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

[Ted Stern]

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.


On Sat, Jan 25, 2020 at 5:15 PM Forest Simmons <fsimmons at pcc.edu> wrote:

> We to things:
>
> 1. A standard way of converting rankings to pseudo-ratings.
> 2. A standard way of identifying "runner-ups" in single winner elections.
>
> For now suppose that these problems have been taken care of. I made some
> suggestions in previous messages, but they are only tentative, so don't
> stop thinking!
>
> Our main heuristic for the method is this: when a candidate has won a
> round in the election and thereby qualified to be seated with the other
> winners [there was a brief typo that said "sinners" instead of "winners."]
> no ballot that strictly preferred the runner-up to the winner should be
> penalized in any degree, and even a ballot that ranked the winner ahead of
> or equal to the runner-up, should only be de-weighted in fair accord with
> the level of support indicated by the rating or pseudo rating of that
> winner on the ballot.
>
> So if your ballot rates the winner at 100 percent, then it will suffer the
> max de-weighting for that round.  If your ballot rates the winner at zero,
> it dodges the de-weighting bullet unscathed for that round.  If your ballot
> rates the winner at 50%, then the de-weighting depends on whether or not
> the runner-up is ranked strictly ahead of the winner or not. If so, there
> is no de-weightiing; if not, the de-weight effect is half of the full
> possibility.
>
> That's the idea. Now for a few details. The de-weighting scheme is based
> on what we could call "satisfaction," or "relief" or "blame" depending on
> your point of view.  The (additive) increment delta to the "satisfaction" S
> of a particular ballot B by a round of the election with winner W and
> runner-up R is computed as follows:
>
> If R is ranked strictly above W, on ballot B, then delta is zero.
> Else delta is the (pseudo) rating of W on ballot B.
>
> At some stage, let S be the cumulative satisfaction on one particular
> ballot from the results of the previous stages, i.e. the sum over all of
> the winners so far of the ballot satisfaction increments for each of them.
> Then the weight of that ballot in the next round is 1/(1+S).
>
> That's it!
>
> Why is this proportional?
>
> Because in a "colour (or color) election," which is one standard test case
> for proportionality, the voters will vote only at the extremes of equal top
> and equal bottom, AND in this case the method precisely reduces to
> Sequential Proportional Approval Voting (SPAV) which is known to satisfy
> the color criterion (specifically with D'Hondt quotas).
>
> BTW this is true no matter what method may be used to convert rankings to
> pseudo-ratings as long as it leaves equal top at the top, and equal bottom
> at the bottom.
>
> Nor does the choice of runner-up affect this result, since no matter who
> the runner-up may be, if the winner is top rated then the increment delta
> will be one, and if the winner is bottom rated the increment will be zero
> in accordance with the recipe given above (and repeated here so you can
> check it):
>
> If R is ranked strictly above W, on ballot B, then delta is zero.
> Else delta is the (pseudo) rating of W on ballot B.
>
> In particular, if W is rated at zero, in either case (R > W or both bottom
> rated), delta will be zero.
>
> It took a few sleepless nights to get this just right before too many EM
> enthusiasts started exposing the mistakes and weaknesses of the first
> version. Pressure of pride!  It comes before the fall, but it can also
> hasten the time of face-saving redemption!
>
>
>
>
> ----
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> info
>
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