<div dir="ltr">
<p class="MsoNormal">Here’s an example (with variations):</p>
<p class="MsoNormal">41 C</p>
<p class="MsoNormal">30 A>B</p>
<p class="MsoNormal">29 B</p>
<p class="MsoNormal">Candidate A’s pairwise approval is 30 relative to every
candidate.</p>
<p class="MsoNormal">Candidate B’s smallest pairwise approval is 29 against
candidate A.</p>
<p class="MsoNormal">Candidate<span> </span>C’s
pairwise approval is 41 relative to every candidate.</p>
<p class="MsoNormal">So the MaxMinPA winner is C.</p>
<p class="MsoNormal">Now suppose that we replace 41 C with 41 C>B:</p>
<p class="MsoNormal">This raises B’s pairwise approval relative to A from 29 to
70 so that B’s new min PA is 59 (relative to C)</p>
<p class="MsoNormal">Now B is the MaxMinPA winner.</p>
<p class="MsoNormal">It appears that if 41 C>B is the sincere preference, then
truncation to 41 C is a successful maneuver to change the MaxMinPA winner from
B to C.<span> </span></p>
<p class="MsoNormal">Q. What defense is there to preserve the win for the sincere
CW?<span> </span></p>
<p class="MsoNormal">Ans. The 30 A>B voters can raise B to unconditional
approval A=B.<span> </span>This works because now there
are only two occupied levels (top and bottom), in which case MaxMinPA reduces
to ordinary Approval.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">What if (in the original preference profile) we replace 29 B
with 29 B>A ?</p>
<p class="MsoNormal">Then A’s PA relative to C increases to 59 and A’s PA
relative to itself increases to 44.5, but its PA relative to B stays at 30, so
its MinPA does not increase, so C is elected rather than the CW.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">This example merely shows that the method does not satisfy
the Condorcet Criterion.<span> </span>If the
Condorcet Criterion is more important than the FBC, then we could go with
Smith//MaxMinPA.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">If we’re not willing to give up the FBC, then at least
eleven members of the 29 B>A faction should raise A to the top tier: 11 B=A,
18 B>A.<span> </span>Then A and C are tied for the
MaxMinPA=41, but A’s other PA’s dominate C’s.</p>
<p class="MsoNormal">So again the CW is rescued by raising it to top on some of
the ballots.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">It may turn out that approval strategy is optimal for
MaxMinPA.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Another idea: instead of MaxMinPA apply game theoretic
principles to the PA matrix defined as the matrix whose entry in row i and
column j is the pairwise approval of candidate i relative to candidate j.</p>
<p class="MsoNormal">In this example (assuming that candidates one through three
are respectively candidates A through C) the three rows are respectively<span> </span>[44.5, 30, 59], [29, 44, 59], and <span> </span>[41, 41, 41].</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">Note that the third entry of the last row is dominated by
the third entries of the other two rows.</p><p class="MsoNormal"><span><br></span></p><p class="MsoNormal"><span> </span>From a game theoretic point of view this allows us to eliminate the
third candidate. <br></p><p class="MsoNormal">The submatrix that remains has respective rows [44.5, 30] and
[29, 44]. Their respective min’s are 30 and 29, so the first candidate (A)
wins.</p><p class="MsoNormal"><br></p>
<p class="MsoNormal">So I’m not sure that MaxMinPA is the best way to make use of
the PA matrix, but it is one way that guarantees FBC compliance.<span> <br></span></p><p class="MsoNormal"><span><br></span></p><p class="MsoNormal"><span> </span>In any case I think we should not throw out
the PA matrix, because I believe that its entries are useful estimates of how
much approval candidate i would receive if the contest were largely between
candidates i and j.<span> </span>This information
should be useful in deciding who the over-all sincere approval winner should
be.</p><p class="MsoNormal"><br></p><p class="MsoNormal">Forest<br></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<div class="gmail_extra"><br><div class="gmail_quote">On Thu, Oct 13, 2016 at 9:30 AM, Michael Ossipoff <span dir="ltr"><<a target="_blank" href="mailto:email9648742@gmail.com">email9648742@gmail.com</a>></span> wrote:<br><blockquote style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex" class="gmail_quote"><p dir="ltr">I meant W is the offensively strategizing faction.</p><span class="gmail-HOEnZb"><font color="#888888">
<p dir="ltr">Michael Ossipoff</p>
</font></span><div class="gmail_quote"><span class="gmail-">On Oct 12, 2016 2:17 PM, "Forest Simmons" <<a target="_blank" href="mailto:fsimmons@pcc.edu">fsimmons@pcc.edu</a>> wrote:<br type="attribution"></span><blockquote style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex" class="gmail_quote"><div><div class="gmail-h5"><div dir="ltr">
<p class="MsoNormal">The following method is based on score or range style
ballots.<span> </span>I believe it satisfies the FBC,
Plurality, the CD, Monotonicity, Participation,<span>
</span>Clone Independence, and the IPDA.<span>
</span>It reduces to ordinary Approval when only the extreme ratings are used
for all candidates.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">I call it MinMaxPairwiseApproval or MinMaxPA for short.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">It is based on a concept of “pairwise approval.”<span> </span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">A zero to 100% cardinal ratings ballot contributes the
following amount to the “pairwise approval of candidate X relative to candidate
Y”: </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The amount is either …</p>
<p class="MsoNormal">100% if X is rated strictly above Y, or</p>
<p class="MsoNormal">Zero if X is rated strictly below Y, or</p>
<p class="MsoNormal">Their common rating if they are rated equally.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">According to this definition, the ballot’s contribution to
the pairwise approval of X relative to itself is simply the ballot’s rating of
X, since it is rated equally with itself.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The method elects the candidate whose minimum pairwise
approval (relative to all candidates including self) is maximal.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The motivation for this idea is the question, “If candidates
X and Y were the only two candidates with any significant chance of winning the
election, what is the probability that the ratings ballot voter would want X
approved (in a Designated Strategy Voting system, say)?”</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">If the voter rated X over Y, this probability would be 100
percent.</p>
<p class="MsoNormal">If the voter rated Y over X, this probability would be zero.</p>
<p class="MsoNormal">If the voter rated both X and Y at 100 percent, this
probability would be 100 percent.</p>
<p class="MsoNormal">If the voter rated them both at zero, she would want neither
of the approved.</p>
<p class="MsoNormal">If she rated them both at 50%, then our best guess is that there
is a fifty-fifty chance that she would approve X.</p>
<p class="MsoNormal">Etc.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Whatever nice properties the method has depends solely on
its definition, not the motivation for the definition, so please explore it
with an open mind.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Tomorrow, when I have more time, I’ll give some examples.</p><p class="MsoNormal"><br></p><p class="MsoNormal">Enjoy,</p><p class="MsoNormal"><br></p><p class="MsoNormal">Forest</p><p class="MsoNormal"><br></p><p class="MsoNormal">P.S.</p><p class="MsoNormal"><br></p><p class="MsoNormal">The rules can be modified for ranked preference ballots:</p><p class="MsoNormal"><br></p><p class="MsoNormal">The amount (per ballot) of approval of X relative to Y is either ...</p><p class="MsoNormal"><br></p><p class="MsoNormal">100 percent if X is ranked ahead of Y or equal top with Y<br></p><p class="MsoNormal">zero if Y is ranked ahead of X or equal bottom with X<br></p><p class="MsoNormal">50 percent if both are ranked equally and strictly between top and bottom.<br></p>
<div class="gmail_extra"><br></div><div class="gmail_extra">Smith//MaxMinPA may be a nice method that trades the FBC and possibly other nice properties for the Condorcet Criterion.<br></div><div class="gmail_extra"><br></div></div>
<br></div></div><span class="gmail-">----<br>
Election-Methods mailing list - see <a target="_blank" rel="noreferrer" href="http://electorama.com/em">http://electorama.com/em</a> for list info<br>
<br></span></blockquote></div>
</blockquote></div><br></div></div>