[EM] Framework for adapting Single Winner methods to Multiwinner PR

[Kristofer Munsterhjelm]

On 27/01/2020 21.26, Ted Stern wrote:
> I encountered another PR method recently that I think is worth adding to
> the discussion:
> 
> https://electowiki.org/wiki/Sequential_Monroe
> 
> In other words, for each candidate, determine the quota threshold rating
> using
> 
> score_total = 0
> approve_total = 0
> threshold_surplus = 0
> threshold_margin = 0
> for threshold_rating in maxscore:0:-1 :
>    score_total_minus_1 = score_total
>    score_total += rating * score[threshold_rating]
> 
>    approve_total_minus_1 = approve_total
>    approve_total += score[threshold_rating]
> 
>    if ( approval_total > quota ) then:
>         threshold_surplus = approve_total - quota
>         threshold_margin = quota - approve_total_minus_1
>         break
> 
> end for loop
> 
> threshold_score = score_total_minus_1 + threshold_margin * threshold_rating
> 
> If the candidate's threshold_rating is > 0, then that candidate gets a
> composite rating of
> 
>     (1, threshold_score/quota, approve_total)
> 
> Otherwise, the candidate gets a composite rating of
> 
>    (0, threshold_score / quota, approve_total)
> 
> Then sort these composite ratings in descending order.
> 
> To reweight ballots, Parker Friedland (the originator of the method)
> uses a relatively complex method of setting ballots giving the winner a
> rating above threshold_rating a weight of zero, while multiplying the
> ballots scoring the winner exactly at the threshold_rating by a factor of
> 
>    factor = ( 1 - (threshold_margin / score[threshold_rating]) )
> 
> (or equivalently, factor = threshold_surplus / score[threshold_rating] )
> 
> Alternatively, one could use a simpler factor for all ballots at and
> above the rating, 
> 
>     factor = ( 1 - (quota / approve_total) )
> 
> The reason I bring this up is that if you use a Droop quota of 
> N_ballots / (M_seats + 1), this method isn't really what I would look at
> as a good single-winner method.  Instead, it is trying to find the
> strongest preference in the top quota of voters for each candidate,
> *NOT* the strongest preference of the total remaining weight of
> ballots.  But I think this is more of what one is trying to achieve for
> all but the last seat of the multiwinner election, and certainly if you
> are are trying to satisfy Droop proportionality.
> 
> If one used a Hare quota, this method reduces to Score for the last
> seat.  But I think it might be just as reasonable to use a Condorcet
> method such as Score Sorted Margins for that last seat.

The most obvious ways to do Range Monroe sequentially I can think of is:

1. Let the starting set of winners be empty.
2. For each possible non-winner candidate, provisionally put that
candidate in the set and maximize an assignment of voters to candidates
in the set so that at most a Hare quota of voters is assigned to each
candidate, and the total score of voters to the candidate assigned to
them is maximized. (This can be done using the Hungarian algorithm or
linear programming.) Voters who are not assigned to any candidate don't
contribute to the objective function.
3. Find the candidate that optimizes this measure and permanently
include him in the winner set. If you have as many winners as there are
seats, you're done, otherwise go to 1.

To do a house-monotone method, just have the quota be of the current
number of seats (including the provisional candidate) rather than the
numer of desired seats.

As for Hare vs Droop, I have the feeling that the distinction is...
under Hare, each representative represents the whole quota cluster;
under Droop, each representative is elected by a majority. The remaining
Droop quota doesn't get its own representative, but it influences who
that majority is through all the other seats. So Droop will tend to be
more majoritarian and Hare is more representative, but Droop is also
less fragmented. In party list systems, Droop usually translates to a
bonus to the largest party, but in a Condorcet PR system (or Range for
that matter), its majoritarian/utilitarian logic can bias all the seats
in the direction of the true center instead of just the strongest bloc.