In score voting (also called range voting), each voter can assign a score to each candidate. The scores come from some range and granularity defined by the election designers (e. g., zero up through one by hundredths). A voter need not score all the candidates. The average score for each candidate across the voters who scored that candidate, is calculated (arithmetic mean). The candidate with the highest average score wins. In cases where the electorate is large, about 1000 artificial ballots should be thrown in giving all the candidates the lowest score. This prevents a small contingent from electing a candidate about whom no one else knows anything.
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Weighted Net Approval voting is a type of range or score voting, but uses a net total rather than an arithmetic mean, and treats direct opposition to a candidate as the additive inverse of direct support for that same candidate. One vote = one ballot, but on that one ballot
the voter may express any of a range of positions from full opposition to full support or endorsement for each and any candidate and need not express an opinion about EVERY candidate. The indications of support or opposition are converted into numbers according to a predetermined scale, such as for example +7 for full support, +6 for very strong support, +5 for strong support, +4 for moderate support, +3 for weak support, +2 for very weak support, +1 for almost no support, 0 for neutral, -1 for almost no opposition, -2 for very weak opposition, -3 for weak opposition, -4 for moderate opposition, -5 for strong opposition, -6 for very strong opposition, and -7 for full opposition. These could just as easily be real numbers ranging from -1 for absolutely against to +1 for absolutely in support of a candidate winning, but the range and granularity must be predetermined and agreed upon, and the same for all voters and all candidates. It is also MY recommendation that words rather than numbers be used on the actual ballot, except for a list of conversions available to the voter so that one may cast an “informed” vote, and that words be carefully chosen to be as unambiguous as possible and the conversion to numeric form be shown directly on the ballot to remove any possible ambiguity, with the numbers making the order obvious while the words remove any doubt as to the usage of each number. For example, one of the problems with a non-negative number only range starting at 0 is that many voters understand 0 to be full direct opposition while others see 0 as weak opposition and yet others see 0 as identical to no number being indicated. When calculating a mean, an indication of 0 is OBVIOUSLY NOT the same as “not voting” but the exact meaning of a “mean of zero” may still be left undefined. In #WeightedNetApproval voting, 0 is neutral and unambiguous, and may be indicated as simply as “NEUTRAL” or “NEITHER SUPPORTING NOR OPPOSING” on the ballot.
I wrote this response because I was asked for a simple algorithm, and this page was sent to me on twitter ( @DonaldKronos ) as an example. There are many already out there in various forms, and I am now working on a simulation as one more form, but the problem is that the mere idea of voting against someone has been so differently twisted in so many minds as to make it difficult to find a single explanation which works best for all people. Voters know SOMETHING is wrong, but they go right on voting for the “lesser evil” to keep someone from winning, or showing their opposition by “not voting” and thereby not getting counted, and feeling more and more like the whole concept is flawed and fair elections are a mathematical and physical impossibility. Well… They’re not.