-weakest" proposal: Someone pointed out that, since we're only dropping pairwise defeats because they can't all exist together, because they are in conflict in regards to choosing a winner--it would make more sense to only drop from among those defeats that are the ones that are actually in conflict for choosing a winner.
A cycle is the reason why the defeats cannot all exist together, why they contradict eachother, and the reason why there's no unbeaten candidate, and therefore the reason why we need to drop defeats. So why not only drop from cycles? Here's a simple rule that does that:
Drop the weakest defeat that's in a cycle. Repeat till there's an unbeaten candidate.
An example has been found in which SD violates the Monotonicity Criterion. PC, and SSD have been demonstrated to pass Monotonicity. RP has never been shown to violate Monotonicity. It probably doesn't but even if it did have a so difficult-to-find Monotonicity failure example, then for all practical strategic purposes it doesn't fail Monotonicity. RP is a very popular method among Condorcet advocates.
SD's violation of Monotonicity requires an unlikely special situation, whereas IRV violates Monotonicity much more easily. When offering Condorcet as an alternative to IRV, one is offering Condorcet to people who can accept IRV, and who must also accept IRV's Monotonicity violation. Therefore, SD's Monotonicity violation doesn't count against it in a comparison with IRV. Still, SD's violation of Monotonicity is an embarrassment, and could be used against it when SD is being considered as an alternative to Plurality or Borda. That's a reason why RP seems a better public proposal.
Monotonicity is defined at the Academic Criteria page, but, briefly, a method meets Monotonicity if voting a candidate higher will never make him lose, and if voting a candidate lower will never make him win.
The method defined next, SSD (Schwartz Sequential Dropping ), is the Monotonic relative of SD. It meets SD's criteria, while also meeting Monotonicity. Also, SSD is a refined drop-weakest approach, and a slightly different interpretation of what Condorcet meant in his drop-weakest proposal, making SSD interesting even though its wordy definition makes it a less feasible public proposal.
If there's a set of candidates none of whom are beaten by anyone outside that set, then there's nothing to contradict or conflict their defeats of the candidates outside that set. The only defeats in conflict for choosing a winner are the defeats among the candidates in that set.
Let me state a brief & simple definition about such a set:
1. An "unbeaten set" is a set of candidates none of whom is beaten by anyone outside that set. 2. An innermost unbeaten set is an unbeaten set that doesn't contain a smaller unbeaten set. 3. The "Schwartz set" is the set of candidates who are in innermost unbeaten sets.
In public elections, where there are no pairwise ties, there will only be one innermost unbeaten set. Under all conditions there will be at least one.
So the Schwartz set is the innermost set of candidates unbeaten from without. Those are the candidates whose defeats are in conflict for choosing a winner, and so no defeat other than those needs to be dropped, because the candidates outside that set have defeats that aren't contradicted by other defeats.
That leads to the following method definition:
The "current Schwartz set is the Schwartz set based only on defeats that haven't yet been dropped.
Drop the weakest defeat among the members of the current Schwartz set. Repeat till there's an unbeaten candidate.
(The strength of B's defeat by A is measured by how many people have voted A over B).
SD & SSD, like RP, meet SFC, WDSC, GSFC, & SDSC.
Though PC meets SFC & WDSC, it hasn't been proven yet whether or not PC meets GSFC &/or SDSC.
Incidentally, it also hasn't been demonstrated yet whether or not these Condorcet versions can fail FBC, the criterion that requires that no one ever have incentive to vote someone over his/her favorite.
Instant Runoff (the Alternative Vote) and Plurality (First Past The Post) fail FBC in simple, obvious examples.
Briefly, a method is clone-independent if, under sincere voting, parties or factions don't gain or lose advantage by running several identical candidates. RP is clone-independent under all conditions. SSD is clone-independent in public elections, where there are many voters, but not in small committees. But a simple change of SSD's stopping rule makes SSD clone-independent under all conditions. Instead of stopping when someone is unbeaten, stop when there are no defeats among the members of the current Schwartz set.
The resulting method is called "Cloneproof SSD" (CSSD).
Let me write the full CSSD definition here:
If no one is unbeaten, drop the weakest defeat among the candidates of the current Schwartz set. Repeat till there are no defeats among the members of the current Schwartz set.
Of course typically we get a current Schwartz set containing only one member, and so of course then there are no defeats among the members of that set. CSSD avoids the clone-strategic-incentives that SSD could have in a small committee election.
I've defined SD, SSD, and CSSD in a way that's clear, without being fully explicit. Let me now word CSSD's definition in an entirely explicit way, a program-style wording:
1. Calculate the Schwartz set, based only on undropped defeats.
2. If there are no defeats among the members of that set, then the members of that set are the winners, and the count ends.
3. Otherwise, drop the weakest defeat among the members of that set. Go to 1.
CSSD is equivalent to another popular method-- BeatpathWinner. That means that CSSD and BeatpathWinner will always pick the same winner.
2. The strength of a beatpath is measured by the strength of its weakest defeat.
3. If the strongest beatpath from X to Y is stronger than the strongest beatpath from y to X, then X has a beatpath win against Y.
4. A candidate wins if no one has a beatpath win against him.
Of course Beatpath will produce only one winner, except under the rare tie conditions that could make any method give more than 1 winner.
As I said, Beatpath Winner and CSSD are equivalent, meaning that they always give the same result.
I like those 2 methods for recommending to committees and organizations, because (via the BeatpathWinner implementation) they have a relatively simple, brief, & elegant computer program. In organizations and committees, the people being asked to adopt a voting system are much closer to the actual count procedure, as compared to a voter in public elections. So, while brief definition seems most important in public elections, a simpler count program seems more important for committees & organizations.
BeatpathWinner & CSSD win in that category, due to the relative simplicity of the BeatpathWinner algorithm.
CSSD follows from a natural and obvious motivation and justification. But there's a case for saying that BeatpathWinner does too:
The reason why we don't have a BeatsAll winner is because of a cycle. Say, in that cycle, A beats B beats C beats D beats A. If the voters have indicated that A is better than B, and that B is better than C, then that's a statement that A is better than C. The A to C beatpath says that. But there's also a C to A beatpath contradicting it.
As before, when the defeats are mutually contradictory, it makes sense to start dropping the weakest ones, till the contradiction is gone.
The preceding 2 paragraphs lead us to BeatpathWinner.
So, BeatpathWinner, like CSSD, has natural and obvious motivation and justification. The two wordings for the method, the BeatpathWinner wording and the CSSD wording, show that motivation in two different ways.
Now, someone might object to our way of measuring the strength of a defeat. Some would rather go by the margin of defeat. That's counter to Condorcet's proposal, because Condorcet drops the defeat with the smallest majority, not the one with the smallest difference between majority & minority.
Also, we're only dropping defeats because we have to, because they're mutually contradictory & conflicting. When we do that, we're overruling some people's pairwise preferences, something that shouldn't be done lightly. If A beats B when the pairwise preferences are tallied, and we drop that defeat, we're overruling the voters who voted A over B. We can't overrule the voters who voted B over A. They were already overruled by the count of pairwise votes, which the A-beats-B voters won. Therefore, the voters who voted for B's defeat by A are the ones whom we have to consider when we try to overrule as few individual pairwise preferences as possible. That's a basic ethical justification for measuring pairwise defeats as we do. Another reason to measure them as we do is because of the criterion complicances that it confers. For instance, PC, RP, SD, SSD, CSSD, and BeatpathWinner meet SFC & WDSC only because they measure defeats in the way that I've described.