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