I won't beat Demorep or Richard on this, but here are my two entries: I. In the first method, the left bit is the approval bit, and is the only instrumental bit, i.e. the only bit with any possible effect on the outcome. The right bit is purely expressive, for psychological, educational and mandating purposes. The candidate with the greatest number of left column ovals marked is the winner. II. The second method (my favorite two bit method) is similar to Ranked Pairs, only instead of winning votes or margin size determining the strength of each pair, it's the number of ballots on which the pair straddles the big gap, i.e. the number of ballots on which the two candidates in the pair have ratings that differ in the most significant bit. If there is no pairwise unbeaten candidate, then weak pairs are zeroed out (starting at the weakest) until there is an unbeaten candidate. In this method it is useful for the voter to think of the two bit ratings as base ten expressions: 00=zero < 01=one << 10=ten < 11=eleven The gap sizes count for nothing if there is a pairwise unbeaten candidate. Otherwise, the relative gap sizes (big vs small or zero) help determine the strength of the pairs. It is necessary to compute two matrices: the pairwise margin matrix, as well as the pair strength matrix. Both of these matrices are n by n matrices, where n is the number of candidates. The margin matrix is antisymmetric, and the strength matrix is symmetric with zeros down the main diagonal. So there are only n(n-1) numbers to be carried in the precinct summaries. This method satisfies both the Condorcet Criterion and the FBC. Because of the symmetry properties of the margin and strength matrices it should also satisfy the Reverse Symmetry Criterion. The reason I prefer this method above Demorep's two bit method is that for more than three candidates, this method is likely to resolve all cycles before reducing all of the way down to Approval. On the other hand, Demorep's method is easier to explain. In a future posting I'll try to come up with some good examples that illustrate the advantages of the second method. Meanwhile, feel free to try to discredit it with some favorite example of your own. Forest
