[EM] AT-TV, a monotonic, ratings-based, median-like, PR, STV-like multiwinner method

[Jameson Quinn]

I have developed a new PR method which I call Approval Threshold
Transferrable Votes or AT-TV.

---- *Procedure* ---- (see also python code, attached.)
If you'd rather understand the purpose before you look at the inner
workings, you can skip over this for now and look back at it after you read
the properties below:

Ballots are rated ballots from 0 (worst) to N (highest) rating; 2<=N<=5 is
recommended. Thus on each ballot B, candidate C has rating 0<=R(BC) <=N

One droop quota Q is the total number of ballots divided by the number of
seats to be elected.

Approval threshold T starts at N.
Ordered list of elected candidates starts empty
While T>0:

All ballots are given a "approval score weighting" S[B] and a "droop score
weighting" W[B] which are both (re)set to 1
"approval score" A[C] is defined as ∑[B]: S[B] H(R[BC] - N) , where H() is
the Heaviside step function. That is, the sum of the score weightings of the
ballots which rate C at or above the threshold N. This will be
recalculated/adjusted whenever any S[B] changes.
Similarly, "droop score" D[C] is defined as ∑(B): W[B] H(R[BC] - N). That
is, the sum of the droop weightings of the ballots which rate C at or above
the threshold N. This will be recalculated whenever any W[B] changes.
The already-elected candidates are re-added in their original order,
discounting the droop weighting W[B] (but not S[B]) of all ballots which
approve of the elected candidate C by a factor sufficient to reduce droop
score D[C] by one Droop quota. (that is, multiplying by (D[C]-Q)/D[C] )
While there are candidates with at D[C] greater than one Droop quota Q:

Of the candidates with D[C]>Q, the one with highest A[C] (or in case of
ties, the highest D[C]), is appended to elected candidate list

The droop weighting W[B] of any ballot which approves of the elected
candidate C is discounted by a factor sufficient to reduce droop score D[C]
by one Droop quota. (that is, multiplied by (D[C]-Q)/D[C] )
The approval score weighting S[B] of any ballot which approves of the
elected candidate C is discounted by a factor sufficient to reduce approval
score A[C] by one Droop quota. (that is, multiplied by (A[C]-Q)/A[C] )

T (Approval threshold) is reduced by one

While there are still unfilled seats:

T is set to 1
The candidate with highest droop score D(C) is added to elected candidates
All ballots which approve that candidate (that is, rank them above 0) are
given W(B) of 0

---- *Motivation and Properties* ----
This procedure reduces to a sequential representativeness-based procedure
(in the terminology of Kilgour [1]) in the two-rating (that is, N=1,
approval-style ballot) case. Thus, when each candidate is elected (except in
the "unfilled seats" fall-through), they are assigned one droop quota of
fractional voters who "approve" of that candidate (that is, rate them at or
above the current threshold). Although votes are assigned fractionally, if
the droop quota is an integer, after candidates are elected the fractional
votes could be reassigned so that a droop quota of whole ballots are
assigned to each candidate.

This procedure reduces to a median-based system in the single-winner case.
That is because a droop quota in that case is 50%. So as soon as the
threshold drops to the median of one or more candidates, they will have a
droop quota, and the one who surpasses that quota by the most will be
elected.

Thus, this is a sequential procedure which (implicitly) assigns a droop
quota of fractional voters to each candidate, and tries to maximize the
lowest rating of a voter for (any of) their representative(s). This
least-satisfied of the represented voters is thus one droop quota from the
bottom: in a 9-seat election, it maximizes the satisfaction of the
10th-percentile voter with their representative. This makes its relationship
to median obvious: in a 1-seat election, it would maximize the satisfaction
of the 50th-percentile voter.

A sequential procedure is the only way to solve this problem. A
globally-maximizing procedure would be NP-hard even for just approval-style
ballots (or, in AT-TV, for just one round), because it's easy to map the
vertex cover problem into this domain. A sophisticated linear programming
algorithm would be impossible to "code" into a legal statute. And since the
parameter being maximized is the rating of the single lowest representative
vote, ties are common, and thus it is not possible to simply resolve the
NP-hard problem by allowing anyone to submit a proposed slate and choosing
the best solution submitted. That leaves sequential procedures as the only
solution.

This system is not subject to Woodall free-riding (top-ranking a useless
candidate to sheild your vote from being reweighted during the initial
pre-elimination elections), as candidates are never eliminated, and all
approvals cause the same reweighting.

It is subject to Hylland free-riding (not ranking a candidate that can win
without your vote), as any PR system must be to some extent. However, by
separating the approval score (not discounted for candidates approved at
earlier thresholds) separate from the droop score (which is), voters whose
vote is exhausted by electing a candidate at an higher threshold can still
have their vote help decide between two candidates who just attained a Droop
quota at this level. And by resetting the ballot weights and re-discounting
at each threshold, votes will be discounted less as the threshold drops and
new approvals for a given candidate are added. The two factors should have
the respective effects of of helping to encourage Hylland free rider
strategists to move the candidates they honestly approve of, but expect to
win without needing more votes, to a higher score, rather than a lower one;
and of reducing the urgency of Hylland free riding.

This process, since it is a rated rather than ranked process, is monotonic
in each candidate. Raising candidate A does not affect other candidates
unless A is elected. In a similar sense, this procedure is independent of
irrelevant alternatives.

It is, of course, subject to the "Alabama paradox" (increasing the number of
seats may result in removing certain previously-elected candidates), like
other vote-transferring or representativeness systems.

I believe it is a promising system.

[1]
http://books.google.com.mx/books?hl=en&lr=&id=TzUVIpXRusQC&oi=fnd&pg=PA105&ots=fogvO2r0Ot&sig=0F4FQYFbPlxryHZagJi_-oKTX7U#v=onepage&q&f=false
 ,
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