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

[Chris Benham]

> 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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