I have developed a new PR method which I call Approval Threshold Transferrable Votes or AT-TV. <div><br><div>---- <b><font size="4">Procedure</font></b> ---- (see also python code, attached.)</div><div>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:<br>
<div><br></div><div>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<span style="border-collapse:collapse;font-family:arial, sans-serif">(BC) <=N</span></div>
<div><div style="border-collapse:collapse;font-family:arial, sans-serif"><br></div>
<div style="border-collapse:collapse;font-family:arial, sans-serif">One droop quota Q is the total number of ballots divided by the number of seats to be elected.</div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse"><br>
</span></font></div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">Approval threshold T starts at N. </span></font></div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">Ordered list of elected candidates starts empty</span></font></div>
<div><font face="arial, sans-serif"><span style="border-collapse:collapse">While T>0:</span></font></div></div></div></div><blockquote style="margin:0 0 0 40px;border:none;padding:0px">
<div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">All ballots are given a "approval score weighting" S[B] and a "droop score weighting" W[B] which are both (re)set to 1</span></font></div>
</div></div></div><div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">"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.</span></font></div>
</div></div></div><div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">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. </span></font><span style="border-collapse:collapse;font-family:arial, sans-serif">This will be recalculated whenever any W[B] changes.</span></div>
</div></div></div><div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">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 </span></font><span style="border-collapse:collapse;font-family:arial, sans-serif">droop score </span><span style="border-collapse:collapse;font-family:arial, sans-serif">D[C] by one Droop quota. (that is, multiplying by (D[C]-Q)/D[C] )</span></div>
</div></div></div><div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">While there are candidates with at D[C] greater than one Droop quota Q:</span></font></div>
</div></div></div></blockquote><blockquote style="margin:0 0 0 40px;border:none;padding:0px"><blockquote style="margin:0 0 0 40px;border:none;padding:0px">
<div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">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</span></font></div>
</div></div></div></blockquote><blockquote style="margin:0 0 0 40px;border:none;padding:0px"><div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">The droop weighting W[B] </span></font><span style="border-collapse:collapse;font-family:arial, sans-serif">of any ballot which approves of the elected candidate C </span><span style="border-collapse:collapse;font-family:arial, sans-serif">is discounted by a factor sufficient to reduce </span><span style="border-collapse:collapse;font-family:arial, sans-serif">droop score</span><span style="border-collapse:collapse;font-family:arial, sans-serif"> D[C] by one Droop quota. (that is, multiplied by (D[C]-Q)/D[C] )</span></div>
<font face="arial, sans-serif"><span style="border-collapse:collapse">The approval score weighting S[B] </span></font><span style="border-collapse:collapse;font-family:arial, sans-serif">of any ballot which approves of the elected candidate C </span><span style="border-collapse:collapse;font-family:arial, sans-serif">is discounted by a factor sufficient to reduce </span><span style="border-collapse:collapse;font-family:arial, sans-serif">approval score </span><span style="border-collapse:collapse;font-family:arial, sans-serif">A[C] by one Droop quota. (that is, multiplied by (A[C]-Q)/A[C] )</span></div>
</div></div></blockquote></blockquote><blockquote style="margin:0 0 0 40px;border:none;padding:0px"><div><div><div>
<div><font face="arial, sans-serif"><span style="border-collapse:collapse">T (Approval threshold) is reduced by one</span></font></div></div></div></div></blockquote><div>
<div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">While there are still unfilled seats:</span></font></div></div></div></div><blockquote style="margin:0 0 0 40px;border:none;padding:0px">
<div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">T is set to 1</span></font></div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">The candidate with highest droop score D(C) is added to elected candidates</span></font></div>
</div></div></div><div><div><div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">All ballots which approve that candidate (that is, rank them above 0) are given W(B) of 0</span></font></div>
</div></div></div><div><font face="arial, sans-serif"><span style="border-collapse:collapse"><span style="font-family:arial;border-collapse:separate"><br>
</span></span></font></div></blockquote>---- <b><font size="4">Motivation and Properties</font></b> ----</div>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.<div>
<br></div><div><div>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.</div>
<div><br></div><div>Thus, this is a sequential procedure <span style="border-collapse:collapse;font-family:arial, sans-serif;font-size:13px">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.</span></div>
<span style="border-collapse:collapse;font-family:arial, sans-serif;font-size:13px"><div><br></div><div>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.</div>
</span><div><br></div><div>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.</div>
<div><br></div><div>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<span style="border-collapse:collapse;font-family:arial, sans-serif;font-size:13px"> (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 </span><span style="border-collapse:collapse;font-family:arial, sans-serif;font-size:13px">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 </span><span style="border-collapse:collapse;font-family:arial, sans-serif;font-size:13px">reducing the urgency of Hylland free riding.</span></div>
<div><span style="border-collapse:collapse;font-family:arial, sans-serif;font-size:13px"><br></span></div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">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.</span></font></div>
<div><font face="arial, sans-serif"><span style="border-collapse:collapse"><br></span></font></div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">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.</span></font></div>
<div><font face="arial, sans-serif"><span style="border-collapse:collapse"><br></span></font></div><div><font face="arial, sans-serif"><span style="border-collapse:collapse">I believe it is a promising system.</span></font></div>
<div><br></div><div>[1] <span style="border-collapse:collapse;font-family:arial, sans-serif;font-size:13px"> <a href="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" style="color:rgb(92, 69, 32)" target="_blank">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</a> ,</span></div>
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