[EM] Quota Limited Weighted Approval PR

[Chris Benham]

  Participants,
I  have discovered that  QLWA (Quota Limited Weighted Approval), 
appropiately adapted, makes an  excellent  
ranked-ballot PR  method  .

Single-Winner  QLWA:
Voters rank the candidates. Equal preferences and truncation ok.
(1) Symetrically complete the ballots.
(2) Based on these now symetrically completed ballots, give each 
candidate a weight of  1 for each ballot on which it is
ranked in first place. (The total weight of  the candidates will now be 
equal to the total number of  original before-step-1
ballots.  Any candidate with a weight  equal to or greater than half 
 the total weight of all the candidates wins).
(3) Each ballot fully approves the highest-ranked  candidates whose 
combined weight is less than half  the total weight of
all the candidates. Each ballot also fractionally approves the next 
highest-ranked candidate, so that  the combined weight
of  the candidates approved by each ballot  is equal to half  the total 
weight of all the candidates.
(4) The candidate with the highest  approval score wins.

QLWA  Proportional  Representaion:
Voters rank the candidates. Equal preferences and truncation ok.
(1) Symetrically complete the ballots.
(2) Based on these now symetrically completed ballots, give each 
candidate a weight of  1 for each ballot on which it is
ranked in first place. (The total weight of  the candidates will now be 
equal to the total number of  original before-step-1
ballots.)
(3) Each ballot fully approves the highest-ranked  candidates whose 
combined weight is less than one Droop quota. Each
ballot also fractionally approves the next highest-ranked candidate, so 
that  the combined weight of  the candidates
approved by each ballot equals  one Droop quota.
(4) Elect  the candidate with the highest score.
(5) Reduce the value of  the ballots  that approved   the previously 
elected candidate by  a  total value of  one Droop quota,
in proportion to their contribution to that candidate's  approval score.
(6) Based on the ballots as  they are now valued,  give the remaining 
candidates new weights as in  step (2). A remaining
candidate who  was ranked second on a ballot that approved a previous 
winner, inherits first place on those ballots.
(7) Using the new weights  and the newly re-valued ballots, repeat step 
(3)  and  then step (4).
(8) If  there remain seats unfilled, repeat in turn steps (5),(6) and 
(7)  until  all seats are filled.

I  believe this method retains all the important good features of  the 
single-winner version, like  Clone Independence, Mono-raise
(Monotonicity) and  Participation.  It  is less chaotic and capricious 
than other PR  ranked-ballot methods.

I  have been told that in this example  QLWA  agrees with Sequential STV.
400 ballots, 7 candidates, 3 seats,  Droop Quota =  100.
96:A>C1>C2>C3
96:B>C1>C2>C3
96:D>C1>C2>C3
88:E>C1>C2>C3
08:C1>C2>C3
08:C2>C3>C1
08:C3>C1>C2
QLWA  elects  C1, C2, C3.

Normal  STV  simply culls the C candidates for  not having enough first 
preferences, and then  then  elects ABD.
A, B, D, and E's voters all rank each other's favourites as their bottom 
three and so can't put together a Droop quota.
C1 is ranked first or second on 98% of the  ballots, and with 7 
candidates and no mutual Droop quotas,  C1's case is
compelling. I  think the case for picking one or two of  ABD at random 
 is weak.  

This from a "Voting Matters"  (issue 15, June 2002)  article on 
 Sequential STV:

"With 5 candidates for 2 seats, consider the voting pattern

        104 ABCD
        103 BCDA
        102 CDBA
        101 DBCA
          3 EABCD
          3 EBCDA
          3 ECDBA
          3 EDCBA
            

Plain STV elects BC. Sequential STV chooses BC as probables, then tests 
BCD, BCE and BCA in that order. BC win each time and are elected.

Suppose, however, that the voters for A, B, C and D had all put in E as 
second preference to give (the example used in reference 1).

        104 AEBCD
        103 BECDA
        102 CEDBA
        101 DEBCA
          3 EABCD
          3 EBCDA
          3 ECDBA
          3 EDCBA

            
This evidently makes E a very much stronger candidate, for if any one of
A, B, C or D had not stood, E would have been the first elected, but plain
STV takes no notice, electing BC just as before. Sequential STV chooses BC
as probables but then tests BCD, where BC stay as probables and D goes to
the end of the queue, followed by BCE where BE become the new probables and
C goes to the end of the queue. It then tests BEA and BED, BE winning each
time. There is no need to test BEC again as that result is already known,
so BE are elected."

QLWA again gives the same results as Sequential STV.

Here is an example of Condorcet Loser Elimination STV being a bit unstable and giving an
order-reversal incentive.

300 votes, 3 seats, Droop quota = 75.
74 A B C D X E
39 B A C D X E
75 C
37 D X E C B A
73 E D X C B A
02 X E D C B A

CLE elects CBD, but if the two X voters reverse their top 2 preferences, then E has a quota
and the result changes to CEB (a preferable result for the two Compromisers).
QLWA elects CEA in both cases (like normal STV).

Chris Benham












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