<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>Part of my demonstration of many methods' failure of the Unmanipulable Majority</DIV>
<DIV>criterion has inspired me to suggest another strategy criterion: </DIV>
<DIV> </DIV>
<DIV>"Push-over Invulnerability":<BR>*It must not be possible to change the winner from candidate X to candidate Y by<BR>altering some ballots (that vote Y above both candidates X and Z) by raising Z above<BR>Y without changing their relative rankings among other (besides X and Z) candidates.*</DIV>
<DIV> </DIV>
<DIV>I might later suggest a more elegant re-wording, and/or suggest a simplified approximation</DIV>
<DIV>that is easier to test for.</DIV>
<DIV> </DIV>
<DIV>25: A>B<BR>26: B>C<BR>23: C>A<BR>26: C<BR><BR>B>C 51-49, C>A 75-25, A>B 48-26<BR><BR>Schulze/RP/MM/River (WV) and Approval-Weighted Pairwise and DMC and MinMax(PO)<BR>and MAMPO and IRV elect B.<BR><BR>Now say 4 of the 26C change to A>C (trying a Push-over strategy):<BR><BR>25: A>B<BR>04: A>C<BR>26: B>C<BR>23: C>A<BR>22: C<BR></DIV>
<DIV>B>C 51-49, C>A 71-29, A>B 52-26<BR></DIV>
<DIV>Now Schulze/RP/MM/River (WV) and AWP and DMC and MinMax(PO) and MAMPO<BR>and IRV all elect C. </DIV>
<DIV> </DIV>
<DIV>For a long time I thought that only "non-monotonic" methods like IRV and Raynaud (that</DIV>
<DIV>fail mono-raise) were vulnerable to Push-over, so therefore there was no need for a separate</DIV>
<DIV>"Push-over Invulnerability" criterion.</DIV>
<DIV><BR>But now we see that the Schulze, Ranked Pairs, MinMax, River algorithms (all equivalent with 3<BR>candidates) using Winning Votes are all vulnerable to Push-over (as my suggested criterion</DIV>
<DIV>defines it).</DIV>
<DIV> </DIV>
<DIV>Now I know that Winning Votes' failure can be seen as functionally "really" a failure of Later-no-help,</DIV>
<DIV>because those C-supporting strategists could more safely achieve the same end just by changing</DIV>
<DIV>their votes from C to C>A instead of from C to A>C. But that is hardly a bragging point for WV.</DIV>
<DIV> </DIV>
<DIV>I think this Pushover criterion can be seen as a kind of "monotonicity" criterion, in the sense that all</DIV>
<DIV>else being equal methods that meet it must be in some way "more monotonic" than those that don't.</DIV>
<DIV> </DIV>
<DIV>I have shown that WV fails "Pushover Invulnerability". I strongly suspect (but not at present up to</DIV>
<DIV>proving) that both Margins and Schwartz//Approval (ranking) meet it.</DIV>
<DIV> </DIV>
<DIV>Can anyone please give an example (or examples) that show that either or both of Margins and<BR>S//A(r) fail my suggested "Push-over Invulnerability" criterion?</DIV>
<DIV> </DIV>
<DIV>Chris Benham<BR></DIV></div><br>
<hr size=1>
Start your day with Yahoo!7 and win a Sony Bravia TV. <a href="http://au.rd.yahoo.com/hppromo/mail/tagline2/*http://au.docs.yahoo.com/homepageset/?p1=other&p2=au&p3=tagline" target=_blank>Enter now</a>.</body></html>