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

Fri Jun 27 03:40:54 PDT 2025

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."

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.

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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