<div dir="ltr"><div><div><div><div><div><div><div><div><div>Now I remember the interesting example that shows that IA-MPO can fail Plurality when equal ranking at top is allowed:<br><br></div>33 A<br></div></div>16 C=A<br></div><div>02 C<br></div>16 C=B<br></div>33 B<br></div><div><br></div>The IA-MPO score for both A and B is 49-49=0, while the score for C is 34-33=1, so C wins.<br><br></div>This is a failure of Plurality because A (for example) is top ranked on 49 ballots, while C is ranked on only34 ballots.<br><br></div>However, any configuration in issue space that could give rise to this ballot set would be more faithfully reflected in a ballot like<br><br>33 A<br></div><div>16 C>A<br><div>02 C<br></div>16 C>B<br></div><div>33 B<br><br></div><div>Why would C voters raise A and B to top if they didn't really like them as well as the Condorcet Winner C?<br><br></div><div></div><div>It could be that (through typical disinformation) voters thought that C didn't have a chance compared to the two main party candidates A and B. They raised their lesser evil compromise candidates to hedge their bets.<br><br></div><div>It turns out that IA-MPO does satisfy a modified version of Plurality: If A is ranked top above C on more ballots than C is ranked, then C cannot be the IA-MPO winner.<br><br></div><div>In any case where this Plurality' would allow C to win while an ordinary Plurality requirement would preclude C's right to win, the C voters would (under perfect information conditions) rightly have an incentive to change each instance of C=A to C>A .<br><br></div><div>Proof that IA-MPO satisfies this modified Plurality':<br><br></div><div>First note that the IA winner cannot have a negative IA-MPO score, because it is ranked on as many (or more) ballots than any other candidate, including the one that gives it max opposition.<br><br></div><div>Next note that if A is ranked top above C on more ballots than C is ranked, then A's pairwise opposition against C is greater than C's IA score, therefore C's IA-MPO score is negative, and therefore smaller than the IA-MPO score of the IA, winner, and therefore not maximal.<br><br></div><div>In a way IA-MPO automatically compensates for voters' hypercautious raising of compromises to equal top status. This should attract voters that don't like Approval because they know that (under Approval) approving their compromise can take the win away from a Condorcet Winner.<br><br></div><div>For this reason, I suggest that even on two slot approval style ballots, we use the Approval-MPO score to determine the winner instead of Approval alone.<br><br></div><div>Forest<br></div><div><br></div></div><div class="gmail_extra"><br><div class="gmail_quote">On Mon, Jun 1, 2015 at 7:10 PM, Forest Simmons <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><div>Chris,<br><br></div>it is interesting to me that IA-MMPO (implicit
approval minus max pairwise opposition) gives the same results as
Approval Seeded by MinGS (etrw) in the four examples that you offered:<br><div class="gmail_extra"><br><div class="gmail_quote"><span class=""><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<br>
46 A<br>
44 B>C<br>
10 C<br>
</blockquote></span><div> The respective IA-MPO scores for A, B, and C are 46-54, 44-46, and 54-46, the only positive one.<br></div><span class=""><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">
46 A<br>
44 B>C (sincere might be B or B>A)<br>
05 C>A<br>
05 C>B</blockquote></span><div>The respective IA-MPO scores are 51-54, 49-51, and 54-46, again the only positive one.<br><br></div><span class=""><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
46 A>C<br>
10 B>A<br>
10 B>C<br>
34 B=C<br></blockquote></span><div>The respective IA-MPO scores are 56-54, 54-46, and 90-56. C wins again.<br><br> <br></div><span class=""><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
40 A>C<br>
15 B>A<br>
20 B<br>
15 C>A<br>
10 C<br></blockquote></span><div>The respective scores are 70-35, 35-65, and 65-55. This time A wins with an IA-MPO score of 35 compared to C's 10.<br><br></div><div>This IA-MPO method does satisfy the FBC, but is not chicken proof. However, small defensive moves can thwart chicken threats.<br><br></div><div>It seems like I might have suggested IA-MPO before, but we were trying for something fancier at the time.<span class="HOEnZb"><font color="#888888"><br></font></span></div></div><span class="HOEnZb"><font color="#888888"><br></font></span></div><span class="HOEnZb"><font color="#888888"><div class="gmail_extra">Forest<br></div><div class="gmail_extra"><br></div></font></span></div>
</blockquote></div><br></div>