Here's the simplest idea along these lines that seems promising to me. I'm more excited about it than I have been any of my other ideas in the last few weeks:
Each voter votes a CR ballot on which one of the candidates is the virtual Borderline Acceptable Candidate (BAC). On each ballot BAC's rating is considered the nominal approval cutoff. [Conversely, if the CR ballots allow approval cutoff specification in some other manner, then it is assumed that BAC receives the rating so specified on any such ballot.] The first round approvals are based on the nominal approval cutoff. In this first round the BAC candidate is considered to have exactly 50 percent approval, since any candidate with more than 50 percent approval beats BAC head-to-head, and any candidate with less than 50 percent approval loses to BAC head-to-head, and BAC itself neither beats nor loses to BAC head-to-head. The second round approval cutoff is given by the expression (r2+2*r1)/3 where r1 and r2 are the ratings of the top two approval candidates in the first round. The key new idea here is that the virtual candidate BAC may or may not be one of these top two candidates. There are (excluding ties) three cases: (1) BAC is the first round approval winner. (2) BAC is the second place approval winner in the first round. (3) BAC doesn't place in the top two in the first round. These three cases can be recast without reference to BAC: (1) No flesh and blood candidate gets more than 50 percent nominal approval. (2) Exactly one candidate gets more than fifty percent nominal approval. (3) More than one candidate gets more than fifty percent nominal approval. Observe how the method exerts a pressure against low utility candidates: The fewer candidates that achieve more than 50 percent nominal approval, the more the user specified approval cutoff counts. If the two highest approval winners beat BAC, then the user specified approval cutoff is not needed in the second round to bolster the utility. The whole thing can be formulated without using or referring to the BAC, but this BAC formulation shows the rhyme and reason behind the apparently ad hoc rules of the non BAC formulation below: Voters fill out CR ballots, including (perhaps voter specified) nominal approval cutoffs. First round approval counts are tallied. If no candidate gets more than fifty percent nominal approval, then the final approval cutoff on each ballot is given by the expression (2*a1+r1)/3 where a1 is the nominal approval on the ballot, and r1 is the rating on the ballot of the (real) candidate with the highest nominal approval. If exactly one candidate gets more than fifty percent nominal approval, then on each ballot the final approval cutoff is determined by the expression (a1+2*r1)/3 where a1 is the nominal approval indicated on the ballot, and r1 is the rating on the ballot of the candidate who achieved more than fifty percent approval in the first round. If two candidates achieve more than fifty percent nominal approval, then on each ballot the final cutoff is calculated as (2*r1+r2)/3 where r1 and r2 are the respective ratings on the ballot of the top two nominal approval candidates. If the CR is on a scale from zero to 100 percent, and if the nominal approval cutoffs are taken as fifty percent (as opposed to user specified cutoffs), then the three final cutoff values in the respective cases are given by (100% + r1)/3, (50% + 2*r1)/3, and (2*r1 + r2)/3 . The method is summable in (n+1) by (n+1) matrices, where n is the number of real candidates. I would be very interested in any example where this method appears to give the win to the wrong candidate or where it would give incentive to (insincerely) collapse ratings in high resolution CR. Forest ---- Election-methods mailing list - see http://electorama.com/em for list info
