<div dir="ltr">Hi Chris,<div><br></div><div>Well, the intuition that "if there is no CW, then the candidate who was (in some sense) the closest to be the CW should win" is a high-level rationale behind a lot of rules (Minimax, Kemeny-Young, Dodgdon, Ranked Pairs, Schulze, etc.) introduced by different people over time. I understand that you don't share this intuition and prefer different methods, but it's quite radical to call it "very weak" and "bizzarre".</div><div><br></div><div>"Any close election (Condorcet or not) can be "unstable" in this way."</div><div><br></div><div>Yes, but for me there is a difference whether the result of the closest (least stable) comparison between A and B decides between the election of A or B (which is natural) or between the election of B or C (which is weird). </div><div><br></div><div>"By what bizarre stretch of the imagination has extra (second place) votes for C strengthened any candidate other than C? The winner should either still be A or change to C."</div><div><br></div><div>You think this is bizarre because of the violation of IIA or the violation of LNH? In both cases this is unavoidable under any Condorcet rule, so I'm a bit surprised by this argument. Since you had to alter a lot of votes to get this effect (and in fact obtain a completely new election), I wouldn't say it is a particularly outrageous failure of these axioms.</div><div><br></div><div>In general, the discussion about "which method is least vulnerable to strategy" is quite arbitrary and hand-wavy for me in the situation where all the methods are vulnerable and there is no single objective measure of this vulnerability. And the arguments based on that don't justify sacrificing the quality of the winner under sincere votes. For your example with 46: A, 44: B>C, 10: C, I really can't convince myself that electing C is justified. The argument that "B's supporters could have a preference of B or B>A instead of B>C, and then B would have lost" is not convincing to me if we only have the actual results of the election and don't know if such an alternative scenario was even seriously considered by B's supporters. Your argument with "Possible Approval Winner" is more convincing, but I have two problems with it: </div><div>(1) A practical one: if you want to use AV as a justification, you additionally need to explain AV to people, convince them that AV is a good method (so that the possibility of being the AV winner is a good justification) and at the same time, convince them that it is a bad method (so that you do not advocate for it but for Condorcet). </div><div>(2) A theoretical one: using AV as a quality measure, requires us to assume that people have objective "approval sets" in mind. I don't believe so, but even if we take this for granted, then it is arbitrary to assume that they are non-empty. It is perfectly possible that some of A's supporters have a weak preference of A>B=C but in fact do not like anyone, and the most approved candidate is B. </div><div><br></div><div>"I look forward to reading someone's argument that electing A in my other example is justified."</div><div><br></div><div>My honest and totally subjective opinion about this example is that the preferences there are quite weird and (if they are sincere) I have little intuition for or against any of these candidates. It's clear to me that B is a better candidate than A, but it's also at least equally clear that C is better than B and A is better than C. And WV would elect B in this example even if you change 17: B>C to 17: B=C, where I would strongly lean towards either A or C.</div><div><br></div><div>Best,</div><div>Grzegorz</div></div><br><div class="gmail_quote gmail_quote_container"><div dir="ltr" class="gmail_attr">czw., 26 cze 2025 o 23:31 Chris Benham <<a href="mailto:cbenhamau@yahoo.com.au">cbenhamau@yahoo.com.au</a>> napisał(a):<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><u></u>
<div>
<p><br>
</p><blockquote type="cite">
<div>46: A</div>
<div>44: B>C</div>
<div>10: C</div>
WV elect C here, while margins elect B. In fact, if the above
preferences are honest, then B is clearly the best candidate,
since he is the closest to be the Condorcet winner. </blockquote>
<br>
I don't see "closest to the Condorcet winner" as being necessarily
especially positive, let alone the compelling consideration. The
actual voted CW has a strong case to be elected and of course must
be in a Condorcet method. But "close to" doesn't mean anything.<br>
<br>
Imagine you are an A supporter, or simply a sane sensible person
(preferably one who has never heard of Condorcet or Margins). Who
do you think should win this election?<br>
<br>
46 A<br>
44 B<br>
10 C<br>
<br>
Let me guess that you agree with me that the answer is A. Now
let's change that a little bit to this:<br>
<br>
46 A<br>
44 B>C <br>
10 C<br>
<br>
By what bizarre stretch of the imagination has extra (second
place) votes for C strengthened any candidate other than C ? The
winner should either still be A (the Hare and Benham winner) or
change to C (the WV and Margins Sorted Approval(implicit) and
Smith//Approval(implicit) winner).<br>
<br>
<blockquote type="cite">Electing A or (especially) C would be
extremely unstable - if just one voter changes his preference
from A to B, the result would switch to B under any Condorcet
rule.</blockquote>
<br>
I find this to be a very weak and bizarre argument. Any close
election (Condorcet or not) can be "unstable" in this way. <br>
<br>
<blockquote type="cite">Moreover, B has much broader support than
C (assuming that A's supporters are truly indifferent between
both).</blockquote>
<br>
Only C is voted above bottom on more than half the ballots. There
was a criterion suggested called something like "Possible Approval
Winner" that said that if the voters all inserted an approval
cutoff in their rankings either only approving those candidates
they vote below no others or all except those they vote below no
others or anywhere in between, then a candidate who can't possibly
be the most approved candidate can't win.<br>
<br>
In this example the most approved candidate can only be A or C.<br>
<br>
My favourite Condorcet method is Margins Sorted Approval
(explicit):<br>
<br>
*Voters rank however many candidates they wish and also indicate
an approval threshold. Initially order the candidates according to
their approval scores. Check the pairwise result of the adjacent
pair of candidates with smallest difference in their approval
scores.(If there is a tie for this then the lowest-ordered pair
among the tied pairs.) If the lower-ordered of the two pairwise
beats the higher-ordered candidate, then those two candidates
change places in the order. Repeat this procedure to the end. The
candidate at the top of the final order is the winner.* <br>
<p></p>
<p>(The "implicit" version is the same except that ranking is
interpreted as approval.)<br>
<br>
In this example, depending on whether or not the B>C voters
approve C, the initial order (based on approval scores) is
either A>B>C or C>A>B. In neither case is any pair
of adjacent candidates out of order pairwise, i.e. in the first
case A pairwise beats B and B pairwise beats C and in the second
case C pairwise beats A and A pairwise beats B. So either way
the initial order is the final order and so the winner is either A
or C.<br>
<br>
"Benham" is the simplest and best of the Hare-Condorcet hybrids.
<br>
<br>
*Voters strictly rank from the top however many candidates they
wish. Before any and each elimination we check for a
pairwise-beats-all candidate among the remaining candidates and
elect the first one we find. Until then we one-at-a-time eliminate
the candidate that is the highest voted remaining candidate on the
smallest number of ballots.*<br>
<br>
(Allowing above-bottom equal ranking makes Push-over strategy
easier. I suggest interpreting ballots that have more than one
candidate at the same rank as having truncated just above that
rank. I have the same opinion about Hare.)<br>
<br>
These methods I prefer to Winning Votes. Margins is beyond the
pale. I look forward to reading someone's argument that electing A
in my other example is justified.<br>
<br>
46 A>C <br>
17 B<br>
17 B>C<br>
20 B=C <br>
<br>
Chris Benham<br>
<br>
<br>
</p>
<div>On 26/06/2025 9:12 pm, Grzegorz
Pierczyński wrote:<br>
</div>
<blockquote type="cite">
<div dir="ltr">Hi all,
<div><br>
</div>
<div>Thanks for all your comments, axioms and explanations! From
what I see, the justification of WV is indeed rather pragmatic
and strategy-oriented, which is quite a problem for me. I
would really prefer to avoid answering the question: "Why did
your rule elect a bad candidate in this election?" by saying
"Well, because you might have been dishonest in some specific
way, and then this candidate wouldn't be so bad". I also agree
with Juho that "in large public real life Condorcet elections
it is very difficult to implement and coordinate successful
malicious strategies".</div>
<div><br>
</div>
<div>For example, the second example of Chris rather convinces
me to support margins and oppose WV, than the other way
around. Let's see:</div>
<div>46: A</div>
<div>44: B>C</div>
<div>10: C</div>
<div>WV elect C here, while margins elect B. In fact, if the
above preferences are honest, then B is clearly the best
candidate, since he is the closest to be the Condorcet winner.
Electing A or (especially) C would be extremely unstable - if
just one voter changes his preference from A to B, the result
would switch to B under any Condorcet rule. Moreover, B has
much broader support than C (assuming that A's supporters are
truly indifferent between both). I really can't find a logical
justification of electing C here if the voters are honest.</div>
<div><br>
</div>
<div>On the other hand, if we assume that voters were strategic
and the honest opinion of the middle voters is B or B>A,
then it means that a massive number of voters colluded to vote
strategically, in a situation where (1) the result of the race
between A and B was unpredictable before the election and B
had real chances to win anyway, (2) a lot of voters had a
fragile preference of either B=A or B=C, and such a "dirty"
operation of B could easily change their minds to
(respectively) A>B and C>B. I just don't see this
happening in practice. I can agree that such a theoretical
possibility is bad, because violating strategyproofness
generally is bad, but there's nothing particularly worrisome
for me here.</div>
<div><br>
</div>
<div>Best,</div>
<div>Grzegorz</div>
<div><br>
</div>
<div><br>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">czw., 26 cze 2025 o
05:57 Chris Benham <<a href="mailto:cbenhamau@yahoo.com.au" target="_blank">cbenhamau@yahoo.com.au</a>>
napisał(a):<br>
</div>
<blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><br>
There is also the Non-Drastic Defense criterion, which says
that if more <br>
than half the voters vote X above Y and X no lower than
equal-top then <br>
Y can't win.<br>
<br>
46 A>C (maybe sincere is A or A>B)<br>
17 B<br>
17 B>C<br>
20 C=B (maybe sincere is C>B)<br>
<br>
B>A 54-46, A>C 63-37, C>B 46-34.<br>
<br>
Here B is above A and no lower than equal-top on more than
half the <br>
ballots, but Margins elects A. Winning Votes elects B.<br>
<br>
Also Margins can fail Later-no-Help especially egregiously and
elect the <br>
weakest candidate:<br>
<br>
46 A<br>
44 B>C (sincere might be B or B>A)<br>
10 C<br>
<br>
Margins elects B (failing the Plurality criterion). How does
the B <br>
voters ranking C remotely justify switching the win from A to
B?? A <br>
pairwise beats and positionally dominates B, and C is ranked
above <br>
bottom on the most number of ballots. I can't accept any
method that <br>
elects B here. (Or A in the previous example.)<br>
<br>
I have long since decided that resolving Condorcet top cycles
by <br>
deciding (on some basis or another) that some pairwise defeats
are <br>
"weaker" than others is a dead end. I vastly prefer 3 other
Condorcet <br>
methods: Margins Sorted Approval(explicit), Margins Sorted
Approval <br>
(implicit), and "Benham".<br>
<br>
They all resist Burial better than Margins or Winning Votes,
and Margins <br>
Sorted Approval is very elegant.<br>
<br>
Chris Benham<br>
<br>
On 26/06/2025 1:50 am, Kevin Venzke via Election-Methods
wrote:<br>
> Hi Grzegorz,<br>
><br>
>> 1. What exactly are the axioms that Condorcet rules
with WV satisfy, but with<br>
>> margins do not? (I'm only aware of the Plurality
criterion)<br>
> Very few have been articulated, but:<br>
><br>
>> 2. I have sometimes read that WV are better to
prevent strategic behavior of<br>
>> the voters (without much details),<br>
> I do use the minimal defense criterion, which represents
the notion that a full<br>
> majority of voters can always get their way if they want
to, so it will reduce<br>
> compromise strategy for the majority if you just give
them their way when you<br>
> know what it is.<br>
><br>
> To me, WV resolution is an approximation of an ideal. I
made a webpage that<br>
> attempts to show what options are available for electing
from a provided cycle,<br>
> with the aim of avoiding compromise incentive when you
can:<br>
><br>
> <a href="https://votingmethods.net/check" rel="noreferrer" target="_blank">https://votingmethods.net/check</a><br>
><br>
> This doesn't always favor WV, and sometimes there are no
actual solutions.<br>
><br>
>> but do you have any idea how to justify WV<br>
>> more "intuitively" or "philosophically", assuming
sincere votes? Margins are<br>
>> very easy to justify. I came up with two possible
justifications for WV here<br>
>> (described below), but I'm not sure how convincing
they could be for the<br>
>> general audience.<br>
> Here I'm not sure. I guess by "sincere votes" you mean
that absence of a<br>
> pairwise preference indicates an expression that two
candidates are equal. Or<br>
> maybe that truncation is not different from explicit
equal ranking.<br>
><br>
>> 3. Don't you think it is "ugly" that the WV measure
applied e.g., to Schulze<br>
>> or RP/MAM requires us to artificially exclude "50%
vs. 50%" ties between<br>
>> candidates from consideration (or equivalently, to
mark them as the weakest)<br>
> That's never occurred to me actually. All non-wins are
excluded from<br>
> consideration.<br>
><br>
>> --- and that a victory "50%+1 vs. 50%-1" is rapidly
considered to be quite<br>
>> strong, stronger than e.g., a "45% vs. 1%" victory
(with 54% voters who rank<br>
>> both candidates equally)? Under margins, ties or
close ties are naturally<br>
>> considered the weakest. How would you refute this
argument?<br>
> Ideally by some kind of rephrasing. I don't know if this
is possible, but it<br>
> would be nice if the matter could be presented without
making it feel like the<br>
> defeats themselves have an interest in being respected.<br>
><br>
> Alternatively, you want to find a explanation where
losing votes are just<br>
> meaningless, because for the practical purposes (the
strategic incentive ones),<br>
> they are. You don't obtain a valid complaint against the
method by losing a<br>
> close race, you can only get one by winning races and
losing anyway because you<br>
> didn't lie.<br>
><br>
> (In a 51:49 matchup, those on the losing side have no
power to lie and change<br>
> the outcome (we hope), while there is considerable
possibility that those on the<br>
> 51 side *could* lie and win (i.e. if they had not),
because they comprise more<br>
> than half the voters. With 45:1, there are decent odds
that those on the 45%<br>
> side could win by lying; your method could determine this
to be sure, if you<br>
> wanted, before ruling for instance that 45:1 prevails
over a win of 40:39. WV is<br>
> just making a mathematically easy "best guess.")<br>
><br>
>> Regarding pt. 2, here are my ideas for a high-level
intuitive principle behind<br>
>> WV:<br>
>> (1) "It is much harder (infinitely harder?) to
convince a voter to change his<br>
>> mind from B<A to A>B, than it is to change his
mind from A=B to A>B". Then, in<br>
>> particular, it is more probable that a "45% vs. 1%"
victory would become a<br>
>> "45% vs. 55%" defeat, than that a "51% vs. 49%"
victory would become a defeat.<br>
> That has some familiarity to me. If the winning side has
a full majority then we<br>
> "know" it is right. In fact if you entertain the concept
of an overall "median<br>
> voter" it suggests to us something about what that voter
thinks.<br>
><br>
> Though I understand that you want to suppose that the
equalities are in fact<br>
> sincere.<br>
><br>
> In that case, if it's 45% A>B, 54% A=B, 1% B>A, my
observation would be that the<br>
> median position is that A and B are equal. The 54% aren't
just abstaining, are<br>
> they? I don't think that's what the assumption of
sincerity implies.<br>
><br>
> Your second idea is kind of suggestive of this
actually... You're just focusing<br>
> more on voters' desire for how the matchup is handled.<br>
><br>
>> (2) "If a voter votes for A=B, then he is not
neutral, but he is actively<br>
>> voting against treating the resolution of the matchup
between A and B as<br>
>> important". Then, in particular, in the case of a
"45% vs. 1%" victory, we in<br>
>> fact have 45% of voters who consider it important to
resolve the matchup in a<br>
>> particular direction, and 55% of voters who think
otherwise. This is a smaller<br>
>> number than for a "51% vs. 49%" victory.<br>
> I view this possibility of voters having such a
sentiment, and acting on it in<br>
> this way, more as something useful that WV enables. I
don't think we can say<br>
> it's intuitively the case that voters are meaning to do
this.<br>
><br>
> Kevin<br>
> <a href="http://votingmethods.net" rel="noreferrer" target="_blank">votingmethods.net</a><br>
><br>
> ----<br>
> Election-Methods mailing list - see <a href="https://electorama.com/em" rel="noreferrer" target="_blank">https://electorama.com/em</a>
for list info<br>
</blockquote>
</div>
</blockquote>
</div>
</blockquote></div>