<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; "><DIV>The "Possible Approval Winner" criterion looks actually quite natural in the sense that it compares the results to what Approval voting could have achieved.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>The definition of the criterion contains a function that can be used to evaluate the candidates (also for other uses) - the possibility and strength of an approval win. This function can be modified to support also cardinal ratings.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>In the first example there is only one entry (11: A>B) that can vary when checking the Approval levels. B can be either approved or not. In the case of cardinal ratings values could be 1.0 for A, 0.0 for C and anything between 0.0001 and 0.9999 for B. Or without normalization the values could be any values between 0.0. and 1.0 as long as value(A) > value(B) > value (C). With the cardinal ratings version it is possible to check what the "original utility values" leading to this group of voters voting A>B could have been (and if the outcome is achievable in some cardinal ratings based method, e.g. max average rating).</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>The max average rating test is actually almost as easy to make as the PAW test. Note that my description of the cardinal ratings for candidate B had a slightly different philosophy. It maintained the ranking order of the candidates, which makes direct mapping from the cardinal values to ordinal values possible. The results are very similar to those of the approval variant but the cardinal utility values help making a more direct comparison with the "original utilities" of the voters.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Now, what is the value of these comparisons when evaluating the different Condorcet methods. These measures could be used quite straight forward in evaluating the performance of the Condorcet methods if one thinks that the target of the voting method is to maximise the approval of the winner or to seek the best average utility. This need not be the case in all Condorcet elections (but is one option). There are several utility functions that the Condorcet completion methods could approximate. The Condorcet criterion itself is majority oriented. Minmax method minimises the strength of interest to change the selected winner to one of the other candidates. Approval and cardinal ratings have somewhat different targets than the majority oriented Condorcet criterion and some of the common completion methods, but why not if those targets are what is needed (or if they bring other needed benefits like strategy resistance).</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>I find it often useful to link different methods and criteria to something more tangible like concrete real life compatible examples or to some target utility functions (as in the discussion above). One key reason for this is that human intuition easily fails when dealing with the cyclic structures (that are very typical cases when studying the Condorcet methods). In this case it seems that PAW and corresponding cardinal utility criterion lead to different targets/utility than e.g. the minmax(margins) "required additional votes to become the Condorcet winner" philosophy. Maybe the philosophy of PAW is to respect clear majority decisions (Condorcet criterion) but go closer to the Approval/cardinal ratings style evaluation when the majority opinion is not clear. You may have different targets in your mind but for me this was the easiest interpretation.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Juho</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>P.S. One example.</DIV><DIV>1: A>B</DIV><DIV>1: C</DIV><DIV>Here B could be an Approval winner (tie) but not a max average rating winner in the "ranking maintaining style" that was discussed above (since the rating of B must be marginally smaller than the rating of A in the first ballot).</DIV><DIV><BR class="khtml-block-placeholder"></DIV><BR><DIV><DIV>On Mar 7, 2007, at 16:28 , Chris Benham wrote:</DIV><BR class="Apple-interchange-newline"><BLOCKQUOTE type="cite"> <BR> Juho wrote (March7, 2007):<BR> <BLOCKQUOTE type="cite"> <PRE wrap="">The definition of plurality criterion is a bit confusing. (I don't
claim that the name and content and intention are very natural
either :-).)
- <A class="moz-txt-link-freetext" href="http://wiki.electorama.com/wiki/Plurality_criterion">http://wiki.electorama.com/wiki/Plurality_criterion</A> talks about
candidates "given any preference"
- Chris refers to "above-bottom preference votes" below
</PRE> </BLOCKQUOTE> <BLOCKQUOTE type="cite"><P><EM>If the number of ballots ranking <I>A</I> as the first preference is greater than the number<BR> of ballots on which another candidate <I>B</I> is given any preference, then <I>B</I> must not be elected.</EM></P> </BLOCKQUOTE> <BLOCKQUOTE type="cite"> <PRE wrap="">Electowiki definition could read: "If the number of voters ranking A
as the first preference is greater than the number of voters ranking
another candidate B higher than last preference, then B must not be
elected".</PRE> </BLOCKQUOTE> Yes it could and to me it in effect does (provided "last" means "last or equal-last") The criterion come <BR> from Douglas Woodall who economises on axioms so doesn't use one that says that with three candidates <BR> A,B,C a ballot marked A>B>C must always be regarded as exactly the same thing as A>B truncates. He <BR> assumes that truncation is allowed but above bottom equal-ranking isn't.<BR> <BR> A similar criterion of mine is the "Possible Approval Winner" criterion:<BR> <BR> "Assuming that voters make some approval distinction among the candidates but none among those<BR> they equal-rank (and that approval is consistent with ranking) the winner must come from the set of <BR> possible approval winners".<BR> <BR> This assumes that a voter makes some preference distinction among the candidates, and that truncated<BR> candidates are equal-ranked bottom and so never approved. <BR> <BR> Looking at a profile it is very easy to test for: considering each candidate X in turn, pretend that the<BR> voters have (subject to how the criterion specifies) placed their approval cutoffs/thresholds in the way<BR> most favourable for X, i.e. just below X on ballots that rank X above bottom and on the other ballots<BR> just below the top ranked candidate/s, and if that makes X the (pretend) approval winner then X is<BR> in the PAW set and so permitted to win by the PAW criterion.<BR> <BR> <PRE wrap="">11: A>B
07: B
12: C</PRE> So in this example A is out of the PAW set because in applying the test A cannot be more approved<BR> than C.<BR> <BR> IMO, methods that use ranked ballots with no option to specify an approval cutoff and rank among<BR> unapproved candidates should elect from the intersection of the PAW set and the Uncovered set<BR> <BR> One of Woodall's "impossibility theorems" states that is impossible to have all three of Condorcet,<BR> Plurality and Mono-add-Top. MinMax(Margins) meets Condorcet and Mono-add-Top.<BR> <BR> Winning Votes also fails the Possible Approval Winner (PAW) criterion, as shown by this interesting<BR> example from Kevin Venzke:<BR> <BR> <PRE wrap="">35 A
10 A=B
30 B>C
25 C
A>B 35-30, B>C 40-25, C>A 55-45
Both Winning Votes and Margins elect B, but B is outside the PAW set{A,C}.
Applying the test to B, we get possible approval scores of A45, B40, C25.
ASM(Ranking) and DMC(Ranking) and Smith//Approval(Ranking) all meet the Definite
Majority(Ranking) criterion which implies compliance with PAW. The DM(R) set is
{C}, because interpreting ranking (above bottom or equal-bottom) as approval, both
A and B are pairwise beaten by more approved candidates.
Chris Benham
</PRE> <BR> </BLOCKQUOTE></DIV><BR></BODY></HTML>