<div dir="ltr">See inserted reply below:<div class="gmail_extra"><br><div class="gmail_quote">On Sun, Feb 4, 2018 at 6:53 AM, Kristofer Munsterhjelm <span dir="ltr"><<a href="mailto:km_elmet@t-online.de" target="_blank">km_elmet@t-online.de</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><span>On 12/26/2017 12:43 AM, Ted Stern wrote:<br>
</span><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><span>
Chris Benham proposed IBIFA in May and June, 2010, on the<br>
election-methods mailing list:<br>
<br>
<a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-May/091807.html" rel="noreferrer" target="_blank">http://lists.electorama.com/pi<wbr>permail/election-methods-elect<wbr>orama.com/2010-May/091807.html</a><br></span>
<<a href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-May/091807.html" rel="noreferrer" target="_blank">http://lists.electorama.com/p<wbr>ipermail/election-methods-elec<wbr>torama.com/2010-May/091807.htm<wbr>l</a>><br>
<br>
<a href="http://election-methods.electorama.narkive.com/KdBxpweB/irrelevant-ballots-independent-fallback-approval-ibifa" rel="noreferrer" target="_blank">http://election-methods.electo<wbr>rama.narkive.com/KdBxpweB/irre<wbr>levant-ballots-independent-<wbr>fallback-approval-ibifa</a><br>
<<a href="http://election-methods.electorama.narkive.com/KdBxpweB/irrelevant-ballots-independent-fallback-approval-ibifa" rel="noreferrer" target="_blank">http://election-methods.elect<wbr>orama.narkive.com/KdBxpweB/irr<wbr>elevant-ballots-independent-<wbr>fallback-approval-ibifa</a>><br>
<br>
<a href="http://wiki.electorama.com/wiki/IBIFA" rel="noreferrer" target="_blank">http://wiki.electorama.com/wik<wbr>i/IBIFA</a><span><br>
<<a href="http://wiki.electorama.com/wiki/IBIFA" rel="noreferrer" target="_blank">http://wiki.electorama.com/wi<wbr>ki/IBIFA</a>><br>
<br>
IBIFA is, as originally stated, a "Bucklin-like method meeting Favorite<br>
Betrayal and Irrelevant Ballots." Its key principle is to compare the<br>
ballots voting for a candidate at-or-above a particular rating to the<br>
most-approved candidate on the complementary ballots. When the former<br>
exceeds the latter, a meaningful threshold has been crossed, unlike the<br>
arbitrary 50% threshold of median rating methods. This is what enables<br>
IBIFA to yield the same result if irrelevant ballots are added or<br>
dropped. By construction, IBIFA is cloneproof.<br>
<br>
With this in mind, I realized that a minor modification of IBIFA would<br>
make it more like Majority Judgment, reducing later-harm and improving<br>
Condorcet consistency (though not completely), while satisfying the same<br>
criteria as MJ.<br>
</span></blockquote>
<br>
Do you think it's possible to generalize that strategy to handle multiwinner elections as well? I can't see any obvious ways, but it would be nice if one could make multiwinner methods with implicit thresholds as well, since the vast majority uses explicit thresholds (usually the Droop quota).</blockquote><div><br></div><div>Single-winner rated methods can be adapted to an ER-Bucklin-Droop multiwinner method. Just choose each seat using the single-winner method and weighted ballots.</div><div><br></div><div>Then reweight the ballots by finding the score at-and-above which at least one Droop quota of voters approves the seat winner, and reweight those ballots to remove one Droop quota.</div><div><br></div><div>The difference from straight ER-Bucklin multiwinner is that the quota-finding score is <i>not</i> the one used to determine the seat winner.</div><div> <br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><span><br>
<br>
<blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
EXACT does require several N^2 arrays for summable storage, but note<br>
that no sorting of the ballots is required as with pairwise methods.<br>
</blockquote>
<br></span>
Pairwise methods don't strictly require that ballots are sorted. E.g. the following will produce a Condorcet matrix (if there's no truncation or equal rank):<br>
<br>
for i = 1 to num ballots<br>
for j = 1 to num candidates<br>
c1 = candidate ranked at jth place on the ith ballot<br>
<br>
for k = j+1 to num candidates<br>
c2 = candidate ranked at kth place on the ith ballot<br>
matrix[c1 beats c2] += weight of ballot i<br>
<br>
It's a bit of a nitpick, though :-)<br>
</blockquote></div><br></div><div class="gmail_extra">I apologize for my earlier assertion. When I tried coding the pairwise array myself, using the numpy package in Python, I found a very compact way to do so, much along those lines. Using the matrix optimized numpy package, imported as "np",</div><div class="gmail_extra"><br></div><div class="gmail_extra"><div class="gmail_extra"><font face="monospace, monospace"> for ballot, w in zip(ballots,weight):</font></div><div class="gmail_extra"><font face="monospace, monospace"> for v in range(1,maxscore): # Assuming all rated candidates are approved</font></div><div class="gmail_extra"><span style="font-family:monospace,monospace"> A += w * np.multiply.outer(np.where(ballot==v,1,0),</span><br></div><div class="gmail_extra"><font face="monospace, monospace"> np.where(ballot<=v,1,0))</font></div><div><br></div></div><div class="gmail_extra"><br></div><div class="gmail_extra">In other words, the matrix update to the pairwise array is, for each rating v, the outer product of (a) a vector with ones where candidates have score v, with (b) another vector with ones where candidates have scores less than or equal (or less than) v.</div></div>