[EM] The rationale under the "winning votes" defeat strength measure

[Grzegorz Pierczyński]

Chris,

Once again -- your argumentation is based on the assumption that
preferences 46: A, 44: B>C, 10: C are not honest and the "real" preferences
are 46: A, 44: B, 10: C. I see no reason for a voting rule to presume that,
having only the actual results in hand. But even if we accept this
assumption, it only demonstrates that Margins can violate IIA/Later-no-help
(bad, but unavoidable for Condorcet) if you alter preferences in 44 votes
out of 100 (quite a lot). In the same way you could e.g., presume that the
real preferences are 46: A, 44: B>C, 2: C>B, 8: C and then WV would violate
Later-no-harm with only altering 2 votes out of 100, while Margins would be
resistant to that. In fact, the "stability" of Margins provides us (as a
side effect) that, since it is harder than under WV to alter the result by
changing preferences of a tiny fraction of the voters, it is also harder to
successfully strategize by a tiny fraction of the voters.

In general, I haven't seen a convincing argument that C is a good winner
under honest votes in this election and to me it would be more egregious to
say to B's supporters "the rule elected C instead of B, because it presumed
that your preference C>A was dishonest" than to say to A's supporters: "No
rule could respect all pairwise defeats between A, B and C, and the one
between A and B was the least decisive. If you only cared about having A
elected, then sorry, but 54 voters preferred C. And according to your
preferences B is as good choice as C, so if you preferred C to be elected,
you should have voted A>C, not A".

I don't believe our discussion is decidable so maybe we have reached the
point where we should just "agree to disagree" on that matter.

Grzegorz

niedz., 29 cze 2025, 02:46 użytkownik Chris Benham <cbenhamau at yahoo.com.au>
napisał:

> Grzegorz,
>
> 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.
>
>
> No, not a "completely new election".  Say the first one is the result of a
> completely accurate poll. Say all the voters really want their favourites
> to win and their preferences among their non-favourites are very weak or
> non-existent.
>
> 46  A
> 44  B
> 10  C
>
> Come the actual election the A supporters think "We are the largest
> faction and we know this method fails Later-no-Harm, so we shall (quite
> sincerely) truncate."  The B supporters think "If we (sincerely) truncate
> then we will almost certainly lose to A. We know this method fails
> Later-no-Help, so why don't we rank C in second place and see what
> happens?  This can't do any harm because C is on 10% and so presumably
> can't win, and/or if our favourite B can't win then we don't care who
> wins."  The C voters think "We don't like or care about A or B. We are just
> here to fly the flag for our candidate with a view to maybe being
> competitive in a future election."
>
> So in the actual election we get:
>
> 46  A
> 44  B>C
> 10  C
>
> And Margins elects B. Yes all Condorcet methods fail Later-no-Help, but
> this is an especially egregious and simple example. And it is combined with
> a failure of the Plurality criterion, which says that if A has more
> first-place votes than B has any (above-bottom) votes then B can't win.  I
> like something similar, that says if A both positionally dominates B and
> pairwise beats B, then B can't win. By "positionally dominate" I mean that
> A has more first place votes, more first and second place votes, and so on
> down to more above-bottom votes.)
>
> So forget about C for the time being and just focus on the A>B pairwise
> comparison. To any person who doesn't fetishise the Margins algorithm and
> has some common sense, there is no case for A losing to B.  When the A
> supporters ask you "How did our candidate lose to B?? We understand this is
> some sort of preferential system, but B got no second-place votes and A got
> more first-place votes" you tell them what?   Do you really think that they
> will and should be satisfied with some mumbo-jumbo about B being "closer to
> being the CW"?
>
> You and Juho like to talk about "stability".  Do you really think that (if
> the stakes are high)  that this (social stability) is enhanced by you
> openly shafting the largest faction??
>
> Hopefully I have now got it through your  skull that B is an unacceptable
> winner due to A.    So what about the C>A comparison?  The WV philosophy is
> that if there is no voted CW and enough truncation then it is possible that
> there is a sincere CW due to some sincere preferences that the truncation
> is concealing and so it is important that we elect one of the candidates
> who could be that sincere CW.
>
> C has a  pairwise win over A that can't be undone by filling in some
> truncated  ballots in a way that favours A, so A can't be this
> (hypothetical, imaginary) "sincere CW".  But C's pairwise loss to B could
> go away if the A truncating ballots were filled in (changed) to A>C.
> So WV elects C.
>
> But I am not  on board with this philosophy. If voters choose not to
> express some of their pairwise preferences I don't see how doing anything
> other than simply assuming they don't exist is justified.
>
> It could be that the only insincerity is the C faction truncating against
> B, so B is the sincere CW and electing C is letting that faction get away
> with defecting from the presumed BC coalition.
>
> That is one of the main reasons I like Margins Sorted Approval (explicit).
> If the B<C voters have beating A no-matter-what as a high priority then
> they can approve C.  If on the other hand they were expecting the C
> supporters to return the favour and vote C>B and they want to ensure that
> they can't steal the election from B by defecting then they can approve B
> only.
>
> 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.
>
>
> Possibly, but why do you assume that this approach  is correct?
>
> Chris Benham
>
> On 27/06/2025 8:10 pm, Grzegorz Pierczyński wrote:
>
> Hi Chris,
>
> 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".
>
> "Any close election (Condorcet or not) can be "unstable" in this way."
>
> 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).
>
> "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."
>
> 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.
>
>
>  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.
>
> 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:
> (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).
> (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.
>
> "I look forward to reading someone's argument that electing A in my other
> example is justified."
>
> 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.
>
> Best,
> Grzegorz
>
> czw., 26 cze 2025 o 23:31 Chris Benham <cbenhamau at yahoo.com.au>
> napisał(a):
>
>>
>> 46: A
>> 44: B>C
>> 10: C
>> 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.
>>
>>
>> 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.
>>
>> 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?
>>
>> 46  A
>> 44  B
>> 10  C
>>
>> Let me guess that you agree with me that the answer is A.  Now let's
>> change that a little bit to this:
>>
>> 46  A
>> 44  B>C
>> 10  C
>>
>> 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).
>>
>> 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.
>>
>>
>>  I find this to be a very weak and bizarre argument. Any close election
>> (Condorcet or not) can be "unstable" in this way.
>>
>> Moreover, B has much broader support than C (assuming that A's supporters
>> are truly indifferent between both).
>>
>>
>> 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.
>>
>> In this example the most approved candidate can only be A or C.
>>
>> My favourite Condorcet method is  Margins Sorted Approval (explicit):
>>
>> *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.*
>>
>> (The "implicit" version is the same except that ranking is interpreted as
>> approval.)
>>
>> 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.
>>
>> "Benham" is the simplest and best of the Hare-Condorcet hybrids.
>>
>> *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.*
>>
>> (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.)
>>
>> 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.
>>
>> 46  A>C
>> 17  B
>> 17  B>C
>> 20  B=C
>>
>> Chris Benham
>>
>>
>> On 26/06/2025 9:12 pm, Grzegorz Pierczyński wrote:
>>
>> Hi all,
>>
>> 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".
>>
>> For example, the second example of Chris rather convinces me to support
>> margins and oppose WV, than the other way around. Let's see:
>> 46: A
>> 44: B>C
>> 10: C
>> 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.
>>
>> 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.
>>
>> Best,
>> Grzegorz
>>
>>
>>
>> czw., 26 cze 2025 o 05:57 Chris Benham <cbenhamau at yahoo.com.au>
>> napisał(a):
>>
>>>
>>> There is also the Non-Drastic Defense criterion, which says that if more
>>> than half the voters vote X above Y and  X no lower than equal-top then
>>> Y can't win.
>>>
>>> 46  A>C (maybe sincere is A or A>B)
>>> 17  B
>>> 17  B>C
>>> 20  C=B (maybe sincere is C>B)
>>>
>>> B>A  54-46,   A>C  63-37,  C>B  46-34.
>>>
>>> Here B is above A and no lower than equal-top on more than half the
>>> ballots, but Margins elects A.   Winning Votes elects B.
>>>
>>> Also Margins can fail Later-no-Help especially egregiously and elect the
>>> weakest candidate:
>>>
>>> 46  A
>>> 44  B>C (sincere might be B or B>A)
>>> 10  C
>>>
>>> Margins elects B (failing the Plurality criterion).  How does the B
>>> voters ranking C remotely justify switching the win from A to B??  A
>>> pairwise beats and positionally dominates B, and C is ranked above
>>> bottom on the most number of ballots.  I can't accept any method that
>>> elects B here.  (Or A in the previous example.)
>>>
>>> I have long since decided that resolving Condorcet top cycles by
>>> deciding (on some basis or another) that some pairwise defeats are
>>> "weaker" than others is a dead end. I vastly prefer 3 other Condorcet
>>> methods: Margins Sorted Approval(explicit),  Margins Sorted Approval
>>> (implicit), and "Benham".
>>>
>>> They all resist Burial better than Margins or Winning Votes, and Margins
>>> Sorted Approval is very elegant.
>>>
>>> Chris Benham
>>>
>>> On 26/06/2025 1:50 am, Kevin Venzke via Election-Methods wrote:
>>> > Hi Grzegorz,
>>> >
>>> >> 1. What exactly are the axioms that Condorcet rules with WV satisfy,
>>> but with
>>> >> margins do not? (I'm only aware of the Plurality criterion)
>>> > Very few have been articulated, but:
>>> >
>>> >> 2. I have sometimes read that WV are better to prevent strategic
>>> behavior of
>>> >> the voters (without much details),
>>> > I do use the minimal defense criterion, which represents the notion
>>> that a full
>>> > majority of voters can always get their way if they want to, so it
>>> will reduce
>>> > compromise strategy for the majority if you just give them their way
>>> when you
>>> > know what it is.
>>> >
>>> > To me, WV resolution is an approximation of an ideal. I made a webpage
>>> that
>>> > attempts to show what options are available for electing from a
>>> provided cycle,
>>> > with the aim of avoiding compromise incentive when you can:
>>> >
>>> > https://votingmethods.net/check
>>> >
>>> > This doesn't always favor WV, and sometimes there are no actual
>>> solutions.
>>> >
>>> >> but do you have any idea how to justify WV
>>> >> more "intuitively" or "philosophically", assuming sincere votes?
>>> Margins are
>>> >> very easy to justify. I came up with two possible justifications for
>>> WV here
>>> >> (described below), but I'm not sure how convincing they could be for
>>> the
>>> >> general audience.
>>> > Here I'm not sure. I guess by "sincere votes" you mean that absence of
>>> a
>>> > pairwise preference indicates an expression that two candidates are
>>> equal. Or
>>> > maybe that truncation is not different from explicit equal ranking.
>>> >
>>> >> 3. Don't you think it is "ugly" that the WV measure applied e.g., to
>>> Schulze
>>> >> or RP/MAM requires us to artificially exclude "50% vs. 50%" ties
>>> between
>>> >> candidates from consideration (or equivalently, to mark them as the
>>> weakest)
>>> > That's never occurred to me actually. All non-wins are excluded from
>>> > consideration.
>>> >
>>> >> --- and that a victory "50%+1 vs. 50%-1" is rapidly considered to be
>>> quite
>>> >> strong, stronger than e.g., a "45% vs. 1%" victory (with 54% voters
>>> who rank
>>> >> both candidates equally)? Under margins, ties or close ties are
>>> naturally
>>> >> considered the weakest. How would you refute this argument?
>>> > Ideally by some kind of rephrasing. I don't know if this is possible,
>>> but it
>>> > would be nice if the matter could be presented without making it feel
>>> like the
>>> > defeats themselves have an interest in being respected.
>>> >
>>> > Alternatively, you want to find a explanation where losing votes are
>>> just
>>> > meaningless, because for the practical purposes (the strategic
>>> incentive ones),
>>> > they are. You don't obtain a valid complaint against the method by
>>> losing a
>>> > close race, you can only get one by winning races and losing anyway
>>> because you
>>> > didn't lie.
>>> >
>>> > (In a 51:49 matchup, those on the losing side have no power to lie and
>>> change
>>> > the outcome (we hope), while there is considerable possibility that
>>> those on the
>>> > 51 side *could* lie and win (i.e. if they had not), because they
>>> comprise more
>>> > than half the voters. With 45:1, there are decent odds that those on
>>> the 45%
>>> > side could win by lying; your method could determine this to be sure,
>>> if you
>>> > wanted, before ruling for instance that 45:1 prevails over a win of
>>> 40:39. WV is
>>> > just making a mathematically easy "best guess.")
>>> >
>>> >> Regarding pt. 2, here are my ideas for a high-level intuitive
>>> principle behind
>>> >> WV:
>>> >> (1) "It is much harder (infinitely harder?) to convince a voter to
>>> change his
>>> >> mind from B<A to A>B, than it is to change his mind from A=B to A>B".
>>> Then, in
>>> >> particular, it is more probable that a "45% vs. 1%" victory would
>>> become a
>>> >> "45% vs. 55%" defeat, than that a "51% vs. 49%" victory would become
>>> a defeat.
>>> > That has some familiarity to me. If the winning side has a full
>>> majority then we
>>> > "know" it is right. In fact if you entertain the concept of an overall
>>> "median
>>> > voter" it suggests to us something about what that voter thinks.
>>> >
>>> > Though I understand that you want to suppose that the equalities are
>>> in fact
>>> > sincere.
>>> >
>>> > In that case, if it's 45% A>B, 54% A=B, 1% B>A, my observation would
>>> be that the
>>> > median position is that A and B are equal. The 54% aren't just
>>> abstaining, are
>>> > they? I don't think that's what the assumption of sincerity implies.
>>> >
>>> > Your second idea is kind of suggestive of this actually... You're just
>>> focusing
>>> > more on voters' desire for how the matchup is handled.
>>> >
>>> >> (2) "If a voter votes for A=B, then he is not neutral, but he is
>>> actively
>>> >> voting against treating the resolution of the matchup between A and B
>>> as
>>> >> important". Then, in particular, in the case of a "45% vs. 1%"
>>> victory, we in
>>> >> fact have 45% of voters who consider it important to resolve the
>>> matchup in a
>>> >> particular direction, and 55% of voters who think otherwise. This is
>>> a smaller
>>> >> number than for a "51% vs. 49%" victory.
>>> > I view this possibility of voters having such a sentiment, and acting
>>> on it in
>>> > this way, more as something useful that WV enables. I don't think we
>>> can say
>>> > it's intuitively the case that voters are meaning to do this.
>>> >
>>> > Kevin
>>> > votingmethods.net
>>> >
>>> > ----
>>> > Election-Methods mailing list - see https://electorama.com/em for
>>> list info
>>>
>>
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