This is the final article in a series analyzing the 12 best voting-systems. See the introductory article for the series here.

Below, is a long and detailed comparison of 12 voting-systems by 4 standards. I recommend it only if you’re interested in something that’s quite long, detailed, & thorough. Otherwise, regard it as a reference, for such time as there’s something in it that you want to look up. Or you could check it for verification or justification of my recommendation for, and statements about, MDDA & MDDAsc.

The 12 methods will be compared in the following regards:

1. Chicken-Dilemma.
2. Protection of ranked candidates against unranked.
3. Protection of top-ranked candidates against all others.
4. Protection middle (not top or bottom) ranked candidates against each other.

1. Chicken-Dilemma

This section discusses the chicken-dilemma in Approval, Score, Bucklin, and some methods that get rid of the chicken-dilemma, or provide a particularly easy, convenient & reliable way to deal with it. Here’s a list of the latter:

• IRV
• Benham
• MDDA
• MDDAsc
• MMPOsc
• IC,MMPO
• MAW
• MAPW2

Approval & Score will be discussed first. IRV & Benham will be discussed last, because they’re so completely different from the other methods.

Chicken-Dilemma Definition:

First, let me re-state the chicken-dilemma, as defined at the election-methods mailing-list:

Briefly it’s a situation in which one or both of two similar factions fear that if they support the other faction, then that other faction will take advantage of their co-operation by not reciprocating it, and thereby take the win from them (even if the co-operative faction is the  larger one).

In a little more detail:

There are 3 candidates, A, B, & C.

The A & B factions greatly prefer both A & B to C.

The A & B factions add up to more than half of the voters.

The order of the factions’ sizes is: C>A>B

The C faction are indifferent between A & B.

The A faction have reason to believe that if they support the B faction, the B faction might not support A, and might thereby take the win from A.

If neither faction supports the other, then C wins—the worse outcome for the A & B factions.

If one faction co-operates, by supporting the other faction, and the other faction doesn’t, then the non-co-operating faction, the defecting faction, wins.

The message is “You help, you lose.”

What follows will be chicken-dilemma comparisons between the methods. It will be discussed from the point of view of a member of the A faction, in the above definition.

Approval:

(The discussion of chicken-dilemma for Approval is long, because Approval doesn’t have the quick, convenient, easy & reliable way to avoid chicken-dilemma that some of the rank-methods have. But, as described below, chicken-dilemma isn’t really a problem with Approval, and needn’t keep anyone from advocating or adopting Approval—Much of this long section is devoted to telling why that is.)

There are a number of reasons why the chicken-dilemma isn’t the problem that it might seem.

1. It will be uncommon or rare:

The chicken-dilemma occurs because the C voters don’t prefer A to B, or B to A.

If, for example, they preferred B to A, then B would be the middle candidate. Then, either B is the “Condorcet winner” (the candidate who has a majority over each one of the other candidates), or else C or A has a 1^st-choice majority, in which case there’s no problem anyway, and it doesn’t matter how anyone else votes.

And, if B has a majority over everyone, including A, then the A voters have no reason to expect the B faction to approve A. Why should they? And the A voters have no reason to not approve B. B is the rightful winner.

It would be unusual for the C faction to not have a preference between A & B.

So the chicken-dilemma will be uncommon or rare.

• If either A faction or the B faction is known to probably be larger than the other, then the larger faction is the rightful winner. They have no reason to approve the other faction. The other faction should approve them, if the other faction wants to defeat C. Both factions know that. There’s no chicken-dilemma.
• If either the A faction or the B faction is clearly the one with a more moral candidate or positions, or if one of those factions is clearly the more selfish or less ethical one—maybe the one whose policies are in support of the rich 1%, then it will be obvious to all which one is more desirable.

Or maybe it’s known to all that the A faction is more principled in their policies.

In any of those cases, it’s clear that the A voters aren’t going to approve B, out of principle, and that therefore the B voters would best approve A.

For C voters to be indifferent between A & B, and for there to be no known size-difference between the A faction & the B faction, and for there to be no difference between them as regards morality, principle, favoring of the 1% or the 99%,  etc. –would be highly improbable.

That’s why a chicken dilemma will be uncommon or rare.

2. Widespread intent to defect can’t be kept secret:

If the B faction are going to defect on a large scale, then there’ll be talk about that. Either in conversations everywhere, or even in organizing messages in the media. With that known to the A voters, the A voters will defect too (unless, as noted above, B is Condorcet winner, or the B voters are more principled, etc.).  If the obviously impending B defection is merely a strategic defection, to take advantage of the A faction’s co-operation, then the A voters will defect too, so as to not reward the B voters’ defection.

The B voters know that , if they defect, then they won’t get co-operation from the A faction in the next election. In fact there’s a systematic strategy like that—The Tit-For-Tat strategy:

• Always do what the other faction did in the previous election (co-operate or defect)
• That will efficiently eliminate defection, and the chicken-dilemma

3. Anti-defection strategies that don’t involve future elections:

The A voters could give to B, just enough approvals to help B win if B is the larger of A  & B.

Two questions:

• How can that lesser number of approvals be given?
• How would it be determined how much to give?

How would the A voters give some lesser number of approvals to B? Each B voter could probabilistically give to B a fractional approval?  As follows:

For example, if it’s desired to give half-approval to B, then each A voter could flip a coin, to determine whether to approve B. If it’s desired to give ¾ approval to B, then each A voter could flip a coin twice, and if tails comes up both times, don’t approve B.

There are, of course, innumerable ways it could be done. If it’s desired to give 2/3 approval to B, then each A voter could put 3 identical paper squares into a bag. One of the squares has a mark on it. Draw a paper square from the bag. If it doesn’t have a mark on it, then approve B. For something more general, generate a 2-digit number as follows:

On each of 10 paper squares, write one of the numerals from 0 to 9. Put the squares in a bag.

Draw a paper from the bag. Write its numeral down. Put the paper back in the bag, shake the bag up again, and again draw a paper from the bag. Write its numeral down directly to the right of the 1^st numeral that you wrote down.

Now you’ve written a 2-digit number somewhere from 0 to 99.

Now, say that you want to give .71 approval. Well, the numbers that you could generate are equally likely to be any number from 0 to 99. That’s a hundred numbers that are equally likely to come up.

So pick any 71 of those 100 numbers, and approve B if one of those 71 numbers comes up.

The most convenient way to do that is to say, “Approve B if the number that comes up is from 1 to 71. (Note that that doesn’t include 0).

How would it be determined how much to give?

1. a) Guess:

You could just take a wild guess about the right fraction of an approval to give B. With that guess, you’re trying for an amount that would be just enough to help B beat C, if B is the larger of A & B.

Guessing is really all you’ve got available if there’s no predictive information about the faction sizes. If the A voters don’t know, then neither do the B voters. So if the A voters make it clear that they’re trying for giving B just enough to win if B is larger, then maybe the B voters should do the same, in case it’s A, not B, that’s larger.

But, with a little predictive information, there’s a better way:

1. b) Calculate the right amount to give B:

Suppose that the best available information &/or estimates says that the C faction will have 40% of the voters.

Then here’s what the A voters could say to the B voters:

“C probably has 40% of the voters. Suppose that the A & B factions are about equal in size.

“Suppose that you, the B faction haves 31% of the voters, and that we, the A faction, have 29% of the voters.

“You need 10% more, in order to beat C’s 40%.

“We’re going to give you an amount of approval equivalent to being approved by 10% of the voters. Given that we’re 29% of the voters, we’re going to accomplish that by each of us approving you with a probability of 10/29.

“Then you’ll have a total of 41% approval, enough to beat C.

“But if you’re not the larger faction, if you’re the faction with only 29%, that won’t make you win.

“Suppose we have 31%, and you have 29%.

“31 X (10/29) = 10.69

“We give you 10.69%

“29 + 10.69 = 39.69

“That isn’t enough to beat C’s 40%

“So, in case you’re the smaller of our 2 factions, each of you should likewise approve A with a probability of 10/29, so that, if you can’t beat C, we can.

“In this way, the more popular of our 2 candidates will be the winner.”

—

So both factions’ members each approve the other faction’s candidate with a probability of 10/29, and the result is that the larger faction’s candidate wins.

Something similar could be done in Score.

How would you approve B with 10/29 probability?

One way would be to prepare 29 paper squares, with 10 of them having a mark on them.

Draw a paper from the bag, and approve B if that paper has a mark on it.

That would be the best way, because then the probability is exactly 10/29

But, if you already have the 10 paper squares numbered from 0 to 9, and you want to do it without making more paper squares, then you could just use the general method described above:

10/29 is about 34.5%

Generate, as described above, a 2-digit number from 0 to 99.

Approve B if the number generated is from 1 to 34.

Or, more accurately, because we want 34.5%, you should approve B if the number is from 1 to 34. But if the number is 35, then flip a coin to decide whether to approve B.

Or, alternatively, of course you could generate a 3-digit number from 0 to 999. Then approve B if the number is from 1 to 345.

If that sounds like a lot of trouble, then I emphasizes that the probability-determination only has to be done once, by one person in each faction.  And it avoids the chicken-dilemma, in the simple, minimal Approval voting-system.

Yes, each voter has to draw the numbers from a bag, but why not? Why not have some fun, to vote so as to avoid the chicken-dilemma. Regard it as an entertainment, a game.

And I re-emphasizes that chicken-dilemma will be uncommon in Approval.

…and chicken-dilemma is the only thing that Approval has that even resembles a problem.

Score:

Like Approval, except that, instead of being achieved probabilistically,  the fractional rating can be achieved more easily, by Score’s more flexible ratings.

MDDA:

But, if we’re going to use a rank-method, then why not use one that has a feature that can easily & reliably avoid chicken-dilemma.

The A faction co-operates and the B faction defects:

If you’re an A voter, then, while sincerely ranking B, deny hir approval.

By ranking B over C, you give C a majority-disqualification. The two un-disqualified candidates are A & B. Among those two, A (having a larger faction) has more approvals, and thereby wins.

Both factions co-operate:

Both factions rank eachother’s candidate sincerely above C, and both factions deny approval to eachother’s faction—as a defense against defection.

Just as above, C is majority-disqualified, and A wins because of having more approvals, due to its larger faction.

How well is candidate B protected?:

If you’re an A voter, and B is top-ranked by you, and you deny approval to B, you’re still fully protecting B against burial or truncation.   …because any candidate top-ranked by you receives that protection from you, for reasons discussed below, in the “Protection of Top” section.

If B, to whom you deny approval, is middle-ranked (not top or bottom ranked) by you, then your protection of hir is less:

Because MDDA is fully truncation-proof, B is still protected against truncation.

Truncation of a candidate consists of refusal to rank hir. Under some circumstances, with MDDAsc, truncation can take the win from an MCW (a candidate who is majority-preferred to the other candidates in separate pairwise elections). But MDDA is fully truncation-proof.

But, if you middle-rank B without approving hir, you aren’t giving hir any protection against burial (from any candidate). Burial of a candidate consists of insincerely ranking someone else over hir.

Well, that means that you’re reluctantly cutting B loose from burial-protection because hir faction have acted in a way that makes it necessary to take the precaution of protecting against defection by them.

But, as I said, if you deny approval to a candidate whom you top-rank, your top-ranking of hir fully protects hir against truncation and burial, as described above.  …while your denial of approval to B protects A from defection by hir faction.

MDDAsc:

Same as MDDA, except that MDDAsc doesn’t have full truncation-proofness. So, if you middle-rank B, and withhold approval from hir, you’re cutting hir loose from truncation-protection as well as from burial-protection.

The following paragraph in the MDDA section can be slightly modified to say:

“Well, that means that you’re reluctantly cutting B loose from truncation & burial protection because hir faction have acted in a way that makes it necessary to take the precaution of protecting against defection by them.”

Maybes in MDDA, & especially in MDDAsc, middle ranking someone, and withholding approval from them, is something that you’d reserve for a candidate who isn’t very important or valuable to you–someone barely worth ranking above bottom.

Middle-ranking, without withholding approval, is something you’d do with a candidate with whom you don’t expect a chicken-dilemma, but whom you still want your top-ranked candidates to beat.

As I’ve said, ordinarily there’s no reason to expect a chicken-dilemma.

And it remains true that if, as an A voter, you top-rank B and deny hir approval (to defend against chicken-dilemma defection), you’re still (because you’re top-ranking hir) giving hir full protection against burial & truncation, in both MDDA & MDDAsc.

MMPOsc & IC,MMPO:

Plain MMPO has a problem that makes it quite unusable by itself. Used by itself, plain MMPO could elect a nearly universally disliked candidate, even if only 2 voters, out of millions, ranked him/her. But the MMPO versions defined & discussed in this article avoid that problem.

The MMPO versions included in this article are IC,MMPO & MMPOsc. This section discusses those 2 methods with regard to the chicken-dilemma.

Though Plain MMPO is unusable by itself, the chicken-dilemma avoidance of IC,MMPO & MMPOsc is really that of ordinary MMPO. So it’s only necessary to speak of how, and how well, MMPO avoids chicken-dilemma.

IRV, MDDA, & MMPO avoid chicken-dilemma in 3 different ways:

In IRV, your 2^nd choice doesn’t have any support from you until your favorite is eliminated, due to being favorite of fewest. That makes a chicken-dilemma problem impossible.

In MDDA, if one faction ranks the other over the worst candidate, then that majority-disqualifies the worst candidate. …and if neither faction gives an approval to the other, then the largest of those 2 factions will be the one that wins with the most approvals. The withholding of approval thwarts defection.

In MMPO, if one faction ranks the other over the worst candidate, then, as in MDDA, that gives that worst candidate a majority pairwise-opposition. S/he’s the only candidate with majority pairwise opposition, and therefore can’t win.

The other 2 candidates, the chicken-dilemma rivals, both share the same maximum pairwise-opposition—from the preferrers of the worst candidate. MMPO’s tie-rule says that, when 2 candidates have equal max pairwise-opposition, then we compare their 2^nd-greatest pairwise oppositions. …in this case, the 2 rival candidates have their 2^nd-greatest pairiwise opposition from eachother. …and the smaller faction has more of that, so the larger faction wins.

MMPO doesn’t use approval, and its chicken-dilemma protection is completely automatic, and protects any higher-ranked candidate against any lower-ranked candidate.

And, because there’s no approval-denial needed for chicken-dilemma protection, it doesn’t weaken your protection for middle-ranked candidates against unranked ones.  But the catch is that there’s no such thing as positive protection of ranked against unranked, because there’s always a possibility that burial could succeed. More about that later, in the appropriate section.

Also, and worse, there’s no chicken dilemma protection for one top-ranked candidate over another. That’s a big loss & a big disadvantage, in comparison to MDDA & MDDAsc.

More about these other protections in the sections devoted to them later.

MAPW & MAPW2:

In the chicken-dilemma scenario, no candidate is majority-approved, and so it becomes just a count by MDDA, MDDAsc, MMPOsc, or IC,MMPO count, and so what was said about those methods applies here.

IRV:

The reason for IRV’s freedom from chicken-dilemma was discussed in the MMPO section, above. IRV has no chicken dilemma.

Benham’s method:

Benham’s method, based on IRV, shares IRV’s freedom from chicken-dilemma, for the same reason as IRV.

Bucklin:

Because Bucklin is stepwise Approval, the way of dealing with chicken-dilemma in Approval would work in Bucklin too.

But there’s an additional means that could be used: The candidate of the distrusted faction could be ranked several skipped ranks below top. That would give the top-ranked candidates sufficient rounds to get whatever later-choice votes they’re going to get, before your ranking gives a vote to the candidate of the distrusted faction. That might be better than the method of the previous paragraph, if the faction-size information isn’t reliable enough for that other method.

MAM:

MAM has chicken-dilemma, and doesn’t have an easy, convenient and reliable way to avoid it.

With MAM, chicken-dilemma can be dealt with as in Approval.  …and will be uncommon, & unlikely to be a problem, for the same reasons as with Approval.

But, because the best rank-methods provide a way to easily, conveniently and reliably avoid the chicken-dilemma, I suggest that, if a rank-method is going to be adopted, it should be one that provides a reliable, easy & convenient way to avoid chicken-dilemma. That suggests a method such as MDDA or MDDAsc, which additionally meet FBC  (MAM doesn’t).

MAM’s FBC failure could cause a need for some overcompromisers to rank their regrettable compromise over their favorite.

Having said that, it’s possible (as I was saying) in MAM, to deal with chicken-dilemma as one would in Approval.

It’s the same problem in MAM, as in Approval. The A voters in MAM should decide whether to rank B, in the same way that the A voters in Approval should decide whether to approve B.

In the sections below, I refer to top-ranked, middle-ranked, & unranked candidates. “Middle-ranked” candidates are just candidates who are neither top-ranked nor unranked.

4. Protection of ranked &/or approved candidates against those not:

Approval:

In this minimal method, the protection of approved against unapproved is obvious. If a majority approve (only) a certain set of candidates, then the winner must come from that set.

Score:

Same as Approval, if a majority top-rate a set of candidates.

MDDA:

If a majority rank a certain set of candidates over everyone else, then all of those other candidates are majority-disqualified. If voting is sincere & there’s no indifference, then there will be some candidate who majority-disqualifies each of the other candidates, by being ranked over hir by a majority. I’ll call hir the Majority Condorcet Winner (MCW).

In order for the offensive strategies of truncation (refusal to rank) or burial (insincerely ranking someone over) to make a majority-disqualified candidate win, they’d have to result in everyone being majority-disqualified. If the truncation or burial can majority-disqualify the MCW, then everyone is majority-disqualified, and then the most approved candidate wins.

1. a) In MDDA, truncation can’t possibly majority-disqualify the MCW, which means that s/he wins.
2. b) If burial succeeds in making everyone, including the MCW, majority-disqualified, and if there’s a set of candidates approved by a majority, then obviously one of those will win. Burial can’t take the win from that majority-approved set.

So, when you rank (& approve) a set of candidates, you’re fully protecting their win, against candidates whom you don’t rank or approve.   …just as surely as you are in Approval.

MDDAsc:

Same, except that a) doesn’t hold. Truncation could majority-disqualify the MCW (as could burial, in MDDA or MDDAsc). But MDDAsc remains as invulnerable to truncation & burial as MDDA is to burial. So its protection of approved & ranked candidates, against unapproved & unranked, is still complete.

The loss of full truncation-proofness is somewhat of a disadvantage, but not enough of a disadvantage to lose MDDA’s most important and powerful properties.

MMPOsc & IC,MMPO:

With MMPOsc & IC,MMPO, protection of ranked candidates against unranked ones is more iffy than with MDDA & MDDAsc. As for truncation, IC,MMPO is fully truncation-proof, and MMPOsc is not. What’s iffy is IC,MMPO’s protection against burial, and MMPOsc’s protection against burial or truncation.

Instead of using an approval-count, MMPO just elects the candidate with the least pairwise opposition. So, successful burial just means giving the MCW a strong pairwise opposition, resulting in everyone having more pairwise opposition than the buriers’ candidate.

So MMPOsc & IC,MMPO don’t have MDDA’s & MDDAsc’s solid and positive protection by approval.

Referring to the candidate whom the buriers insincerely rank over the MCW, let’s call hir the “burying-candidate”.

The burying-candidate might very well have a majority-disqualification from another candidate in addition to the MCW. If so, then when the buriers majority-disqualify the MCW, they’ve achieved the majority-disqualification of everyone.  It depends only on whether the burying-candidate has another majority-disqualification, by someone other than the MCW.

The good thing is that it might be difficult for would-be buriers to know that. The bad thing is that, if the situation is favorable, in that way, to the burial, then your middle-ranked candidates aren’t protected from burial (in the event that one of them is MCW).

It could be argued that, because it’s difficult for the buriers to know whether the burying-candidate has the necessary majority-disqualification by someone other than the MCW, then burial is at least somewhat deterred. That’s because (unlike with MDDA) if a burial is foiled, it’s also penalized by the election of someone the buriers like less than the MCW.

That could be counted as an advantage for MMPOsc & IC,MMPO over MDDA & MDDAsc.

But that deterrence is a bit questionable, because, if the MCW is in your strong bottom-set, then you’re completely or nearly indifferent between hir and your less-liked candidates, and so you have little or no deterrence from risking burying hir. That could lead to the “perpetual burial fiasco”.

Then, MDDA’s & MDDAsc’s positive & solid protection of ranked & approved candidates against unranked & unapproved candidates is a big improvement, because, though unsuccessful burial isn’t penalized or deterred, successful burial is definitely, positively & solidly prevented against majority-approved candidates against majority-unapproved candidates.

That’s why I suggest that MDDA & MDDAsc protect ranked candidates against unranked candidates more solidly than MMPOsc & IC,MMPO do.

In the chicken-dilemma section, I mentioned that MDDA, but not MMPO, can protect against chicken-dilemma defection by a candidate whom you’ve top-ranked (and thereby given full protection against burial & truncation).

So these two advantages of MDDA over these 2 MMPO versions add up to a lot of improvement by MDDA over MMPO.

Bucklin:

In this department, Bucklin’s protection is as good as MDDA’s. A majority who only rank down to a certain candidate have the assurance of electing hir or someone they rank higher.  …while still ranking in order of preference. (But Bucklin lack’s MDDA’s particularly easy chicken-dilemma avoidance).

5. Protection of top-ranked candidates against everyone else:

MDDA, MDDAsc, MMPOsc, & IC,MMPO:

In both MDDA  & MMPO, it’s impossible for burial or truncation to get a majority pairwise vote against a candidate top-ranked by a majority. Even with somewhat fewer people top-ranking someone, burial against hir would be made very questionably-feasible. That’s because the buriers need help to make that majority against hir, and they aren’t getting any help from people who top-ranking her—because such people aren’t ranking anyone over hir.

That’s why I say that, in the MDDA methods  & the MMPO methods, top-ranking a candidate fully protects her from burial & truncation, if a majority do so. (and helps that prevention a lot if any significant number of people do so).

Approval & Score:

Of course Approval & Score don’t offer this protection.

IRV & Benham:

IRV doesn’t allow top-ranking more than 1 candidate. But if a majority top-rank a candidate, it can’t lose. But maybe that doesn’t help you a whole lot, when you aren’t allowed to top-rank more than 1 candidate.

Same with Benham.

Bucklin:

This protection, too, is available in Bucklin by skipping rankings. If you want to protect your top-ranked from lower-ranked candidates, then skip some rankings between top-ranked & middle-ranked.  …to give the top-ranked candidates a sufficient number of rounds to get the lower-choice votes coming to them, before giving anything to your middle-ranked candidates.

Note that, though Bucklin’s skipping is a versatile way to give various kinds of protections, it requires more judgement and good estimating than is needed in MDDA’s & MMPO’s easy and reliable protection of top from middle.

MAPW & MAPW2:

They automatically immediately elect a majority-approved candidate if there is one. And, if not, they have the protections of MDDA or MMPO, whichever one is chosen as what they resort to otherwise.

As for top vs middle protection, of course that isn’t looked at when looking for a majority-approved candidate. But under the circumstances when they use MDDA or MMPO, then they of course have that method’s top vs middle protection.

Their emphasis is more on approval than on pairwise-count or rankings, and so the protection of some approved candidates against others isn’t their specialty.

6. Protection of middle-ranked candidates against eachother:

Of course, again, this doesn’t apply to Approval & Score.

Bucklin:

As with other kinds of protection, in Bucklin this can be accomplished by skipping ranks.  …between the candidates between whom you want to provide protection.  …as described above for other kinds of protection.

IRV:

IRV meets the Later-No-Harm criterion, and so your ranking of additional candidates below someone can’t hurt hir chance of winning. IRV isn’t without advantages, under conditions where its problems aren’t problems. Regrettably those conditions don’t currently obtain.

MDDA:

MDDA’s full truncation-proofness protects its middle-ranked candidates from truncation by eachother’s preferrers. But because its winner when everyone is majority-beaten depends on an approval-count, then of course it doesn’t give anti-burial protection among middle-ranked candidates.

But the important protection is the protection of your ranked & approved candidates against your unranked & unapproved.  And your protection of top against middle ranked. Protecting middle against eachother is less important.

MDDAsc:

Like MDDA, but without the full truncation-proofness.   …and therefore even less protection of middle-ranked candidates against eachother. Of course the last paragraph in the MDDA section above applies to MDDAsc too.

You’re really better off with MDDA than with MDDAsc, unless people reject MDDA because of (the operationally-unimportant cosmetic-criterion) Mono-Add-Plump.

MMPOsc:

Like MDDAsc, it lacks full truncation-proofness.

Burial or truncation among your middle-ranked candidates is succeed-able, but uncertain. Failure is penalized, and so there’s some deterrence. That burial deterrence applies everywhere, but, as mentioned above, it won’t deter someone to whom the MCW is in their strong bottom-set.

So, with some ability to foil and even maybe deter burial & truncation, MMPOsc has an advantage of better intra-middle protection than that of MDDAsc.

…at the cost of not doing as well in the other protections described above.

Overall, I suggest that MDDA & MDDAsc (but especially MDDA) do better than the MMPO methods.

IC,MMPO:

Like MMPOsc, but has full truncation-proofness (as does MDDA).

Also meets a form of Condorcet Criterion, always electing a candidate who pair-beats each one of the others, if there is such a candidate.

MDDA & MDDAsc only notice majority pairwise-defeats. MMPOsc, because it looks only at pairwise-opposition, without regard to which candidate, if any, pairbeats the other, doesn’t reliably elect the candidate who pairbeats everyone.

(With all of these 4 methods, other than IC,MMPO, indifference can defeat a candidate who pairbeats everyone else),

But IC,MMPO gains that property at the cost of a wordier definition. And, as mentioned, MDDA’s protections, overall, seem better.

There could be an IC,MDDA, & an IC,MDDAsc, but it hasn’t been proposed. Presumably it would share IC,MMPO’s ability to always elect someone who pairbeats everyone, but with MDDA’s better protections overall.

It might be the ideal best of these MDDA & MMPO methods, at the cost of a wordier definition than MDDA & MDDAsc.

MAPW & MAPW2:

Again, to the extent that they use Approval instead of pairwise-count, these methods don’t do as much protection among approved candidates, because that isn’t their specialty.

MAM:

MAM has full truncation-proofness, but, like MDDA, it doesn’t have burial protection among its ranked candidates. But that’s mitigated by the fact that unsuccessful burial is penalized, and thereby deterred.   …unless the preferred-to-everyone candidate being buried against is in the buriers’ strong bottom-set, in which case there’s little or no deterrence.

But that combination of full truncation-proofness, and burial deterrence makes MAM strategically strong.  But its chicken-dilemma and its FBC failure disqualify it from being counted among the best ranking-methods.