[EM] Multiseat Bucklin? a naiive proposal

[John B. Hodges]

A few days ago I noted that one of the common criticisms of IRV/STV 
is that it is not monotonic. I recalled reading somewhere that any 
procedure that eliminates candidates before selecting a winner fails 
monotonicity. I noted that Bucklin's procedure would be monotonic. I 
wondered if perhaps Bucklin's procedure could be substituted for STV 
in both single-seat and multiseat cases.

Bucklin's procedure was originally intended only for the single-seat 
case. I naiively propose to generalize it for N seats,in this way: 
All voters submit ballots ranking their most-preferred candidates, as 
many or as few as they wish. (I suppose ties could be allowed, each 
tie counting as a fractional vote for each of the tied candidates.)
(1) Count voter's ballots. Each ballot counts as a vote for the 
highest-ranked candidate still in the race, i.e. not yet awarded a 
seat. Add these counts to each candidate's previous total.
(2) Has any candidate reached the winning threshold?
	If not, count voters' next-ranked choices and add these 
counts to the candidates' totals. (If a ballot has nobody ranked 
next, it counts as an abstention. If there are no ballots with anyone 
ranked next, go back to (1).) Go back to (2).
	If so, go to (3).
(3) Are there now more "winners" than seats? I.e. do we now have more 
candidates with vote totals above the winning threshold than there 
are seats remaining to be filled?
	If so, award the seats to the candidates with the largest 
totals. (If two such candidates are tied, break the tie by a Borda 
Count. If they are tied by Borda Count, flip a coin.) You are 
finished.
	If not, go to (4).
(4) Have all seats been filled?
	If not, recalculate the winning threshold for the remaining 
seats. Go back to (1).
	If so, you are finished.

The "winning threshold" I propose to be the Droop quota, i.e. (total 
# of ballots / # of seats plus one.)

STV goes by the principle "one person, one transferable vote". This 
version of multiseat Bucklin goes instead by "One person, one vote 
per round." Everyone gets the same number of votes, unless they fail 
to list a preference. Truncated ballots are allowed but probably 
unwise for this reason; voters should rank candidates until they are 
truly indifferent between those remaining.

The algorithm above bombs out if there are some candidates who get no 
votes, and some seats remaining unfilled after all candidates who DO 
get votes are awarded seats. I expect this to be rare; let the courts 
decide whether to let the seats remain vacant, or fill them with some 
of the remaining candidates by lottery, or what.

I have no idea how well this algorithm would perform, in terms of 
proportionality. I suspect it would overrepresent minorities, because 
candidates accumulate totals over repeated rounds. Undervotes 
accumulate, overvotes vanish. Those candidates who have higher-rank 
votes left over from previous rounds have that much of a head start. 
Higher-rank votes are effectively worth more than lower-rank votes, 
because they are more likely to be repeated. Net result intuitively 
looks to me like a high percentage of the voters should see one of 
their higher-rank choices get a seat, but candidates supported by a 
minority have a head start on getting the remaining seats after those 
supported by a majority have taken theirs. So, this method may be 
less proportional than STV.
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John B. Hodges, jbhodges@  @usit.net
Do Justice, Love Mercy, and Be Irreverent.