<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>Regarding my proposed Unmanipulable Majority criterion:</DIV>
<DIV> </DIV>
<DIV>*If (assuming there are more than two candidates) the ballot <BR>rules don't constrain voters to expressing fewer than three <BR>preference-levels, and A wins being voted above B on more <BR>than half the ballots, then it must not be possible to make B <BR>the winner by altering any of the ballots on which B is voted <BR>above A without raising their ranking or rating of B.*</DIV>
<DIV> </DIV>
<DIV>To have any point a criterion must be met by some method.</DIV>
<DIV> </DIV>
<DIV>It is met by my recently proposed SMD,TR method, which I introduced</DIV>
<DIV>as "3-slot SMD,FPP(w)":</DIV>
<DIV><BR>*Voters fill out 3-slot ratings ballots, default rating is bottom-most<BR>(indicating least preferred and not approved).<BR><BR>Interpreting top and middle rating as approval, disqualify all candidates<BR>with an approval score lower than their maximum approval-opposition </DIV>
<DIV>(MAO) score.<BR>(X's MAO score is the approval score of the most approved candidate on<BR>ballots that don't approve X).<BR><BR>Elect the undisqualified candidate with the highest top-ratings score.*</DIV>
<DIV> </DIV>
<DIV>Referring to the UM criterion: (a) if candidate A has a higher TR score than B<BR>then the B>A strategists can only make B win by causing A to be disqualified.</DIV>
<DIV>But in this method it isn't possible to vote x above y without approving x, so</DIV>
<DIV>we know that just on the A>B ballots A has majority approval. It isn't possible<BR>for a majority-approved candidate to be disqualified, and the strategists can't<BR>cause A's approval to fall below majority-strength. And the criterion specifies</DIV>
<DIV>that none of the B>A voters who don't top-rate B can raise their rating of B to</DIV>
<DIV>increase B's TR score.</DIV>
<DIV> </DIV>
<DIV>(b) if on the other hand B has a higher TR score than A but B is disqualified<BR>there is nothing the B>A strategists can do to undisqualify B.</DIV>
<DIV><BR>So SMD,TR meets the UM criterion.</DIV>
<DIV><BR>93: A<BR>09: B>A<BR>78: B<BR>14: C>B<BR>02: C>A</DIV>
<DIV>04: C<BR>200 ballots</DIV>
<DIV><BR>B>A 101-95, B>C 87-20, A>C 102-20.</DIV>
<DIV>All Condorcet methods, plus MDD,X and MAMPO and ICA elect B.<BR><BR>B has a majority-strength pairwise win against A, but say 82 of the 93A change to<BR>A>C thus:</DIV>
<DIV><BR>82: A>C</DIV>
<DIV>11: A<BR>09: B>A<BR>78: B<BR>14: C>B<BR>02: C>A</DIV>
<DIV>04: C</DIV>
<DIV> </DIV>
<DIV>B>A 101-95, C>B 102-87, A>C 102-20<BR>Approvals: A104, B101, C102</DIV>
<DIV>TR scores: A93, B87, C 20</DIV>
<DIV> </DIV>
<DIV>Now MDD,A and MDD,TR and MAMPO and ICA and Schulze/RP/MinMax etc. using </DIV>
<DIV>WV or Margins elect A. So all those methods fail the UM criterion.<BR></DIV>
<DIV>25: A>B<BR>26: B>C</DIV>
<DIV>23: C>A</DIV>
<DIV>26: C<BR></DIV>
<DIV>B>C 51-49, C>A 75-25, A>B 48-26</DIV>
<DIV> </DIV>
<DIV>Schulze/RP/MM/River (WV) and Approval-Weighted Pairwise and DMC and MinMax(PO)</DIV>
<DIV>and MAMPO and IRV elect B.</DIV>
<DIV> </DIV>
<DIV>Now say 4 of the 26C change to A>C (trying a Push-over strategy):<BR><BR>
<DIV>25: A>B</DIV>
<DIV>04: A>C<BR>26: B>C</DIV>
<DIV>23: C>A</DIV>
<DIV>22: C</DIV>
<DIV> </DIV>
<DIV>
<DIV>B>C 51-49, C>A 71-29, A>B 52-26</DIV>
<DIV><BR>Now Schulze/RP/MM/River (WV) and AWP and DMC and MinMax(PO) and MAMPO</DIV>
<DIV>and IRV all elect C. Since B had/has a majority-strength pairwise win against C, all these<BR>methods also fail Unmanipulable Majority. If scoring ballots were used and all voters score<BR>their most preferred candidate 10 and any second-ranked candidate 5 and unranked candidates<BR>zero, then this demonstration also works for IRNR so it also fails.<BR></DIV>
<DIV>Who knew that such vaunted "monotonic" methods as WV and MinMax(PO) and MAMPO<BR>were vulnerable to Push-over?!</DIV>
<DIV><BR>48: A>B<BR>01: A<BR>03: B>A</DIV>
<DIV>48: C>B</DIV>
<DIV> </DIV>
<DIV>B>A 51-49. Bucklin and MCA elect B, but if the 48 A>B voters truncate the winner changes</DIV>
<DIV>to A. So those methods also fail UM.<BR><BR>49: A9, B8, C0</DIV>
<DIV>24: B9, A0, C0<BR>27: C9, B8, A0</DIV>
<DIV> </DIV>
<DIV>Here Range/Average Ratings/Score/CR elects B and on more than half the ballots B is voted </DIV>
<DIV>above A, but if the 49 A9, B8, C0 voters change to A9, B0, C0 the winner changes to A.<BR>So this method fails UM.</DIV>
<DIV> </DIV>
<DIV>48: A>B>C>D</DIV>
<DIV>44: B>A>D>C</DIV>
<DIV>04: C>B>D>A</DIV>
<DIV>03: D>B>C>A</DIV>
<DIV> </DIV>
<DIV>Here Borda elects B and B is voted above A on more than half the ballots, but if the 48 </DIV>
<DIV>A>B>C>D ballots are changed to A>C>D>B the Borda winner changes to A, so</DIV>
<DIV>Borda fails UM.<BR></DIV>
<DIV>This Unmanipulable Majority criterion is failed by all well known and currently advocated</DIV>
<DIV>methods, except 3-slot SMD,TR!</DIV>
<DIV><BR>Given its other criterion compliances and simplicity, that is my favourite 3-slot s-w method</DIV>
<DIV>and my favourite Favourite Betrayal complying method.</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>Chris Benham</DIV></DIV>
<DIV> </DIV>
<DIV><BR> </DIV></DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV><BR><BR><BR><BR><BR> </DIV></div><br>
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