[EM] Re: completing Condorcet using ratings information

[Chris Benham]

  James G-A,
I   have  given some more thought on the problem of   how best to 
complete Condorcet by using  high-resolution ratings ballots,
and  I've now concluded that the best is  the  method  I  defined in my 
last post  (Tues.Jun.22):

> "Approval Margins":
> High-resolution ratings ballots. Inferring ranking from rating, 
> eliminate the non-members of the Schwartz-set.
> Of the remaining candidates, each ballot approves those candidates 
> rated above average (and half-approves those rated precisely average).
> Use the (inferred) rankings to determine the results of  the pairwise 
> comparisons between the remaining candidates.
> Then  measure the  "defeat strengths" by the differences in the 
> candidates' approval scores. On that basis pick the  Ranked Pairs 
> winner. 

This could perhaps be "Automated-Approval Margins" (AAM)   to distingush 
it from  "(Plain) Approval Margins" which would use
ranked-ballots with an approval cutoff.  Non-members of the Schwartz-set 
are eliminated. Then those ballots that either approve
all or none of the remaining candidates have their approval cutoffs 
moved the minimum distance so that this is no longer the case.
(In other words, those that showed approval for all the remaining 
candidates now approve all except the candidate/s they rank
 last; and those that showed approval for none of the remaining 
candidates now approve the candidate/s they rank first.)
Then, as above, use the rankings to determine the results of  the 
pairwise comparisons (what you would call the "directions of  the
defeats") and the margins between tha candidates' approval scores to 
gauge the "strenghths" of  the defeats, and on this basis use
RP to pick the winner.
(BTW, I don't consider this category of  method, those that use 
ranked-ballots with an approval cutoff, to be nearly as important or
interesting as those that use high-resolution ratings ballots (with many 
more slots than candidates) or those that use plain ranked-ballots.)

So I  no longer  support Condorcet completed by Compressing Ranks, or 
 Condorcet completed by Approval Elimination.
I  think they are unneccessarily drastic. I scratched the  Approval 
 Elimination method when I discovered  that it is vulnerable to
Pushover strategy.  Take this example (from Adam Tarr)  of  sincere 
preferences:

49:R>C>L
12:C>R>L
12:C>L>R
27:L>C>R

C  is  the sincere CW. If  the R voters Burry  C, then normal  (plain 
ranked-ballot) RP/BP/MM  elect  R.
If  instead ratings are used  to complete Condorcet, we might get this:

49:R100>L1>C0
06:C100>R99>L0
06:C100>R1>L0
06:C100>L1>R0
06:C100>L99>R0
27:L100>C99>R0

All candidates are in the Schwartz set, so each ballot approves the 
candidates  rated above average, giving these approval scores:
R:55       C:51      L:33
Approval Elimination eliminates L, and then C  pairwise beats R, and so 
C wins.  (Approval and  Compressing Ranks, equivalent in
the 3-candidate case, both elect  R.)  Both your  Weighted Pairwise and 
 my  Approval Margins method elect  C.
Suppose the R voters, as well as Bury (offensively order-reverse), 
insincerely raise their rating of  L.

49:R100>L99>C0
06:C100>R99>L0
06:C100>R1>L0
06:C100>L1>R0
06:C100>L99>R0
27:L100>C99>R0

Now the approval scores  are  R55,  C51,  L82.  Approval Elimination now 
eliminates C and elects R.
With our favoured methods, the R voters' strategy backfires and  L wins.

I believe that  AAM meets  Minimal Defense/WDSC.  If  a majority prefer 
 candidate y  to candidate x, then they can simply prevent
the election of x  by placing a sufficiciently large  gap  in their 
ratings somewhere between x and y, which they  can do without any 
misrepresentation of  their sincere rankings.

Adapting a Steve Eppley example:
46:X100>Z1>Y0
10:Y100>X1>Z0
10:Y100>Z99>X0
34:Z100>Y99>X0

Because of  the X voters order-reversing, all the candidates are in the 
Schwartz set. The ballots approve the candidates they rate
above average, to give these  approval-scores:   X46,  Y54,  Z44.
The results of the pairwise comparisons (based purely on the inferred 
rankings) are  X>Z,  Z>Y, Y>X.  Determining the "strengths"
of  these defeats by the approval margins, we get:
X>Z  46-44 = +2
Z>Y  44-54 = -10
Y>X  54-46 = +8
RP locks Y>X  and then X>Z,  to give the final order  Y>X>Z.   Y wins.

If instead the X voters raise Z in their ratings:

46:X100>Z99>Y0
10:Y100>X1>Z0
10:Y100>Z99>X0
34:Z100>Y99>X0

the approval-scores become:  X46,   Y54,   Z90.
X>Z  46-90 = -44
Z>Y  90-54 = +36
Y>X  54-46 = +8
RP locks Z>Y and  then Y>X,  to give the order  Z>Y>X.  Z wins, so the X 
voters' strategising has backfired.

Automated-Approval Margins (AAM), gives voters little or no incentive to 
exaggerate their ratings, and gets away from the nonsense
of  voters wanting to rank differently candidates they rate the same, 
and so doesn't have the clumsy feature of  the voters having to fill in
two ballots. Also it  has been shown that Weighted  Pairwise can elect 
the least approved candidate (in a 3-candidate election), and I
doubt that AAM can do that.

I  hope that you (and others) found this interesting.

Chris Benham