[EM] MinMax Losing Votes (equal-ranking whole) Margins

[C.Benham]

Robert,

I'll first explain normal Margins.  Say A pairwise defeats B and we want 
to weigh that defeat so as to
rank it among other pairwise defeats. In normal Margins we use the 
absolute (as distinct from relative) difference
between the number of ballots on which A is voted above B and the number 
of ballots on which B is voted above A.

MMLV(erw)M  determines which candidates pairwise beat which in the same 
way, but has a different way of
weighing/ranking the pairwise defeats.

MMLV(erw)M gives each candidate a MMLV(erw)  score.  When it wants to 
weigh A's pairwise defeat of B it uses
the absolute difference between A's MMLV(erw) score and B' MMLV(erw) 
score.  (This number may be negative).
The lower (or most negative) that number, the weaker the defeat.

The candidates' MMLV(erw) scores are derived from a pairwise matrix, in 
which any candidate X's score against any
candidate Y is the (number of ballots on which X is voted above Y) plus 
(the number of ballots on which X and Y are
voted explicitly equal and above equal- bottom).

Each individual candidate  MMLV(erw) score is, among its scores in this 
pairwise matrix, the lowest one it has against
a candidate that pairwise defeats it.

An example:

46 A>C
10 B>A
10 B>C
34 B=C

Normal Margins just sees  A>C 56-44 (m12),    C>B 46-20 (m26),    B>A 
54-46 (m8)  and so considers that A's pairwise defeat
is the weakest and so elects A.   (With 3 candidates Schulze, Ranked 
Pairs and River are all equivalent to MinMax).

MMLV(erw)M  differs by counting the 34 C=B ballots in the C-B part of 
the pairwise matrix as giving a whole vote to each, so
we have   A>C 56-44,   C>B 80-54,   B>A 54-46.

Then each candidate is given an individual MMLV(erw) score.    B's is 54 
because that is B's pairwise lowest (in this case only)
pairwise score against a candidate that defeats B (C).   A's  is 46 and 
  C's is 44.

Then to use these scores in weighing the pairwise defeats:
A>C  46-44(m2),      C>B 44-54 (m-10),    B>A 54-46 (m8).

B's  "defeat-strength number", negative 10, is the lowest so we judge 
B's defeat to be the weakest and so elect B.

If we are using the Score Margins Sort algorithm, we initially rank the 
candidates according to their scores, from highest to lowest:

B54 > A46 > C44 and then on observing that no candidate pairwise beats 
whichever candidate (if any) is next-highest in this ranking, confirms
this ranking and elects B.

I hope that's now clear. Thanks for taking an interest.

Chris Benham



On 6/18/2014 11:42 AM, robert bristow-johnson wrote:
> On 6/11/14 1:36 PM, C.Benham wrote:
>>
>> On 25 April 2014  I first suggested the  'MinMax Losing Votes
>> (equal-ranking whole) Margins' Condorcet method:
>>
>>> *Voters rank from the top however many candidates they wish Truncation
>>> and equal-ranking is allowed.
>>>
>>> A pairwise matrix is created, giving normal gross scores except that
>>> ballots that explicitly equal rank (not truncate) any two
>>> candidates X and Y give a whole vote to each in that pairwise contest.
>>>
>>> Using this information, give each alternative a score that equals the
>>> smallest number of votes it received in a pairwise loss.
>>>
>>> Henceforth we are only concerned with the direction of the pairwise
>>> defeats and these individual candidate scores.
>>>
>>> Use the Schulze algorithm, weighing each pairwise "defeat" by the
>>> absolute margin of difference between the two candidates'
>>> scores.  (Or use Ranked Pairs or River in the same way if you prefer).
>>>
>>> Or use the candidate scores for the Margins Sort algorithm.*
>>
>> I've given some more thought on how to handle some types of ties.
>>
>> *If we are using the  Score Margins Sort  algorithm and there is an
>> exact tie in the
>> sizes of the margins between pairs of pairwise out-of-order adjacent
>> candidates,
>> make the flip that results in the lowest-ordered candidate being
>> further lowered.
>>
>> If we are using a method that ranks pairwise defeats, and more than
>> one of the pairwise
>> results has the exact same MMLV(erw) score margin, then rank them
>> from strongest to
>> weakest according to the loser's score, i.e. among defeats by the
>> same margin those where
>> the loser has a lower score are we deem weaker than those where the
>> loser has a higher score.*
>>
>
> Chris, could you do me a fav?  could you put this into a sorta flow
> graph or pseudocode step-by-step?  i can't quite grok how it's
> different from the existing Schulze or Ranked Pairs or whatever based
> on Margins.  (or is it *not* based on Margins?)
>
> sorry i'm sorta clueless.
>