User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting: Difference between revisions

The negative counting approach is an alternative method of doing [[pairwise counting]]. It is faster (i.e. requires less marks and tallying), depending on implementation, when voters rank multiple candidates last. Rather than counting a voter's preference for a candidate they ranked (i.e. over lower-ranked candidates), it counts that voter's '''lack''' of preference for that candidate (i.e. over the candidate themselves and higher-ranked candidates). In other words, negative pairwise counting a) treats a voter as having "approved" all of the candidates they ranked above last-place, b) counts the voter's pairwise preferences only among the candidates they "approved", and c) aggregates those two pieces of information for all voters (such that for each candidate, you have the total number of voters who "approved" them along with the relevant pairwise preferences), and then calculates the final pairwise totals using the aggregated information.

An example of negative pairwise counting: if 5 voters ranked a candidate (A), and 3 of them didn't rank A above some other candidate (B), then 2 voters must have ranked A above or equal to B. When a voter only ranks candidates as their 1st choice(s) or last choices (i.e. uses equal-ranking with only two ranks), then negative pairwise counting becomes essentially equivalent to [[Approval voting]]'s vote-counting procedure for that voter's ballot.

Semi-negative pairwise counting, which is theoretically even faster, is based on using both of the regular and negative pairwise counting techniques. A [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Regular pairwise counting but done by counting first choices separately|simple variation]] on semi-negative counting is to do regular pairwise counting except that each voter's 1st-choice candidate(s) are simply counted as being marked by those voters, with no need to do pairwise counting for those specific candidates. A number of election examples are provided [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Election example comparisons|below]], with analysis of how the various pairwise counting methods would perform when counting the ballots (the analysis being based off of various [[User:BetterVotingAdvocacy/Negative vote-counting approach for pairwise counting#Formula for counting the required number of marks to be made|formulas]] also provided below.) There are also vote-counting techniques based on similar principles that can be used in non-pairwise contexts, such as for [[Score voting]] and various [[PR]] methods.

The negative counting approach, depending on implementation, can require even more markings when equal-ranking is allowed and it is desired to have traditional pairwise vote totals. Any implementation of negative counting will give accurate information about who won, tied, or lost each matchup (i.e. the pairwise [[margins]]), however. This is because if there are 2 candidates A and B, with the (explicit) votes being:<blockquote>2 A>B

[[Approval voting]] can be thought of as a [[Smith-efficient]] [[Condorcet method]] where, when a voter approves a candidate, they are assumed to vote for them in every head-to-head matchup (see [[Self-referential Smith-efficient Condorcet method]]). Further, approving a candidate can be thought of as ranking them 1st, while disapproving a candidate can be thought of as ranking them last. Given that connection, and that in Approval voting, the vote-counting is done by counting the number of ballots that approve/mark a candidate, rather than by harder [[pairwise counting]], it is clear that a similar counting procedure could be applied to pairwise counting itself. The only complexity is that when voters rank candidates, they are allowed to express that they don't prefer certain candidates in certain matchups; thus, negative votes are necessary to communicate that lack of preference for specific matchups.