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<blockquote type="cite">
<div>My modification is inserted with emphasis added.</div>
<ul>
<li>Find the highest rating R, for which there is at least one
candidate X who is rated <u>at or above level R</u> on more
ballots than any candidate is approved on ballots which rate
X below R.</li>
<li>If there is more than one such candidate X, <i>then if
there is at least one candidate Y who is rated <u>above R</u>
on more ballots than the highest approved candidate on
ballots that rate Y below R, elect the candidate Y with
the most ballots rating Y above R.</i></li>
<li><i>Otherwise, </i>elect the candidate X with the most
ballots rating X at R or above.</li>
<li>If no candidates satisfy the first criterion, for any
approved rating R, elect the candidate with the highest
approval over all ballots.</li>
</ul>
</blockquote>
<br>
To be more clear, shouldn't the second line read something like:<br>
<br>
If there is more than one such candidate X, <i>then<b> among
those candidates</b> if there is at least one candidate Y who
is rated <u>above R</u> on more ballots than the highest
approved candidate on ballots that rate Y below R, elect the
candidate Y with the most ballots rating Y above R.</i><br>
<br>
?<br>
The method looks good, AFICT.<br>
<br>
Chris Benham<br>
<br>
On 26/12/2017 10:13 AM, Ted Stern wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CAHGFzOQDH-X=XbiwTseeJVCeB4YwDUeKDNkFkrYhBRAwvTQExQ@mail.gmail.com">
<div dir="ltr">Chris Benham proposed IBIFA in May and June, 2010,
on the election-methods mailing list:
<div><br>
</div>
<div><a
href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-May/091807.html"
target="_blank" moz-do-not-send="true">http://lists.electorama.com/<wbr>pipermail/election-methods-<wbr>electorama.com/2010-May/<wbr>091807.html</a><br>
</div>
<div><br>
</div>
<div><a
href="http://election-methods.electorama.narkive.com/KdBxpweB/irrelevant-ballots-independent-fallback-approval-ibifa"
target="_blank" moz-do-not-send="true">http://election-methods.<wbr>electorama.narkive.com/<wbr>KdBxpweB/irrelevant-ballots-<wbr>independent-fallback-approval-<wbr>ibifa</a><br>
</div>
<div><br>
</div>
<div><a href="http://wiki.electorama.com/wiki/IBIFA"
target="_blank" moz-do-not-send="true">http://wiki.electorama.com/<wbr>wiki/IBIFA</a><br>
</div>
<div><br>
</div>
<div>IBIFA is, as originally stated, a "Bucklin-like method
meeting Favorite Betrayal and Irrelevant Ballots." Its key
principle is to compare the ballots voting for a candidate
at-or-above a particular rating to the most-approved candidate
on the complementary ballots. When the former exceeds the
latter, a meaningful threshold has been crossed, unlike the
arbitrary 50% threshold of median rating methods. This is
what enables IBIFA to yield the same result if irrelevant
ballots are added or dropped. By construction, IBIFA is
cloneproof.</div>
<div><br>
</div>
<div>With this in mind, I realized that a minor modification of
IBIFA would make it more like Majority Judgment, reducing
later-harm and improving Condorcet consistency (though not
completely), while satisfying the same criteria as MJ.</div>
<div><br>
</div>
<div>IBIFA, simply stated, does the following:</div>
<div>
<ul>
<li>Find the highest rating R, for which there is at least
one candidate X who is rated <u>at or above level R</u> on
more ballots than any candidate is approved on ballots
that rate X below R.</li>
<li>If there is more than one such candidate X, elect the
candidate X with the most ballots rating X at R or above.</li>
<li>If no candidates satisfy the first criterion, for any
approved rating R, elect the candidate with the highest
approval over all ballots.</li>
</ul>
</div>
<div>My modification is inserted with emphasis added.</div>
<div>
<ul>
<li>Find the highest rating R, for which there is at least
one candidate X who is rated <u>at or above level R</u>
on more ballots than any candidate is approved on ballots
which rate X below R.</li>
<li>If there is more than one such candidate X, <i>then if
there is at least one candidate Y who is rated <u>above
R</u> on more ballots than the highest approved
candidate on ballots that rate Y below R, elect the
candidate Y with the most ballots rating Y above R.</i></li>
<li><i>Otherwise, </i>elect the candidate X with the most
ballots rating X at R or above.</li>
<li>If no candidates satisfy the first criterion, for any
approved rating R, elect the candidate with the highest
approval over all ballots.<br>
</li>
</ul>
<div>I call this IBIFA variant "EXACT", because it uses an
EXclusive Approval Comparison Threshold. That is, the
candidate compared to X is the one with maximum approval on
ballots that <u>exclude</u> votes for X at some rating or
above. Like IBIFA, it is also cloneproof.</div>
</div>
<div><br>
</div>
<div>For EXACT, it is convenient to keep track of co-approval:
the approval for candidates X[j] on a ballot containing
candidate X[i] with rating k:</div>
<div><br>
</div>
<div><font face="monospace, monospace"> for ballot in ballots:</font></div>
<div><font face="monospace, monospace"> for candidate i on
ballot with score k:</font></div>
<div><span style="font-family:monospace,monospace"> if k
approved:</span></div>
<div><span style="font-family:monospace,monospace"> for
candidate j on ballot with score m:</span></div>
<div><span style="font-family:monospace,monospace"> if
m approved:</span></div>
<div><span style="font-family:monospace,monospace">
W[k,i,j] += 1</span></div>
<div><br>
</div>
<div>Note that W[k,i,i] is the total approval for candidate X[i]
at rating k, and the total approval for candidate X[i] at
rating k and higher is the sum of W[k,i,i] over all approved
ratings k.</div>
<div><br>
</div>
<div>It should then be clear that the approval for any candidate
j on a ballot that rates X[i] at R or higher is </div>
<div><br>
</div>
<div> Approval[j] - W[R,i,j] - W[R+1,i,j] ... -
W[MaxScore,i,j]</div>
<div><br>
</div>
<div>The EXACT score for a candidate is tuple similar to
Majority Judgment's "majority grade":</div>
<div><br>
</div>
<div>
<blockquote style="margin:0 0 0 40px;border:none;padding:0px">
<div>EXACT score for candidate X = (R, S, T)</div>
<div><br>
</div>
</blockquote>
<blockquote style="margin:0 0 0 40px;border:none;padding:0px">
<div>where R is the rating at which X's votes at or above R
are greater than the highest approved candidate on
ballots excluding X at R or above.;</div>
<div><br>
</div>
</blockquote>
<blockquote style="margin:0 0 0 40px;border:none;padding:0px">If
the number of ballots with X at rating R+1 and above is
greater than those of the highest approved candidate on
ballots excluding X at ratings R and above, then S = R+1,
and T = votes for X at R+1 and above.<br>
<br>
Otherwise, S = R and T = votes for X at R and above.<br>
<br>
</blockquote>
By sorting these tuples in descending order, one gets, as with
Majority Judgment, an EXACT ranking for the candidates.</div>
<div><br>
</div>
<div>EXACT satisfies all the same properties as Majority
Judgment, and in addition, is irrelevant-ballot-immune (IBI).
That is, a ballot containing approval only for non-contending
candidates won't affect the results.</div>
<div><br>
</div>
<div>EXACT does require several N^2 arrays for summable storage,
but note that no sorting of the ballots is required as with
pairwise methods.</div>
</div>
</blockquote>
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