A Cardinal Ratings style ballot is a ballot that allows the voter to rate each candidate on some numerical scale, for example the Olympic scale of zero to ten.
The standard use of such a ballot is to award the win to the candidate with the highest average rating, where the average is taken over all ballots. It is well known that this standard CR voting method is strategically equivalent to Approval, i.e. the intermediate ballot slots are superfluous for the voter who knows how to maximize the influence of his/her ballot. An approval voter in possession of accurate probabilities could maximize the power or likelihood of his/her vote being pivotal (in a favorable direction) by marking the ballot in such a way that the following two conditions are satisfied: (1) No unapproved candidate is preferred by the voter over any approved candidate. (2) The likelihood of an approved candidate winning is as close as possible to the likelihood of an unapproved candidate winning. Ideally, in a close race with many candidates, the odds would be fifty/fifty, i.e. equally likely that one of your approved (versus one of your unapproved) candidates would win. In the case of only two viable candidates it is easy to see that if those two candidate were tied without your vote, then (until your decisive vote was cast) the odds of one or the other winning would be exactly fifty/fifty. Lumping your preferred of the two with your approved and relegating the other among the unapproved would satisfy the above two conditions, and your ballot would tip the scales in a (according to you) favorable direction. In general, when these two conditions are satisfied there is a maximum probability that your ballot will actually tip the scales from a candidate who would have won had you not cast your ballot to one that you prefer. Unfortunately, in most elections it isn't likely that any voter will actually be in possession of the necessary probabilities. But suppose that you could start with a Cardinal Ratings ballot and gradually convert it into an Approval ballot as the candidate viabilities became apparent during a step-by-step process. In particular, suppose that at each stage of the process current CR scores for the candidates were available to you, and that you were required to make one step of progress towards your final approval ballot by amalgamating two adjacent slots of your CR ballot. [Successive steps of this type ensure automatic satisfaction of condition (1) above.] Since your goal is to end up with viability balanced as nearly as possible between the approved and unapproved candidates, it would make sense to absorb the current minimum viability slot into one of its neighbors. If you absorb it (by amalgamation) into a neighboring slot with minimal viability, then you (hueristically) minimize the chance of regretting that they were not on opposite sides of the final ballot. By the way, at each stage you judge the relative viability of candidates by their CR scores at that stage, since that is the most up-to-date estimate available, superseding even the most accurate pre-election polls. This whole process can easily be automated so that you don't have to worry about the calculations or ballot modifications. The automated version is what I call Max Power Cardinal Ratings (MPCR). The voters fill out CR ballots. MPCR converts these ballots into approval ballots according to the above process. The candidate with the most approval on these final ballots is the method winner. Forest ---- Election-methods mailing list - see http://electorama.com/em for list info
