new STV-ish methods (was RE: STV with "uneliminations"?)

[Steve Eppley]

Demorep1 wrote:
-snip-
>With multiple seats especially it would seem natural that the
>candidate getting the fewest first choice votes should lose
-snip-

I don't think this is a "natural" principle.  For some reason you
think first choices are special (in both single-winner and PR).  
I think that a voter simply prefers his/her first choice more than
his/her second choice, by definition, and trying to glean more
meaning about first choices in ranked ballots is fallacious.

Without a computer, it's much easier to tally ballots by
considering only one of their choices at a time as STV and MPV do.
That's the best argument for that principle, imho, but it doesn't
apply to modern, public, computerized elections.

I don't have a significant problem with STV used for prop rep
systems, since the goal is to represent people and people are best
represented by their first choices.  I think there's a difference,
though, between these two principles: 
1. electing candidates with the most first choices 
2. eliminating candidates with the fewest first choices

Suppose there's a candidate with fewest first choice votes who 
happens to be nearly everyone's second choice?  Why should s/he lose?
   25 voters:  A B C D E
   25 voters:  C B D E A
   25 voters:  D B E A C
   25 voters:  E B A C D
Why do you consider B to be a big loser?  Pairwise, B would beat 
everyone else 75 to 25.  With STV and MPV, B wouldn't win a seat
(except for the silly case where 5 seats are being elected).

No candidate reached the quota.  Who should win the seats?  Is it
better to represent a few of the people well, or many of the people
with a compromise (B), when no one has a quota?

Maybe STV can be improved so it elects strong candidates instead
of eliminating weak candidates when no candidate has a quota?

* *

Here's a new prop rep method, which I'll call PRPV (prop rep
preference voting) for the time being.  This method doesn't work by
eliminating "losers".  (The only candidates eliminated are seat
winners.)  As in STV, seats are awarded sequentially:

repeat
   N = 1
   while no candidate has a seat quota (counting for each candidate 
    the ballots in which the candidate is one of the top N choices)
       N = N + 1
   endwhile
   Award the next seat to the candidate who is among the top N choices
    on the most ballots.
   Delete a quota's worth of ballots, starting with the ballots 
    which rank this winning candidate "true first", continuing 
    (if necessary) with the ballots which rank this candidate 
    "true second", etc.
   Eliminate this winning candidate from all ballots, replacing
    him/her with a null placeholder.
until all seats have been awarded.

Example:  Two seats to be elected.  Suppose the seat quota is 40%.
   38:  R > M > L
   28:  M > R=L
   34:  L > M > R
   Seat 1:
     N=1:  No one has the quota.
     N=2:  
        R = 38 + 28          (or 38 + 28/2, if votes are split)
        M = 28 + 38 + 34 
        L = 34
        M wins seat 1, having the largest overquota count.
     A quota of ballots are deleted, and M is eliminated from the 
      ballots:
      31.667:  R > _ > L               38 * (1 - (40-28)/(38+34))
           0:  _ > R=L             <-- These 28 are entirely deleted.
      28.333:  L > _ > R               34 * (1 - (40-28)/(38+34))
   Seat 2:
     R wins seat 2, based on the 31.667+28.333 ballots remaining at 
      the end of the previous step.  (Details omitted; would involve
      a tie-breaker.)

* *

One troublesome characteristic of PRPV, illustrated in the example
above, is that the voters who ranked M first seem overrepresented--
they've got the representative they prefer most, plus a "share" of 
the winner of seat one.  The voters who ranked L first won nada.

Here's a fancier method which might compensate.  It uses PRPV 
repeatedly.  I'll call it Iterating PRPV (IPV), for the time being.

repeat
   Using PRPV, determine a tentative seat awarding order for all
    the seats remaining to be awarded, using the ballots which 
    remain.
   Award the next seat to the candidate with the most first 
    choice votes (counting both "true" and "transferred" choices), 
    considering only the candidates who are included among the 
    tentative seat winners determined in the previous step.
   Delete a quota's worth of ballots, starting with the ballots 
    which rank this winning candidate "truly first", continuing 
    (if necessary) with the ballots which rank this candidate 
    "truly second", etc.
   Eliminate this winning candidate from all ballots, replacing
    him/her with a null placeholder.  (The placeholder occupies
    a place during the ballot deletion step, but choices "transfer"
    thru it during the seat awarding step.)    
until all seats have been awarded.

Here's the same example tallied using IPV:
Example:  Two seats to be elected.  Suppose the seat quota is 40%.
   38:  R > M > L
   28:  M > R=L
   34:  L > M > R
   Seat 1:
     As shown above, the initial tentative seat award order is {M,R}.
     Of M and R, R has the most first choice ballots.
      Therefore R wins seat one.
     A quota's worth of ballots are deleted, including all 38 which
      ranked R first, and (Q-38) of the ballots which ranked R 2nd.
      Also, R is eliminated from the ballots:
       0:  _ > M > L            <-- these 38 are entirely deleted
      26:  M > L                    28 - (Q-38)
      34:  L > M
   Seat 2:
     Since there's only one seat remaining to be filled, the 
      tentative order for the remaining seats will have only one
      candidate.  Based on the ballots remaining at the end of
      the previous step, this tentative order is {L}.  (Details 
      omitted; would involve a tie-breaker.)
     L wins seat two.

The results using IPV differed from the results using PRPV:
The voters who ranked R first will be represented by R.  The voters
who ranked L first will be represented by L.  The voters who ranked
M first will be represented, somewhat, by their equally 2nd choices.

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)