"For your eyes only." Warning: do not proceed past this point if you don't like lottery methods. Ballots are approval style. Ballots are counted in two different ways: (1) the approval count and (2) the fractional (cumulative) count, which means the candidates marked on a ballot are assigned equal fractional scores that add up to unity, i.e. if five candidates are approved on a ballot, then that ballot contributes one fifth of a point to the count of each of its approved candidates. Each candidate is assigned a unique color. We create a rainbow with the color bands in the order determined by the first count, and each band thickness determined by the second count. Halfway* through the thickness of the rainbow we make a cut, and throw away the half of the rainbow that has (most of, if not all of) the color of the candidate with lowest approval. A randomly chosen voter (or his proxy) picks one of the colors still represented in the remaining half of the rainbow (even if part of a chosen color band has been excised). The candidate that corresponds to the chosen color wins the election. This method satisfies Pareto, Clone Independence, and Monotonicity. (Proofs supplied on request) If a majority "bullet votes" for the same candidate, then that candidate will surely be elected. * "Halfway" can be adjusted to some other fraction of the way through the rainbow for special purpose methods. For example eighty percent of the way from the high approval side to the low approval side of the rainbow might be appropriate in a country like Rwanda, where there is a marked 80/20 ethnic split in the population. On the other hand, if the cutoff is adjusted extremely close to the high approval side, then almost surely only the color of the approval winner will remain after the discard. Isn't that nifty? Forest
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