[EM] The rationale under the "winning votes" defeat strength measure
Chris Benham
cbenhamau at yahoo.com.au
Sat Jun 28 17:46:16 PDT 2025
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 <http://votingmethods.net>
>> >
>> > ----
>> > Election-Methods mailing list - see
>> https://electorama.com/em for list info
>>
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