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<div class="moz-cite-prefix">In my last EM post I included
Smith//DAC in a list of Condorcet methods "that meets mono-raise".<br>
<br>
A knowledgeable correspondent has cast doubt on this claim, and I
admit that I can't prove that it does.<br>
<br>
But I am fairly sure that any failure example needs there to me
more than three candidates in the top cycle,<br>
and if I'm right about that then I'm not concerned enough to
scratch it (at least) as a quite burial-resistant curiosity<br>
that is far less absurd than "elect the member of the Smith set
that is voted the least desirable".<br>
<br>
However it may be too difficult to sell and explain for it to be a
practical proposal for elections to public office.<br>
<br>
So leaving that method idea aside for the time being, I have
become more convinced that Condorcet methods should<br>
(if they are to compete convincingly with IRV/RCV) use ballots
that allow voters to enter an explicit approval cutoff, with the <br>
default placement being just below the candidate/s ranked below no
others.<br>
<br>
My favourite in this category is Margins-Sorted Approval. It is
the most elegant, as monotonic as any Condorcet method can<br>
be, and its explanation and counting has no need of any concept of
cycles or "Smith sets".<br>
<br>
And like all acceptable methods that use that type of ballot (like
Smith//Approval) it meets "Double Defeat" ( which says that<br>
a candidate that is pairwise beaten by a more approved candidate
can't win).<br>
<br>
This means that if the voters who prefer a purported sincere CW to
a likely Burier's favourite wish to protect the purported <br>
sincere CW from that burial, they can do so (if the purported
sincere CW really does pairwise beat the likely BF) by simply
approving <br>
the purported sincere CW and not approving the likely BF.<br>
<br>
No need for anyone to truncate or insincerely equal-rank.<br>
<br>
If we want still more Burial resistance, and are prepared to put
up with something much less monotonic and more clunky, <br>
I suggest "Double Defeat, Benham".<br>
</div>
<div class="moz-cite-prefix"><br>
*Voters use ranked ballots with an explicit approval cutoff,
default placement just below first place. Above-bottom equal
ranking<br>
not allowed (otherwise Pushover strategising becomes easier and
less risky).<br>
<br>
Elect the CW if there is one. If not disqualify the Doubly
Defeated candidates. If more than one remains qualified then elect
the<br>
qualified candidate that pairwise beats all the others if there is
one. Otherwise proceed with IRV-style eliminations until one<br>
qualified candidate pairwise beats any others remaining, and elect
that candidate.*<br>
<br>
(That could probably put more clearly and succinctly).<br>
<br>
So this method meets both Double Defeat and Unburiable Mutual
Dominant Third. I doubt that an acceptable Condorcet method<br>
can get more Burial resistant than that.<br>
<br>
Chris B.<br>
<br>
<br>
<br>
On 4/11/2023 7:02 am, C.Benham wrote:<br>
</div>
<blockquote type="cite"
cite="mid:3c9f74df-e1ee-489a-8585-eca3d2867583@adam.com.au">
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<p>Forest,<br>
<br>
Douglas Woodall in a 1996 article demonstrated that Condorcet is
incompatible with Later-no-Help.<br>
<br>
By simple logic we know that any method that fails Later-no-Help
must be vulnerable to Burial strategy.<br>
<br>
Therefore your quest to invent a completely burial-proof
Condorcet method is futile.<br>
<br>
<a class="moz-txt-link-freetext"
href="https://www.sciencedirect.com/science/article/pii/S0166218X9600100X"
moz-do-not-send="true">https://www.sciencedirect.com/science/article/pii/S0166218X9600100X</a><br>
<br>
</p>
<blockquote type="cite">Theorem 2. (a) Even if truncated
preference listings are not allowed, CONDORCET is incompatible
with <br>
PARTICIPATION, MONO-RAISE-RANDOM and MONO-SUB-TOP. <br>
<br>
(b) In general, CONDORCET is incompatible with LATER-NO-HELP,
LATER-NO-HARM, MONO-RAISE-DELETE, MONO-SUB-PLUMP and,<br>
in the presence of PLURALITY, MONO-ADD-TOP. <br>
<br>
(c) There is no election rule that sutisfies LATER-NO-HELP and
LATER-NO-HARM,<br>
and that also satisfies CONDORCET whenever there are no
truncated preference listings. <br>
<br>
Election 3. (1 seat)<br>
3 A>B>C<br>
3 B>C>A<br>
3 C>A>B<br>
2 A>C>B<br>
2 B>A>C<br>
2 C>B>A<br>
<br>
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 <br>
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. <br>
<br>
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), <br>
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 <br>
(contrary to LATER-NO-HELP and LATER-NO-HARM together).<br>
<br>
This proves (a), (c) and three parts of (b). </blockquote>
<br>
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<br>
Smith//Approval (ranking), Margins-Sorted Approval (ranking),
and<br>
<br>
Smith//Descending Acquiescing Coalitions.<br>
<br>
Not allowing voters to rank among unapproved candidates increase
the risk for Burial strategists that they will simply elect their
"bus". <br>
<br>
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 <br>
of one or two weak should-be-irrelevant candidates (which is why
I'm wary of Smith,DAC).<br>
<br>
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<br>
the Smith set. <br>
<br>
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.<br>
<br>
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.<br>
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.<br>
<br>
46 A<br>
44 B>C (sincere might be B or B>A)<br>
10 C<br>
<br>
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.<br>
<br>
AC disqualifies B, BC disqualifies A so C wins.<br>
<br>
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<br>
truncating against B.<br>
<br>
Chris B.<br>
<br>
<br>
<div class="moz-cite-prefix">On 4/11/2023 12:05 am, Forest Simmons
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CANUDvfqmnTzk5gMbgYnqCQC_5no1TnspjNYVGF3oOmV7XgYnpw@mail.gmail.com">
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<div dir="auto">
<div><br>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Mon, Oct 30, 2023,
6:21 PM C.Benham <<a
href="mailto:cbenham@adam.com.au"
moz-do-not-send="true" class="moz-txt-link-freetext">cbenham@adam.com.au</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote"
style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<p><br>
Why do we support the Condorcet criterion? For me
there are three reasons:<br>
<br>
(1) Failure to elect a voted CW can give the voters
who voted the CW over the actual winner<br>
a potentially very strong, difficult (if not
impossible ) to answer complaint.<br>
<br>
And those voters could be more than half the total.<br>
<br>
(2) Always electing a voted CW is (among methods
that fail Favorite Betrayal) is the best way to
minimise<br>
Compromise incentive.<br>
<br>
(3) Limited to the information we can glean for pure
ranked ballots (especially if we decide to only
refer<br>
to the pairwise matrix), the voted CW is the most
likely utility maximiser.<br>
<br>
If there is no voted CW , then the winner should
come from the Smith set. Condorcet is just the
logical<br>
consequence of Smith and Clone Independence
(specifically Clone-Winner).<br>
<br>
Some methods are able to meet Condorcet but not
Smith, but hopefully they get something in return.<br>
(For example I think Min Max Margins gets
Mono-add-Top and maybe something else).<br>
<br>
So coming to the question of which individual member
of the Smith set should we elect, I don't see that a<br>
supposed, guessed-at "sincere CW" has an especially
strong claim, certainly nothing compared to an
actual<br>
voted CW.<br>
<br>
Suppose sincere looks like:<br>
<br>
49 A>>>C>B<br>
48 B>>>C>A<br>
03 C>A>>>B<br>
</p>
</div>
</blockquote>
</div>
</div>
<div dir="auto">My favorite burial proof method is to elect
the nemesis of the nemesis of the (repeated) Submidway
Approval Elimination winner.</div>
<div dir="auto"><br>
</div>
<div dir="auto">Max Approval is 52</div>
<div dir="auto">Min Approval is 3</div>
<div dir="auto">Midway is 27.5</div>
<div dir="auto"><br>
</div>
<div dir="auto">So we eliminate C.</div>
<div dir="auto"><br>
</div>
<div dir="auto">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.</div>
<div dir="auto"><br>
</div>
<div dir="auto">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.</div>
<div dir="auto"><br>
</div>
<div dir="auto">There is no way out of it if we want a
Condorcet Criterion Compliant method.</div>
<div dir="auto"><br>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">
<div class="gmail_quote">
<blockquote class="gmail_quote"
style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<p> <br>
Suppose that all voters get about the same utility
from electing their favourites. In that case A is
the big utility<br>
maximiser.<br>
<br>
Now suppose that this is say the first post-FPP
election, and the voters are all exhorted to express
their full<br>
rankings, no matter how weak or uncertain some of
their preferences may be, because we don't want
anything <br>
that looks like the (shudder) "minority rule" we had
under FPP.<br>
</p>
</div>
</blockquote>
</div>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">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).</div>
<div dir="auto"><br>
</div>
<div dir="auto">Those lying (by intimidation) judges are a
disgrace to their office ... and should be stripped of their
holy robes ... imho.</div>
<div dir="auto">
<div class="gmail_quote">
<blockquote class="gmail_quote"
style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<p> <br>
So they vote:<br>
<br>
49 A>C<br>
48 B>C<br>
03 C>A<br>
<br>
C is the voted CW. For some pro-Condorcet zealots,
this is ideal. No sincere preferences were reversed
or <br>
"concealed", resulting in the election of the
"sincere CW".<br>
<br>
(In passing I note that in most places if the
non-Condorcet method IRV/RCV were used, A would be
uncontroversially<br>
elected probably without anyone even noticing that C
is the CW.)<br>
<br>
Backing up a bit, suppose that instead of the voters
being exhorted to fully rank no-matter-what, they
are given the<br>
message "this election is for a serious powerful
office, so we don't want anything like GIGO
("garbage in, garbage out")<br>
so if some of your preferences are weak or uncertain
it is quite ok to keep them to yourself via
truncation or equal-ranking."<br>
</p>
</div>
</blockquote>
</div>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">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.</div>
<div dir="auto">
<div class="gmail_quote">
<blockquote class="gmail_quote"
style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<p> <br>
So they vote:<br>
<br>
49 A<br>
48 B<br>
03 C>A<br>
<br>
Now the voted CW is A. Should anyone be
seriously concerned that, due to so many voters
truncating, that some other<br>
candidate might actually be the "sincere CW"?<br>
</p>
</div>
</blockquote>
</div>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">Your example shows the wisdom of electing the
ballot CW when there is one.</div>
<div dir="auto"><br>
</div>
<div dir="auto">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."</div>
<div dir="auto">
<div class="gmail_quote">
<blockquote class="gmail_quote"
style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<p> <br>
For me, if voters have the freedom to fully rank but
for whatever reason choose to truncate (and/or
equal-rank, assuming that<br>
is allowed) a lot of that is fine and the voting
method should prefer not to know about weak and
uncertain preferences.<br>
<br>
The type of insincere voting that most concerns me
is that which produces outrageous failure of
Later-no-Help, achieving by order-reversal<br>
Burial what could not have been done by simple
truncation.<br>
<br>
46 A<br>
44 B>C (sincere is B or B>A)<br>
10 C<br>
</p>
</div>
</blockquote>
</div>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">Let's see what our Sub Midway Approval
Elimination burial resistant method does here:</div>
<div dir="auto"><br>
</div>
<div dir="auto">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.</div>
<div dir="auto"><br>
</div>
<div dir="auto">C's nemesis is B, and B's nemesis is A.</div>
<div dir="auto"><br>
</div>
<div dir="auto">So A is the winner of our burial resistant
method.</div>
<div dir="auto"><br>
</div>
<div dir="auto">How does this work?</div>
<div dir="auto"><br>
</div>
<div dir="auto">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.</div>
<div dir="auto"><br>
</div>
<div dir="auto">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. </div>
<div dir="auto"><br>
</div>
<div dir="auto">[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.]</div>
<div dir="auto"><br>
</div>
<div dir="auto">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.</div>
<div dir="auto"><br>
</div>
<div dir="auto">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.</div>
<div dir="auto"><br>
</div>
<div dir="auto">[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.]</div>
<div dir="auto"><br>
</div>
<div dir="auto">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.</div>
<div dir="auto"><br>
</div>
<div dir="auto">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!</div>
<div dir="auto">
<div class="gmail_quote">
<blockquote class="gmail_quote"
style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<p> <br>
Electing B here is completely unacceptable. </p>
</div>
</blockquote>
</div>
</div>
<div dir="auto"><br>
</div>
<div dir="auto">Right, and now we have a method designed to
automatically avoid that kind of mistake.</div>
<div dir="auto">
<div class="gmail_quote">
<blockquote class="gmail_quote"
style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<p>Regardless of whether or not the B>C voters are
sincere, there isn't any case that B has a stronger<br>
claim than A.<br>
<br>
I don't like (but it can sometimes be justified) a
larger faction being stung by a successful
truncation Defection strategy of a smaller one, but
apart<br>
from that I consider a lot of truncation to be
normal, natural and mostly desirable.<br>
<br>
More later.</p>
<p>Chris Benham<br>
<br>
<br>
<br>
<br>
<br>
</p>
<blockquote type="cite"><b
style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial">Forest
Simmons</b><span
style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial;display:inline!important;float:none"><span> </span></span><a
href="mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Benefit%20of%20a%20doubt%20runoff%20challenge&In-Reply-To=%3CCANUDvfru_xs%2BEE6kd7Xbb4p%2Bsh3Zijqy-yCmBwNPOdwLP1emgQ%40mail.gmail.com%3E"
title="[EM] Benefit of a doubt runoff challenge"
style="font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal"
target="_blank" rel="noreferrer"
moz-do-not-send="true">forest.simmons21 at
gmail.com</a><br
style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial">
<i
style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial">Sun
Oct 29 21:30:58 PDT 2023</i>
<hr
style="color:rgb(0,0,0);font-family:"Times New Roman";font-size:medium;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;white-space:normal;text-decoration-style:initial;text-decoration-color:initial">
<pre
style="white-space:pre-wrap;color:rgb(0,0,0);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial">Are the beatcycles that sometimes arise from expressed ballot preferences
... are these cycles more likely to arise from occasional inevitable
inconsistencies inherent in sincerely voted ballots? ... or from ballots
that reflect exaggerated preferences from attempts to improve the election
outcome over the one likely to result from honest, unexagerated ballots (?)
Should Condorcet methods be designed on the assumption that most ballot
cycles are sincere? .... or on the assumption that most are the result of
insincere ballots (?)
Some people think that the question is irrelevant ... that no matter the
answer, the best result will be obtained by assuming the sincerity of the
voted ballots. Others think healthy skepticism is necessary for optimal
results. What do you think?</pre>
</blockquote>
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</div>
</blockquote>
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</blockquote>
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