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<DIV><STRONG>Kristofer Munsterhjelm</STRONG> wrote (Sun. Aug.10):<BR></DIV>
<DIV>"There's also the "it smells fishy" that nonmonotonicity - of any kind or <BR>frequency - evokes. I think that's stronger for nonmonotonicity than for <BR>things like strategy vulnerability because it's an error that appears in <BR>the method itself, rather than in the move-countermove "game" brought on <BR>by strategy, and thus one thinks "if it errs in that way, what more <BR>fundamental errors may be in there that I don't know of?". But that <BR>enters the realm of feelings and opinion."</DIV>
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<DIV>Kristopher,<BR>The intution or "feeling" you refer to is based on the idea that the best method/s<BR>must be mathematically elegant and that methods tend to be consistently good</DIV>
<DIV>or consistently bad.<BR></DIV>
<DIV>But in the comparison among reasonable and "good" methods, this idea is wrong.</DIV>
<DIV>Rather it is the case that many arguably desirable properties (criteria compliances)</DIV>
<DIV>are mutually incompatible. So on discovering that method X has some mathematically</DIV>
<DIV>inelegant or paradoxical flaw one shouldn't immediately conclude that X must be<BR>one of the worst methods. That "flaw" may enable X to have some other desirable</DIV>
<DIV>features.</DIV>
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<DIV>To look at it the other way, Participation is obviously interesting and viewed in isolation</DIV>
<DIV>a desirable property. But I know that it is quite "expensive", so on discovering that method</DIV>
<DIV>Y meets Participation I know that it must fail other criteria (that I value) so I don't expect</DIV>
<DIV>Y to be one of my favourite methods. <BR><BR>"I think that all methods that work by calculating the ranking according <BR>to a positional function, then eliminating one or more candidates, then <BR>repeating until a winner is found will suffer from nonmonotonicity. I <BR>don't know if there's a proof for this somewhere, though.<BR><BR>A positional function is one that gives a points for first place, b <BR>points for second, c for third and so on, and whoever has the highest <BR>score wins, or in the case of elimination, whoever has the lowest score <BR>is eliminated.<BR><BR>Less abstractly, these methods are nonmonotonic if I'm right: Coombs <BR>(whoever gets most last-place votes is eliminated until someone has a <BR>majority), IRV and Carey's Q method (eliminate loser or those with below <BR>average plurality scores, respectively), and Baldwin and Nanson (the <BR>same, but with Borda)."</DIV>
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<DIV>That's right, but I think that Carey's method (that I thought was called "Improved FPP")</DIV>
<DIV>is monotonic (meets mono-raise) when there are 3 candidates (and that is the point of it.)<BR><BR>"It may be that this can be formally proven or extended to other <BR>elimination methods. I seem to remember a post on this list saying that <BR>Schulze-elimination is just Schulze, but I can't find it. If I remember <BR>correctly, then that means that not all elimination methods are <BR>nonmonotonic."</DIV>
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<DIV>Of course Schulze isn't a "positional function". Obviously if there are just 3 candidates in</DIV>
<DIV>the Schwartz set then "Schulze-elimination" must equal Schulze, but maybe there is some</DIV>
<DIV>relatively complicted example where there are more than 3 candidates in the top cycle</DIV>
<DIV>where the two methods give a different result.</DIV>
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<DIV>Chris Benham<BR><!--endarticle--><!--htdig_noindex--><BR></DIV>
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