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

[Toby Pereira]

 The way I've always seen it is that margins makes more sense from a purely philosophical or mathematical point of view. As Grzegorz alluded to, winning votes can lead to weird discontinuities, and having 50-0 as a "smaller" win than 51-49 seems weird, to say the least. I see winning votes as purely practical and pragmatic. It purely exists because of real-life strategic concerns, not because of any philosophical or mathematical ideal. It's clunky, ugly, and mathematically illogical, but seems to deal with certain situations better than margins.
Toby
    On Wednesday 25 June 2025 at 17:25:09 BST, Kevin Venzke via Election-Methods <election-methods at lists.electorama.com> 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

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