[EM] Losing Votes (equal-ranking whole)

Sun May 26 08:34:41 PDT 2019

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: A44: B>C10: C
Is the same as:
46: A>B=C44: B>C>A10: 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
Winning Votes or Margins.

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.  
Losing Votes(erw) elects A.

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 
candidate and Losing Votes (erw)
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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