Toby Pereira tdp201b at yahoo.co.uk
Sun Jun 2 09:40:39 PDT 2019

```Chris
Yes, that does seem reasonable. For A to positionally dominate B, I take it that means that for every rank position (above last) A has more of that rank or higher than B. A does look the more reasonable winner than B in your scenario.
Toby

To: Toby Pereira <tdp201b at yahoo.co.uk>; "cbenham at adam.com.au" <cbenham at adam.com.au>; "election-methods at lists.electorama.com" <election-methods at lists.electorama.com>
Sent: Saturday, 1 June 2019, 10:28
Subject: Re: [EM] Losing Votes (equal-ranking whole)

Toby,

I didn't coin the Plurality criterion, and I do somewhat prefer your suggested alternative wording. To take account of equal-top ranking being
allowed, I would specify the fractional interpretation of "number of ballots ranking A as the first preference".  The original coiner of the criterion
was operating on the assumption that no  equal-ranking would be allowed (except at the bottom implied by truncation which would be allowed)
and perhaps also that no-one would needlessly mark a candidate strictly bottom when they could just truncate.

I think in part the criterion is tailor-made for voters accustomed to and content with plurality voting, and after some new election on ballots
that allow voters to rank the candidates is used, they want to know why B won while their favourite candidate A was voted (alone) in first place
on more ballots than B was voted above bottom. And I like the criterion because I agree that there can't be a good enough answer.

A standard  (and possible criterion) I like says that if  A  both positionally dominates and pairwise beats B then B can't win. That implies Plurality.
35: A
10: A=B
30: B>C
25: C
Here no ballots vote A or B below equal-top. A has more top (or first) place votes than B so positionally dominates and pairwise beats B.
Do you think B is an acceptable winner?

Chris Benham

On 29/05/2019 10:37 pm, Toby Pereira wrote:

I don't have a definite answer to the question of equally ranked ballots, and to me I suppose it's still an open question exactly what the best way forwards is, even if you make a good argument against margins.
I don't have an example where the plurality criterion bars from winning the candidate that I think should have won. Looking at the definition on the Wikipedia: "If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is given any preference, then A's probability of winning must be no less than B's.", it's more that I would disagree with the terminology "given any preference."
If the definition was "If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is ranked anything other than last or joint last (either explicitly or through implication on a truncated ballot), then A's probability of winning must be no  less than B's." then I'd be less critical of it. I think the way it's worded implies an approval cut-off even if in practice it makes no difference.
Toby

To: Toby Pereira <tdp201b at yahoo.co.uk>; "cbenham at adam.com.au" <cbenham at adam.com.au>; "election-methods at lists.electorama.com" <election-methods at lists.electorama.com>
Sent: Sunday, 26 May 2019, 20:19
Subject: Re: [EM] Losing Votes (equal-ranking whole)

Toby,

If you try to give that calculator a truncated ballot it will just turn it into the sort of ballot you like.

How do you think equally-ranked ballots should be counted in a pairwise comparison?  A half-vote to
each or zero to both?

So you can't actually point to any election example where the Plurality criterion bars from winning the candidate
that you think should have won?
46: A 44: B>C 10: C Returning to this, are you happy with B winning?  And if not, why not?

Chris Benham

On 27/05/2019 3:38 am, Toby Pereira wrote:

By unranked candidates, I meant the ones that had not had any sort of "vote" - the ones not explicitly listed by the voter. If there are three candidates in an election, A, B, and C, I might like A but absolutely hate the others. My vote might simply be:
A
On the other hand, while I might still absolutely hate B and C, I might still hate C more. So my vote  might be:
A>B
But just because I have ranked B on my ballot, this should not be taken as any sort of endorsement of B or a vote "for" B.

My vote could also be:
A>B>C
Does adding C on the end mean that I have in some sense voted for C? I don't think there would actually be  any methods where adding C on the end would have any effect on how the winner is calculated, but the plurality criterion would presumably in theory find it acceptable to do so.
But this is more a philosophical objection to the assumptions implicit in the plurality criterion than an  an objection to the results that a method obeying the criterion would produce in practice. But anyway, I put my thoughts about the plurality criterion a while ago (as did Juho) here: http://election-methods.5485.n7.nabble.com/EM-Fwd-Ordering-defeats-in-Minimax-td34236.html#a34247
But anyway, thank you for the link to the calculator.
Toby

From: Chris Benham <cbenhamau at yahoo.com.au>
To: "tdp201b at yahoo.co.uk" <tdp201b at yahoo.co.uk>; "cbenham at adam.com.au" <cbenham at adam.com.au>; "election-methods at lists.electorama.com" <election-methods at lists.electorama.com>
Sent: Sunday, 26 May 2019, 18:08
Subject: Re: [EM] Losing Votes (equal-ranking whole)

Toby,

You would like this old online ranked-ballot voting calculator:

https://www.cse.wustl.edu/~legrand/rbvote/calc.html

What do you think are the "false premises" that the Plurality criterion is based on??? It was coined with the assumption
that voters could only strictly rank from the top however many candidates they wish, and those not truncated had in
some sense been "voted for". It says that if A has more first-place votes than B has any sort of votes then B can't win.
No explicit mention of "unranked candidates".

(Adapting it to ballots that allow equal-ranking at the top,?? "first  preferences" refers to first-preference score on the
ballots symmetrically completed, at least at the top, ballots).
To sensibly claim that it is a "mistake" for an algorithm to do (or apparently "assume") something, I think you need to
point to something wrong with an actual result of it doing so.

Chris Benham

On 27/05/2019 1:04 am, Toby Pereira wrote:

I think it's a mistake to assume some sort of approval of a ranked  candidate. If it's not explicitly part of a method then you should not infer it. As far as I'm  concerned:
46: A 44: B>C 10: C
Is the same as:
46: A>B=C 44: B>C>A 10: C>A=B
Presented with these ballots, does this change who you think the winner  should be?
This isn't a defence of margins by me or an argument against anything  else in your post, but I think the plurality criterion, by talking about unranked candidates, is  based on false premises.
Toby

On Sat, 25 May 2019 at 15:31, C.Benham <cbenham at adam.com.au> wrote:   There are several Condorcet algorithms that decide the winner by
weighing "defeat strengths" and they
are all equivalent to MinMax when there are no more than 3  candidates.

The ones I have in mind that are equal or very nearly equal in merit are
River, Schulze, Ranked Pairs, Smith//MinMax.
In public political elections they are very very unlikely to give
different winners. River and Smith//MinMax seem to me
to be the easiest to understand and explain and use. The other two are
perhaps a bit more elegant and have their
enthusiastic supporters.

This is to make the case that measuring pairwise defeat strength by  the
number of votes on the losing side with above-bottom
equal-ranking contributing a whole vote to each side (and otherwise as
with normal Winning Votes) is much better than either

The case for Losing Votes(erw) against Margins is that it (in common
with WV) it meets the Plurality criterion and the Non-Drastic
Defense criterion.

The case for Losing Votes(erw) against Winning Votes is that it meets
the Chicken Dilemma criterion and that is much less likely
to fail to elect a positionally dominant uncovered candidate. (I don't
see how it can fail to elect such a candidate in the 3-candidate case.)

For those who think that Margins might be acceptable:

46: A
44: B>C
10: C

A>B 46-44 (margin=2), B>C 44-10 (margin=34), C>A 54-46 (margin=8).

Using Losing Votes (erw) as the measure of defeat strength, the weakest
defeat is the one with the most votes on the losing side.
That is the C>A defeat so MinMax drops that and A wins. Conversely the
strongest defeat is the one with the fewest votes on the
losing side. That is the B>C defeat so River and Ranked Pairs lock
that. The second strongest is the A>B defeat so those methods
also lock that. All but one candidate has been thereby disqualified so B
wins, or we ignore the third pairwise defeat because that
makes a cycle, so give a final order A>B>C and A wins.

To meet both of the Plurality criterion and the Chicken Dilemma
criterion A must win.

Winning Votes elects C, violating Chicken Dilemma (which it has to do to
meet the previously fashionable Minimal Defense criterion).

Margins elects B. This fails the Plurality criterion because A has more
exclusive first-place votes than B has any sort of above-bottom
votes. It is also an egregious and outrageous failure of Later-no-Help
(assuming that if all the ballots just vote for one candidate we
elect the plurality winner).

To anyone who is remotely positionally or strategically minded or has
any common sense and isn't blind to everything except the
Margins pairwise matrix, B is clearly the weakest candidate and a
completely unacceptable winner.

35: A
10: A=B
30: B>C
25: C

A>B 45-40 (erw, "normally" 35-30, margin=5), B>C 40-25 (margin=15),
C>A 55-45 (margin=10).

Voted at least equal-top (or Top Ratings) scores: A45,  B40, C25.
Voted above bottom (or Approval) scores:  A45,  B40, C55

An old Kevin Venzke example. B is neither the most top-rated candidate
or the most approved candidate and is
pairwise-beaten and positionally dominated by A (the most top-rated).

Winning Votes and Margins both elect the clearly weakest candidate, B.

For those who prefer to have a method comply with Minimal Defense (which
says that if on more than half the ballots
C is voted above A and A no higher than equal-bottom then A can't win)
rather than Chicken Dilemma another method
I prefer to WV is Smith//Approval which here elects C.

25: A>B
26: B>C
23: C>A
26: C

C>A 75-25 (margin=50), A>B 48-26 (margin=22), B>C 51-49 (margin=2).

Voted at least equal-top (or Top Ratings) scores: C49,  B26, A25.
Voted above bottom (or Approval) scores:  C75, B51, A48.

C is an overwhelmingly positionally dominant uncovered candidate.
Margins and Losing Votes elect C.
WV and IRV elect B.

Now say we change 4 of the 26 C ballots to A>C, thereby making C a bit
weaker.

25: A>B
26: B>C
23: C>A
22: C
04: A>C

C>A 71-29 (margin=42), A>B 52-26 (margin=26), B>C 51-49 (margin=2).

Voted at least equal-top (or Top Ratings) scores: C45,  B26, A29.
Voted above bottom (or Approval) scores:  C75, B51, A52.

The weakening of C has caused WV and IRV to change from B to C, now
agreeing with LV and Margins.
Assuming the change was from sincere to insincere, those very lucky
and/or very well informed 4 voters
have pulled off a Push-over strategy.

This is a failure of Mono-raise-delete (more obvious if we reverse the
order of the two situations), which
is one of Woodall's mononicity criteria that he says is incompatible
with Condorcet.

Nonetheless in this case C is still the positionally dominant  uncovered
and Margins both still elect C.

Steve Eppley's old example to illustrate (I think his) Non-Drastic
Defense criterion, which says that if
on more than half the ballots B is voted no lower than equal-top and
above A then A can't win.

46: A>C (sincere may be A>B)
10: B>A
10: B>C
34: C=B (the "defenders", sincere may be C>B)

B>A 54-46 (m=8), A>C 56-44 (m=12), C>B (80-54 erw, "normally" 46-20, m=26).

Voted at least equal-top (or Top Ratings) scores: B54,  A46, C34.
Voted above bottom (or Approval) scores:  B54, A56, C90.

B is the only candidate top-rated on more than half the ballots. More
than half the voters voted B
above A and B not lower than equal-top. Margins and Losing Votes
without my recommended
"above-bottom equal-ranking whole" bit elect A, violating the
Non-Drastic Defense criterion.

Losing Votes (erw) and WV elect B.

If anyone has some counter-examples where they think that Winning Votes
does better than
Losing Votes (erw), I'd be interested in seeing them.

Chris Benham

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