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

[Forest Simmons]

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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