[EM] multiseat pairwise?

[Steve Eppley]

To:            Kevin Hornbuckle <kevinh at efn.org>
Date:          Tue, 19 Mar 1996

Kevin (who I think is not an election-methods subscriber) asked:
>If you use pairwise in a multiseat election, can it be said to yield 
>proportional representation?

I haven't thought about it before; my first thoughts are "no."  
It would seem to be like the system which asks each voter to vote N
distinct choices in an N seat election: the same majority which
elects the top choice can elect the rest, totally shutting out a
minority.

Example:  2 seats up for grabs
  Summary of the ranked ballots:
  26%  Dole, Alexander, Clinton
  25%  Alexander, Dole, Clinton
  49%  Clinton
 ----
 100%

 The pairings:
   Dole over Alexander    by 1% (26-25+0)
   Dole over Clinton      by 2% (26+25-49)
   Alexander over Clinton by 2% (26+25-49)
 The results:
   Dole undefeated, wins seat #1.
   Alexander beat Clinton, and Alexander's worst loss (1%) is smaller 
   than Clinton's worst loss (2%).  Alexander wins seat #2.

Is this proportional?  I don't think so.  The 49% who liked only 
Clinton have no way to get him 1 of the 2 seats.  

So 51% can take all.  Unless the method is somehow modified to
subtract out the ballots which have already served to elect
candidates (Dole) before recalculating the remaining seats
(Alexander vs Clinton), it won't be proportional. I don't know 
if there's a reasonable way to do this. 

Here's an attempt: 
After each seat is awarded, subtract S = Total_votes/N (where N is
the total number of seats) from the seat winner's ballots before
recalculating.  The ballots which are subtracted are the ones which
listed the winner as ranked first.  If there aren't enough ranked-first 
ballots, also subtract ranked-second, etc.  If there are more than
enough ranked-first ballots (i.e., > S), then subtract appropriate
fractions of each.

Reworking the example using this new "multiseat pairwise":
 Dole wins seat #1.
 Since N=2, 50% of the ballots must be eliminated:
   First subtract all 26% {Dole Alexander Clinton} ballots, since 
     these are the ones which ranked Dole 1st.  
   Then subtract 24% from the ballots which ranked Dole 2nd.
 The recalculated pairings:
    1%  Alexander, Clinton  (original25 - 24)
   49%  Clinton
   ---
   50%
 The rest, giving Clinton seat #2, is left as an exercise.  :-)

It's an interesting question; I'm cc:ing it to election-methods-list. 

This looks a bit more complicated than STV.  Is anything gained by
using it instead of STV?  And does this method have an established
name?

--Steve