Forest & Chris—

…

With wv Condorcet, defense against burial, as  you said, requires
defensive truncation:

…

If your candidate might be CW, & you don’t want burial to take the win
from hir & give it to someone you don’t approve, then you & others who
agree with you should refuse to rank anyone you don’t approve.

…

But it’s best to not make a strategic demand on voters. Maybe I
defensively truncate but others in my faction don’t. So, because I regard
our current elections as completely dichotomous, I’d rank all of the
Acceptables together in 1st place, & refuse to rank anyone else.  …so
that I’m not helping anyone’s burial of one of my Acceptables under another
of my Acceptables.

…

So I’d be voting Approval-like.  But it would be much better to avoid
depending on others to defensively-truncate.

…

Hence the desirability of autodeterrent methods.

…

I like Forest’s improvement on CTE by making it into SP with takedown,
where the SP is agenda-ordered by Borda or implicit Approval.

…

For one thing, SP, when there are lots of candidates, is autodeterrent
already, & moreso with takedown. it seems to me. SP with (primary &
secondary) takedown could be called SPT, for SP with Takedown.

…

SP & SPT both seem autodeterrent when there are lots of candidates.  …especially
SPT, because there then are likely more than 1 Bus, which greatly decreases
the likelihood of electing anyone other than a Bus.  …& even moreso when
there are still more buses, as is likely in burial with lots of candidates.

…

But when there are only 3 candidates, SP & SPT, or any autodeterrent
method, depends on predicting which of the 3 candidates is the Bus (whom
it’s desired to elect).

…

That’s the core of the problem, it seems to me.

…

Say there are 3 candidates, & all of one candidate’s preferrers  strategically
bury the CW.

…

What can be said about the pairwise defeats & victories, the wins & loses,
of CW, BF & Bus?

…

BF is in the unique position of having only natural defeats & victories.

…

BF has 1 natural victory & 1 natural defeat.

…

CW has two natural defeats & 1 strategic lowering.

…

Bus has two natural defeats & 1 strategic raising.

…

Do 2 natural defeats or victories affect someone’s Borda or implicit
Approval more, or less, than 1strategic raising or lowering?

…

The natural defeats & victories result from a majority of the electorate
ranking someone low or high.  A strategic lowering or raising results
from just one faction strategically raising or lowering someone.

…

Don’t the two natural victorys or defeats, then ,sound stronger than the
one strategtic lowering or raising?

…

…& so, doesn’t that suggest that implicit-Approval & Borda would be
expected to be in the following order?  :

…

CW>BF>Bus

…

I wanted to check that out, for implicit-Approval & for Borda.

…

I started with implicit-Approval.

…

There are 6 ways that 3 candidates can be ordered. We can assume that all
of the BF-preferrers rank BF>Bus>CW, since the ones who prefer CW to Bus
will bury.

…

So that’s 5 kinds of rankings. Write the rankings in a row, & label them
at the top from A to E.

…

Write the inequality, relating the numbers of voters for A thru E, that
must be satisfied in order for Bus to pairwise-beat CW.

…

Write the inequalities that must be satisfied in order for the implicit
approvals to be in the order:

…

CW>BF>Bus.

…

Check for whether any of those inequalities, together, imply a
contradiction.

…

Unless I made an error, they imply several contradictions.

…

Oh well.  Hopefully I’ll have better luck with Borda.

On Fri, Nov 3, 2023 at 2:50 AM Forest Simmons forest.simmons21@gmail.com
wrote:

On Mon, Oct 30, 2023, 6:21 PM C.Benham cbenham@adam.com.au wrote:

My favorite burial proof  method is to elect the nemesis of the nemesis of
the (repeated) Submidway Approval Elimination winner.

Max Approval is 52
Min Approval is 3
Midway is 27.5

So we eliminate C.

Updating approvals we have 52 for A and 48 for B. Midway is 50. B is the
only Sub Midway candidate so A is the SubMidway Approval winner.

The nemesis of A is C, and C has no nemesis because it not pairbeaten. We
respect the Condorcet Criterion and elect C, a very weak CW.

There is no way out of it if we want a Condorcet Criterion Compliant
method.

This reminds me of the practice omanipulative judges in the US warning
jurors to ignore their sacred right of "Jury Nullification" (inherited from
English Common Law).

Those lying (by intimidation) judges are a disgrace to their office ...
and should be stripped of their holy robes ...  imho.

That warning should be mandatory ... as should an honest effort to "fully
inform" a jury of their most sacred rights predatimg even the Magna Carta.

Your example shows the wisdom of electing the ballot CW when there is one.

In fact, all of our burial resistant methods elect the voted CW when one
exists. But when one does not exist we suspiciously suspect that its
absence was most likely  caused by subversion of a CW, since that is by far
the easiest way to create a beat cycle ... whether innocently or with
"malice aforethought."

Let's see what our Sub Midway Approval Elimination burial resistant method
does here:

The respective max and min implicit approval scores are 54 for C and 44
for B. So Midway is 49. Both A and B have Sub Midway Approval leaving C as
the candidate whose nemesis's nemesis we should elect.

C's nemesis is B, and B's nemesis is A.

So A is the winner of our burial resistant method.

How does this work?

Well, Nanson is very good at fingering the Burial Faction Favorite, but
Poor man's Nanson aka SubMidApproval Elimination is even better at electing
the BFF than ordinary Nanson or RP wv, although they are both pretty good
at it as this example shows.

Once you have the likely BFF you can follow the cycle forward one or
backward two candidates to the "bus" under which the buried candidate was
buried.

[Electing this "bus" is what makes the method backfire on the burying
faction ... so electing the strongest bus (their could be a clone set of
them) is the proper aim of a good burial proof method designer ... to
completely deter potential subverters from even flirting with the idea.]

Like I said you can go in either direction  around the cycle to get to a
"bus" from the BFF candidate .... but it's better psychologically to elect
the syrongest wv defeater of the strongest wv defeater of the Nanson winner
.... as opposed to saying "Elect the candidate with the most losing votes
against the Nanson winner" .. they are almost always the same candidate
even when Smith has more than three members.

Like I say, SubMidAolroval Elimination is more reliable than even ordinary
Nanson or RP wv at finding the BFF candidate.  But for the curjous, you can
always do a sincere runoff between the likely strongest Bus (the nemesis of
the nemesis of the likely BFF), and the Smith candidate with the fewest
losing votes against it (i.e.against the bus) the same as the weakest
Smith  weakest Smith candidate to defeat the BFF.

[It is easy to show that a subverted CW always defeats the BFF, which
beats the bus that beat the buried candidate that precipitated the
insincere cycle resulting from the burial.]

In this example the sincere runoff (were that option to be taken) would be
between the Bus A and the Smith candidate B with the fewest losing votes to
it A.

For the sincere winner of this runoff we have to go clear back to the
original scenario. And it turns out that A is the sincere winner of the
runoff, because C, the sincere CW was acting more like a BFF ... so not a
finalist in the runoff!

Right, and now we have a method designed to automatically avoid that kind
of mistake.

Referring to what I was saying about the order of the 3 too-cycle
candidates’ Borda or implicit approval:

I told why I claim that BF is the most likely one to be middle.

So what’s the most likely order for the top & bottom of that ordering?

Well, top & bottom are the most extreme difference in the ordering “. So,
isn’t that difference more likely to not contradict a pairwise defeat?

Together, those two facts suggest that the most likely order for the 3
candidates’ Borda or implicit approval is:

Bus>BF>CW.

On Fri, Nov 3, 2023 at 06:35 Forest Simmons forest.simmons21@gmail.com
wrote:

Forest,

Douglas Woodall in a 1996 article demonstrated that Condorcet is
incompatible with Later-no-Help.

By simple logic we know that any method that fails Later-no-Help must be
vulnerable to Burial strategy.

Therefore your quest to invent a completely burial-proof Condorcet method
is futile.

https://www.sciencedirect.com/science/article/pii/S0166218X9600100X

Theorem 2. (a) Even if truncated preference listings are not allowed,
CONDORCET is incompatible with
PARTICIPATION, MONO-RAISE-RANDOM and MONO-SUB-TOP.

(b) In general, CONDORCET is incompatible with LATER-NO-HELP,
LATER-NO-HARM, MONO-RAISE-DELETE, MONO-SUB-PLUMP and,
in the presence of PLURALITY, MONO-ADD-TOP.

(c) There is no election rule that sutisfies LATER-NO-HELP and
LATER-NO-HARM,
and that also satisfies CONDORCET whenever there are no truncated
preference listings.

Election 3. (1 seat)
3 A>B>C
3 B>C>A
3 C>A>B
2 A>C>B
2 B>A>C
2 C>B>A

Consider Election 3. By symmetry, the result must be a 3-way tie; but, by
the axiom of discrimination, there must be a profile P
arbitrarily close to this (in the proportions of ballots of each type)
that does not yield a tie. Without loss of generality, suppose a is elected
in P.

But c becomes the Condorcet winner, and so must be elected by CONDORCET,
...if all the abc ballots are replaced by a (contrary to LATER-NO-HELP),
or if all the bac ballots are replaced by a (contrary to MONO-RAISE-DELETE
and MONO-SUB-PLUMP), or if all the abc ballots are replaced by acb
(contrary to LATER-NO-HELP and LATER-NO-HARM together).

This proves (a), (c) and three parts of (b).

The three relatively burial-resistant Condorcet methods that meet
mono-raise and don't ask or allow voters to enter an explicit approval
cutoff that I like are
Smith//Approval (ranking),  Margins-Sorted Approval (ranking), and

Smith//Descending Acquiescing Coalitions.

Not allowing voters to rank among unapproved candidates increase the risk
for Burial strategists that they will simply elect their "bus".

By itself DAC (one of Woodall's inventions) meets Later-no-Help and
Participation (both incompatible with Condorcet) but can behave very oddly
and badly in the presence
of one or two weak should-be-irrelevant candidates (which is why I'm wary
of Smith,DAC).

So defining it thus: Voters rank from the top, equal ranking and
truncation is fine.  Eliminate and drop from the ballots all the candidates
not in
the Smith set.

Ballots "acquiesce" to a candidate or set or subset of candidates (a
"coalition") if they vote no other (outside the set or subset) candidate
strictly above any of them.

Number all the possible coalitions according to how may ballots acquiesce
to them. Start with the highest-numbered and disqualify all the candidate
not in it.
Proceed to the next-highest numbered that contains any not-yet
disqualified candidates and disqualify those not in it, and so on until one
candidate is left undisqualified.

46 A
44 B>C (sincere might be B or B>A)
10 C

In this example all candidates are in the Smith set.  The acquiescing
coalitions are  AC 56,  BC 54,  AB 46,  A 46,  B 44,  C 10.

AC disqualifies B,  BC disqualifies A so  C wins.

That would be great for fans of  the Minimal Defense criterion, but would
be sad if the B>C voters are sincere and the C voters are
truncating against B.

Chris B.

On 4/11/2023 12:05 am, Forest Simmons wrote:

On Mon, Oct 30, 2023, 6:21 PM C.Benham cbenham@adam.com.au wrote:

My favorite burial proof  method is to elect the nemesis of the nemesis of
the (repeated) Submidway Approval Elimination winner.

Max Approval is 52
Min Approval is 3
Midway is 27.5

So we eliminate C.

Updating approvals we have 52 for A and 48 for B. Midway is 50. B is the
only Sub Midway candidate so A is the SubMidway Approval winner.

The nemesis of A is C, and C has no nemesis because it not pairbeaten. We
respect the Condorcet Criterion and elect C, a very weak CW.

There is no way out of it if we want a Condorcet Criterion Compliant
method.

This reminds me of the practice omanipulative judges in the US warning
jurors to ignore their sacred right of "Jury Nullification" (inherited from
English Common Law).

Those lying (by intimidation) judges are a disgrace to their office ...
and should be stripped of their holy robes ...  imho.

That warning should be mandatory ... as should an honest effort to "fully
inform" a jury of their most sacred rights predatimg even the Magna Carta.

Your example shows the wisdom of electing the ballot CW when there is one.

In fact, all of our burial resistant methods elect the voted CW when one
exists. But when one does not exist we suspiciously suspect that its
absence was most likely  caused by subversion of a CW, since that is by far
the easiest way to create a beat cycle ... whether innocently or with
"malice aforethought."

Let's see what our Sub Midway Approval Elimination burial resistant method
does here:

The respective max and min implicit approval scores are 54 for C and 44
for B. So Midway is 49. Both A and B have Sub Midway Approval leaving C as
the candidate whose nemesis's nemesis we should elect.

C's nemesis is B, and B's nemesis is A.

So A is the winner of our burial resistant method.

How does this work?

Well, Nanson is very good at fingering the Burial Faction Favorite, but
Poor man's Nanson aka SubMidApproval Elimination is even better at electing
the BFF than ordinary Nanson or RP wv, although they are both pretty good
at it as this example shows.

Once you have the likely BFF you can follow the cycle forward one or
backward two candidates to the "bus" under which the buried candidate was
buried.

[Electing this "bus" is what makes the method backfire on the burying
faction ... so electing the strongest bus (their could be a clone set of
them) is the proper aim of a good burial proof method designer ... to
completely deter potential subverters from even flirting with the idea.]

Like I said you can go in either direction  around the cycle to get to a
"bus" from the BFF candidate .... but it's better psychologically to elect
the syrongest wv defeater of the strongest wv defeater of the Nanson winner
.... as opposed to saying "Elect the candidate with the most losing votes
against the Nanson winner" .. they are almost always the same candidate
even when Smith has more than three members.

Like I say, SubMidAolroval Elimination is more reliable than even ordinary
Nanson or RP wv at finding the BFF candidate.  But for the curjous, you can
always do a sincere runoff between the likely strongest Bus (the nemesis of
the nemesis of the likely BFF), and the Smith candidate with the fewest
losing votes against it (i.e.against the bus) the same as the weakest
Smith  weakest Smith candidate to defeat the BFF.

[It is easy to show that a subverted CW always defeats the BFF, which
beats the bus that beat the buried candidate that precipitated the
insincere cycle resulting from the burial.]

In this example the sincere runoff (were that option to be taken) would be
between the Bus A and the Smith candidate B with the fewest losing votes to
it A.

For the sincere winner of this runoff we have to go clear back to the
original scenario. And it turns out that A is the sincere winner of the
runoff, because C, the sincere CW was acting more like a BFF ... so not a
finalist in the runoff!

Right, and now we have a method designed to automatically avoid that kind
of mistake.

Woodall has been a great EM pioneer .... and has done us a great favor by
making clear some of the limitations of the UD box, as well has showing us
by his Solid Coalitions methods some of the previously unknown
possibilities of working within that box.

I have suggested some of the "pockets of predictability" within the UD
domain that have not been fully exploited for burial resistant methods ...
at the same time coloring outside the UD lines as you have (eg with
approval cutoffs) to surpass some of the UD limitations.

I'm glad that you did not give up In helpless futility .... nut have
pragmatically pressed on.

In general, impossibility proofs are valuable by making clear the
assumptions that entail impossibility ... so we know which lines we have to
color outside of if we want to avoid dead endends.

The three candidate two stage fresh ballot runoff is one non UD tool
tailor made for finishing elections that surpass the boundaries of UD
possibilities.

Sometimes with manipulated ballot sets we can narrow down to two
candidates that form a satisfactory basis for a two candidate fresh ballot
runoff.

But when you push the boundaries further, a two candidate runoff isn't
good enough.

But in the case of a three candidate Landau set ... the right kind of
fresh ballots runoff is adequate for recovering the buried candidate.

I doubt there has ever been (or ever will be) a real public election
ballot set with more than three uncovered candidates.

It is easy to show that when a CW is buried, it will still be a member of
(not only the Smith set), but also a member of Landau.

Smith may have a large clone set of "buses", but (with statistical
certainty) one of these buses  will cover all of the others.

These tools are outside the scope of Woodall ... who would surely
appreciate their usefulness were he still in the loop.

fws

On Fri, Nov 3, 2023, 1:32 PM C.Benham cbenham@adam.com.au wrote:

Forest—

But the incentive for the potential buriers is the matter of whether
burial’s expectation is positive or negative.

You can have a procedure that probably elects the CW, but that doesn’t
figure in the expectation. What matters for the burial’s expectation is the
relative-probabilities of electing Bus vs BF, & their utilities for the
burier.

We can’t affect the utilities of those two outcomes for the potential
buriers. We can only increase the factor by which the probability of
electing a Bus is greater than the probability of electing BF.

Black (with Borda applied to the Smith-set) & SP, it seems to me, are
more likely to elect Bus than BF.

Likewise, it seems to me, so is your suggestion to elect the candidate wit
the largest wv against the candidate who has the most wv against hir in a
defeat. I call that Greatest Win Against Greatest Loser (GWAGL).

But, for the greatest likelihood of electing a Bus in a many-candidate
election, doe anything else equal your suggestion to apply takedown
(primary & secondary) to SP (agenda-ordered by Borda)?

I call that method TSP.

…partly to stand for Takedown-SP…

…& partly because TSP is also the name for the powerful cleaning alkali,
Trisodium Phosphate.

TSP cleans it up !!!

What’s that? We aren’t allowed to use Trisodium Phosphate?

That’s okay, we aren’t allowed to use an adequate voting-system either.

On Sun, Nov 5, 2023 at 01:14 Forest Simmons forest.simmons21@gmail.com
wrote:

Forest,

Douglas Woodall in a 1996 article demonstrated that Condorcet is
incompatible with Later-no-Help.

By simple logic we know that any method that fails Later-no-Help must
be vulnerable to Burial strategy.

Therefore your quest to invent a completely burial-proof Condorcet
method is futile.

https://www.sciencedirect.com/science/article/pii/S0166218X9600100X

The three relatively burial-resistant Condorcet methods that meet
mono-raise and don't ask or allow voters to enter an explicit approval
cutoff that I like are
Smith//Approval (ranking),   Margins-Sorted Approval (ranking), and

Smith//Descending Acquiescing Coalitions.

Not allowing voters to rank among unapproved candidates increase the
risk for Burial strategists that they will simply elect their "bus".

By itself DAC (one of Woodall's inventions) meets Later-no-Help and
Participation (both incompatible with Condorcet) but can behave very
oddly and badly in the presence
of one or two weak should-be-irrelevant candidates (which is why I'm
wary of Smith,DAC).

So defining it thus: Voters rank from the top, equal ranking and
truncation is fine.  Eliminate and drop from the ballots all the
candidates not in
the Smith set.

Ballots "acquiesce" to a candidate or set or subset of candidates (a
"coalition") if they vote no other (outside the set or subset)
candidate strictly above any of them.

Number all the possible coalitions according to how may ballots
acquiesce to them. Start with the highest-numbered and disqualify all
the candidate not in it.
Proceed to the next-highest numbered that contains any not-yet
disqualified candidates and disqualify those not in it, and so on
until one candidate is left undisqualified.

46 A
44 B>C (sincere might be B or B>A)
10 C

In this example all candidates are in the Smith set.  The acquiescing
coalitions are  AC 56,  BC 54,  AB 46,  A 46,  B 44, C 10.

AC disqualifies B,  BC disqualifies A so  C wins.

That would be great for fans of  the Minimal Defense criterion, but
would be sad if the B>C voters are sincere and the C voters are
truncating against B.

Chris B.

On 4/11/2023 12:05 am, Forest Simmons wrote:

If we want still more Burial resistance, and are prepared to put up with
something much less monotonic and more clunky,
I suggest  "Double Defeat, Benham".

*Voters use ranked ballots with an explicit approval cutoff, default
placement just below first place. Above-bottom equal ranking
not allowed (otherwise Pushover strategising becomes easier and less
risky).

Elect the CW if there is one. If not disqualify the Doubly Defeated
candidates. If more than one remains qualified then elect the
qualified candidate that pairwise beats all the others if there is one.
Otherwise proceed with IRV-style eliminations until one
qualified candidate pairwise beats any others remaining, and elect that
candidate.*

(That could probably put more clearly and succinctly).

So this method meets both Double Defeat and Unburiable Mutual Dominant
Third.  I doubt that an acceptable Condorcet method
can get more Burial resistant than that.

Chris B.

On 4/11/2023 7:02 am, C.Benham wrote:

Forest,

Douglas Woodall in a 1996 article demonstrated that Condorcet is
incompatible with Later-no-Help.

By simple logic we know that any method that fails Later-no-Help must be
vulnerable to Burial strategy.

Therefore your quest to invent a completely burial-proof Condorcet method
is futile.

https://www.sciencedirect.com/science/article/pii/S0166218X9600100X

Theorem 2. (a) Even if truncated preference listings are not allowed,
CONDORCET is incompatible with
PARTICIPATION, MONO-RAISE-RANDOM and MONO-SUB-TOP.

(b) In general, CONDORCET is incompatible with LATER-NO-HELP,
LATER-NO-HARM, MONO-RAISE-DELETE, MONO-SUB-PLUMP and,
in the presence of PLURALITY, MONO-ADD-TOP.

(c) There is no election rule that sutisfies LATER-NO-HELP and
LATER-NO-HARM,
and that also satisfies CONDORCET whenever there are no truncated
preference listings.

Election 3. (1 seat)
3 A>B>C
3 B>C>A
3 C>A>B
2 A>C>B
2 B>A>C
2 C>B>A

Consider Election 3. By symmetry, the result must be a 3-way tie; but, by
the axiom of discrimination, there must be a profile P
arbitrarily close to this (in the proportions of ballots of each type)
that does not yield a tie. Without loss of generality, suppose a is elected
in P.

But c becomes the Condorcet winner, and so must be elected by CONDORCET,
...if all the abc ballots are replaced by a (contrary to LATER-NO-HELP),
or if all the bac ballots are replaced by a (contrary to MONO-RAISE-DELETE
and MONO-SUB-PLUMP), or if all the abc ballots are replaced by acb
(contrary to LATER-NO-HELP and LATER-NO-HARM together).

This proves (a), (c) and three parts of (b).

The three relatively burial-resistant Condorcet methods that meet
mono-raise and don't ask or allow voters to enter an explicit approval
cutoff that I like are
Smith//Approval (ranking),  Margins-Sorted Approval (ranking), and

Smith//Descending Acquiescing Coalitions.

Not allowing voters to rank among unapproved candidates increase the risk
for Burial strategists that they will simply elect their "bus".

By itself DAC (one of Woodall's inventions) meets Later-no-Help and
Participation (both incompatible with Condorcet) but can behave very oddly
and badly in the presence
of one or two weak should-be-irrelevant candidates (which is why I'm wary
of Smith,DAC).

So defining it thus: Voters rank from the top, equal ranking and
truncation is fine.  Eliminate and drop from the ballots all the candidates
not in
the Smith set.

Ballots "acquiesce" to a candidate or set or subset of candidates (a
"coalition") if they vote no other (outside the set or subset) candidate
strictly above any of them.

Number all the possible coalitions according to how may ballots acquiesce
to them. Start with the highest-numbered and disqualify all the candidate
not in it.
Proceed to the next-highest numbered that contains any not-yet
disqualified candidates and disqualify those not in it, and so on until one
candidate is left undisqualified.

46 A
44 B>C (sincere might be B or B>A)
10 C

In this example all candidates are in the Smith set.  The acquiescing
coalitions are  AC 56,  BC 54,  AB 46,  A 46,  B 44,  C 10.

AC disqualifies B,  BC disqualifies A so  C wins.

That would be great for fans of  the Minimal Defense criterion, but would
be sad if the B>C voters are sincere and the C voters are
truncating against B.

Chris B.

On 4/11/2023 12:05 am, Forest Simmons wrote:

On Mon, Oct 30, 2023, 6:21 PM C.Benham cbenham@adam.com.au wrote:

My favorite burial proof  method is to elect the nemesis of the nemesis of
the (repeated) Submidway Approval Elimination winner.

Max Approval is 52
Min Approval is 3
Midway is 27.5

So we eliminate C.

Updating approvals we have 52 for A and 48 for B. Midway is 50. B is the
only Sub Midway candidate so A is the SubMidway Approval winner.

The nemesis of A is C, and C has no nemesis because it not pairbeaten. We
respect the Condorcet Criterion and elect C, a very weak CW.

There is no way out of it if we want a Condorcet Criterion Compliant
method.

This reminds me of the practice omanipulative judges in the US warning
jurors to ignore their sacred right of "Jury Nullification" (inherited from
English Common Law).

Those lying (by intimidation) judges are a disgrace to their office ...
and should be stripped of their holy robes ...  imho.

That warning should be mandatory ... as should an honest effort to "fully
inform" a jury of their most sacred rights predatimg even the Magna Carta.

Your example shows the wisdom of electing the ballot CW when there is one.

In fact, all of our burial resistant methods elect the voted CW when one
exists. But when one does not exist we suspiciously suspect that its
absence was most likely  caused by subversion of a CW, since that is by far
the easiest way to create a beat cycle ... whether innocently or with
"malice aforethought."

Let's see what our Sub Midway Approval Elimination burial resistant method
does here:

The respective max and min implicit approval scores are 54 for C and 44
for B. So Midway is 49. Both A and B have Sub Midway Approval leaving C as
the candidate whose nemesis's nemesis we should elect.

C's nemesis is B, and B's nemesis is A.

So A is the winner of our burial resistant method.

How does this work?

Well, Nanson is very good at fingering the Burial Faction Favorite, but
Poor man's Nanson aka SubMidApproval Elimination is even better at electing
the BFF than ordinary Nanson or RP wv, although they are both pretty good
at it as this example shows.

Once you have the likely BFF you can follow the cycle forward one or
backward two candidates to the "bus" under which the buried candidate was
buried.

[Electing this "bus" is what makes the method backfire on the burying
faction ... so electing the strongest bus (their could be a clone set of
them) is the proper aim of a good burial proof method designer ... to
completely deter potential subverters from even flirting with the idea.]

Like I said you can go in either direction  around the cycle to get to a
"bus" from the BFF candidate .... but it's better psychologically to elect
the syrongest wv defeater of the strongest wv defeater of the Nanson winner
.... as opposed to saying "Elect the candidate with the most losing votes
against the Nanson winner" .. they are almost always the same candidate
even when Smith has more than three members.

Like I say, SubMidAolroval Elimination is more reliable than even ordinary
Nanson or RP wv at finding the BFF candidate.  But for the curjous, you can
always do a sincere runoff between the likely strongest Bus (the nemesis of
the nemesis of the likely BFF), and the Smith candidate with the fewest
losing votes against it (i.e.against the bus) the same as the weakest
Smith  weakest Smith candidate to defeat the BFF.

[It is easy to show that a subverted CW always defeats the BFF, which
beats the bus that beat the buried candidate that precipitated the
insincere cycle resulting from the burial.]

In this example the sincere runoff (were that option to be taken) would be
between the Bus A and the Smith candidate B with the fewest losing votes to
it A.

For the sincere winner of this runoff we have to go clear back to the
original scenario. And it turns out that A is the sincere winner of the
runoff, because C, the sincere CW was acting more like a BFF ... so not a
finalist in the runoff!

Right, and now we have a method designed to automatically avoid that kind
of mistake.