In his response under this subject heading Mr. Lomax seemd to think that I was advocating Cumulative Voting, then he offered what amounted to a plausibility argument for my assertion that Cumulative Voting is strategically equivalent to Plurality. I'm slightly miffed that he would imply that I advocated a method equivalent to Plurality. But the main point of this posting is to offer geometric insight into the equivalences that I mentioned before. If our ballots are vectors of numbers (to be used additively) representing levels of support for candidates, then some limit has to be placed on the numbers or the election game reduces to "who can name the biggest number?" One way to limit the vectors is to limit a norm of the vector. If we limit the max norm, then we get range voting, whether or not we allow negative numbers. If we limit the L_1 norm, a.k.a. the Hamming norm, which is the sum of the absolute values, then we get Cumulative Voting when only positive numbers are allowed, but we get something else if negative numbers are allowed. The geometry is this: In the case of the max norm we get squares, boxes, and their higher dimensional analogues no matter whether negatives are allowed or not. The corners of these boxes are the optimal strategies which amount to approval strategies. The only corners that never yield optimal strategies are two corners corresponding to vectors with all components equal. (all max or all min). In the case of the Hamming norm with negative numbers allowed, we get a diamond, an octogon, and higher dimensional versions of these. The corners of the octogon correspond to the optimal "xor" for xor against strategies. In the case of the Hamming norm with only positive numbers allowed, we get triangles, tetrahedra, and simplexes of different dimensions. The corners (except the all zero corner) correspond to optimal Plurality strategies. When we have three candidates, the approval box has two more corners than the xor octogon, but the two extra ones, (1,1,1) and its opposite, serve no purpose in approval. This is the geometry behind the equivalence of approval and xor in the three candidate case, and their equivalence with their respective continuous versions. Forest
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