I must be missing something. Could you give an example in which the approval winner is not the winner of the method?
Forest On Mon, 10 Mar 2003, Steve Eppley wrote: > On 10 Mar 2003 at 11:36, Kevin Venzke wrote: > > My recent "MinMax" message concluded with a > > half-hearted attempt at a system combining Approval > > and Condorcet. I have a much better proposal now, > > although I'm not entirely certain of its merits. > -snip- > > I have another way of combining Approval and Condorcet, > actually a family of voting methods, plus a criterion they > satisfy that is stronger than Mike Ossipoff's "Strong > Defensive Strategy Criterion" (a variation of which I call > the Minimal Defense criterion). > > 1. Each voter is allowed to (non-strictly) order the > candidates from top to bottom, and optionally > may insert a "dividing line" anywhere in her ordering > (that partitions the candidates into two subsets, > those over the line and those under). > > Given a touchscreen voting interface, it would be > straightforward to implement #1, since the dividing line > could be dragged and dropped into the desired position just > like any other candidate. Given paper ballots that would > be optically scanned, the following format would suffice: > > <--BETTER WORSE--> > Bradley (X) ( ) ( ) ( ) ( ) > Nader (X) ( ) ( ) ( ) ( ) > Gore ( ) (X) ( ) ( ) ( ) > Bush ( ) ( ) (X) ( ) ( ) > Buchanan ( ) ( ) ( ) ( ) ( ) > McCain ( ) (X) ( ) ( ) ( ) > Dole ( ) ( ) (X) ( ) ( ) > Keyes ( ) ( ) (X) ( ) ( ) > DIVIDING LINE: ( ) (X) ( ) ( ) > > Each voting method in the family constructs a social > ordering consistent with the following: > > 2. For all pairs of candidates, say x & y, y is socially > ordered over x if the number of votes that rank "y over > x" exceeds the number of votes that rank "x over y" > and the number of votes that rank "y over the dividing > line over x" exceeds the number of votes that > rank "x over the dividing line over y." > > It's not actually necessary to construct such a social > ordering, as long as the following condition is met: > > For all candidates x, x must not be elected if there > exists a candidate y such that the number of votes > that rank "y over x" exceeds the number of votes > that rank "x over y" and the number of votes that > rank "y over the dividing line over x" exceeds the > number of votes that rank "x over the dividing line > over y." > > Assuming only one winner is to be elected, it's always > possible to satisfy condition 2 (or the revised wording) > since the subset of pairings that meet condition 2 is > acyclic. (I have a proof of acyclicity, but it's tedious so > I won't post it without a request.) > > The strong criterion satisfied by these methods is: > > Sincere Defense: For all subsets X of the candidates, > all subsets C of voters and all candidates y, > if C includes more than half of the voters and > every member of C prefers y over every candidate in X, > then there must exist a way that the members of C > can vote that ensures all candidates in X will lose > and does not require any member of C to misrepresent > any preferences. > > Sincere Defense is stronger than Minimal Defense, which is > stronger than Strong Defensive Strategy Criterion, since > Minimal Defense and SDSC allow a majority coalition to > misrepresent some preferences (by downranking candidate(s) > to ensure their defeat). > > For an example of a preference order method that can be > tweaked to satisfy Sincere Defense, MAM and other > variations of Ranked Pairs can be tweaked to allow each > voter to insert the dividing line in her ranking as in #1 > above, and to give utmost precedence to every pairwise > majority that meets the condition in #2 above. > > I'm concerned that some voters wouldn't use the dividing > line strategically as intended, and instead treat it as > some sort of "sincere approval" dividing line. In that > case, the dividing line may not have much force because > condition 2 wouldn't be met by as many pairwise majorities. > For instance, some Nader voters might rank Gore and Bush > below the line even though ranking Gore over the line would > be more effective (by creating a majority voting "Gore over > the line over Bush" that would ensure the defeat of Bush, > who is the Nader voters' "greater evil"). So even though > many voting methods could be tweaked to be in the family > that satisfies Sincere Defense, only the best methods > should be considered. In particular, by tweaking a method > that satisfies Minimal Defense, the "minimal defensive > strategy" of downranking X can be simultaneously employed > as a second line of defense. > > -- Steve Eppley > > _______________________________________________ > Election-methods mailing list > [EMAIL PROTECTED] > http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com > _______________________________________________ Election-methods mailing list [EMAIL PROTECTED] http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com