<html><head><style type="text/css"><!-- DIV {margin:0px;} --></style></head><body><div style="font-family:times new roman, new york, times, serif;font-size:12pt"><DIV>The examples I used to demonstrate my suggested "Unmanipulable Majority"<BR>criterion have inspired me to suggest another two (very similar to each</DIV>
<DIV>other, and not including my recent "Push-over Invulnerability" suggestion):</DIV>
<DIV> </DIV>
<DIV>"Comprehensive 3-slot Ratings Winner":</DIV>
<DIV><BR>*If no voter expresses more than three preference-levels and the ballot</DIV>
<DIV>rules allow the expression of 3 preference-levels when there are 3 (or </DIV>
<DIV>more) candidates, then (interpreting candidates that are voted above one</DIV>
<DIV>or more candidates and below none as "top-rated", those voted above</DIV>
<DIV>one or more candidates but below all the top-rated candidates as </DIV>
<DIV>"middle-rated" and those not voted above any other candidate and below</DIV>
<DIV>at least one other candidate as "bottom-rated", and interpreting above-</DIV>
<DIV>bottom rating as approval) it must not be possible for candidate X to</DIV>
<DIV>win if there is some candidate Y with simultaneously a higher Top-Ratings</DIV>
<DIV>score, a higher Approval score and a lower Maximum Approval-Opposition<BR>score.*</DIV>
<DIV> </DIV>
<DIV>(Recall that a candidate Y's MAO score is the approval score of the most</DIV>
<DIV>approved candidate only on ballots that don't approve Y.)<BR></DIV>
<DIV>This may seem a bit long-winded and a bit arbitrary, but the standard it<BR>seeks to capture is that in relatively simple situations the obviously strongest</DIV>
<DIV>candidate shouldn't lose, such as C in the following example:</DIV>
<DIV><BR>25: A>B<BR>26: B>C<BR>23: C>A<BR>26: C<BR></DIV>
<DIV>TR scores: C49, B26, A25</DIV>
<DIV>App. scores: C75, B51, A48<BR>MAO scores: C25, B49, A52</DIV>
<DIV> </DIV>
<DIV>I think all determinist methods that don't elect C here must be vulnerable to<BR>Push-over strategy. As noted in my last posting, this is true of IRV and</DIV>
<DIV>Winning Votes and MAMPO and AWP which all elect B.</DIV>
<DIV> </DIV>
<DIV>Unfortunately this criterion is incompatible with Condorcet, as demonstrated<BR>by another example from my UM post:</DIV>
<DIV> </DIV>
<DIV>93: A<BR>09: B>A<BR>78: B<BR>14: C>B<BR>02: C>A<BR>04: C</DIV>
<DIV> </DIV>
<DIV>
<DIV>TR scores: A93, B87, C20</DIV>
<DIV>App. scores: A104, B101, C20<BR>MAO scores: A92, B95, C102<BR></DIV>
<DIV>The "C3RW" criterion says that A must win, but B is the Condorcet winner</DIV>
<DIV>(B>A 101-95, B>C 87-20).<BR></DIV>
<DIV>But I'm still interested in Condorcet methods, so I suggest "Smith-C3RW",</DIV>
<DIV>which specifies that the candidate Y disallowing the election of candidate X<BR>must have a beatpath to X.<BR><BR>*If no voter expresses more than three preference-levels and the ballot
<DIV>rules allow the expression of 3 preference-levels when there are 3 (or </DIV>
<DIV>more) candidates, then (interpreting candidates that are voted above one</DIV>
<DIV>or more candidates and below none as "top-rated", those voted above</DIV>
<DIV>one or more candidates but below all the top-rated candidates as </DIV>
<DIV>"middle-rated" and those not voted above any other candidate and below</DIV>
<DIV>at least one other candidate as "bottom-rated", and interpreting above-</DIV>
<DIV>bottom rating as approval) it must not be possible for candidate X to</DIV>
<DIV>win if there is some candidate Y which has a beat-path to X and </DIV>
<DIV>simultaneously higher Top-Ratings and Approval scores and a lower </DIV>
<DIV>Maximum Approval-Opposition score.*<BR><BR>This clearly failed by Winning Votes and met by Schwartz//Approval.</DIV>
<DIV>(In the first example, C has a beatpath to B.)</DIV>
<DIV> </DIV>
<DIV>As with "Push-over Invulnerability" I strongly suspect that it is met by</DIV>
<DIV>Margins. </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>Chris Benham<BR></DIV>
<DIV> </DIV></DIV><BR><BR><BR><BR></DIV>
<DIV> </DIV></div><br>
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