Identity Note: I got locked out of the account @BetterVoting and the password-recovery emails don't seem to be getting through, so I had to make this new account.

I know there are many criteria for voting methods (such as Majority, Condorcet, etc.) but nobody seems to have described a lot of criteria for vote-counting methods; precinct summability seems to be the only that I can think of off the top of my head. There are some criteria which I have thought about specifically in the context of pairwise counting which I would like to share. If anyone has any resources or ideas regarding vote-counting method criteria, I would be interested to learn more about them as well.

=Pairwise counting=

One of the strange things that I noticed about Condorcet methods is that their vote-counting is unbearably tedious even in scenarios where intuition suggests it ought to be fairly easy to count the votes. Consider the (completely theoretical) example of a voter who bullet votes in a Condorcet election where there are, say, 5 million candidates running. We would hope/expect that a vote-counter processing this voter's ballot would not require a lot of time to do so; however, the vote-counter would have to process almost 5 million pairwise matchups in order to finish counting the voter's ballot to then be in a position to discard the ballot.

• One could say that there ought to be an Independence of unranked candidates criterion stating that a vote-counter should not have to put any additional work in to process a voter's ballot because of "also-ran" candidates the voter didn't rank on their ballot; traditional pairwise counting obviously fails this criterion.

Based off of this, there are a few additional criteria I would propose for vote-counting methods, such as a "Bullet Voting" criterion: when a voter bullet votes (provides full support to 1 candidate and provides no information on their preferences for the other candidates), then the vote-counter should be able to process that voter's ballot using only 1 'mark'. (I will use the term 'mark' to indicate a tally mark, or a vote-counting operation done by the vote-counter). Among pairwise counting methods, the traditional pairwise counting method fails this criterion (in the above example, the vote-counter would need to make 4,999,999 tally marks to process the bullet-voter's ballot), but the "negative pairwise counting" method which I invented, as well as its variants such as "semi-negative pairwise counting", satisfy this.

• Further criteria can be invented like this: suppose we generalize this Bullet voting criterion to be an Approval voting-style ballot criterion: "if a voter votes in such a way that their ballot can be turned into a standard Approval voting ballot (that is, they give full support to some candidates and no support to all others), then the vote-counter should only need to tally 1 mark for every candidate the voter gave full support to, and be able to tally 0 marks for the other candidates."
  • And then we can further generalize this criterion in the context of score ballots to account for the possibility of voters giving less-than-full support to the candidates that they "approve" or more-than-zero support to the candidates they "disapprove" (for example, there is the theoretical scenario where there are only 2 candidates in the election and the majority chooses to compromise by not making a full distinction/strategic vote between the two candidates.)

During the invention of negative pairwise counting, another criterion popped out to me because of something I noticed: in a "pure" implementation of negative pairwise counting, a vote-counter would have to make tally marks for a candidate that a voter had explicitly given no support to on their ballot (i.e. they had ranked the candidate last, but in an explicit way, as opposed to not ranking the candidate at all and thus implicitly ranking the candidate last). This is because such a candidate would be explicitly ranked below all others, and thus 1 tally mark would have to be made for every candidate ranked above the last-ranked candidate, as well as 1 tally mark for the last-ranked candidate themselves. So this leads to two more criteria, which one could consider to be almost like the inverse of the Bullet voting and Approval voting-style ballot criteria: a Solo-no-support criterion stating that a candidate explicitly ranked below all other candidates should not require any tally marks to be made to account for them, as well as a more generalized form of this criterion to deal with multiple candidates who are explicitly given no support.

• To satisfy these two new criteria, I had to come up with a modified form of negative pairwise counting in which last-rank candidates are explicitly ignored by vote-counters (though that in and of itself is an undesirable trait in a vote-counting method, since the vote-counter would have to check which candidates are last-ranked before or while processing a ballot, and thus we could say that the modified negative pairwise counting method fails the imaginary criterion stating that a vote-counter should not have to explicitly know which candidates are last-ranked or not when processing a voter's ballot).

=Miscellaneous=

Other criteria or things for which criteria ought to be invented, or which could maybe be expanded upon:

• When dealing with write-in candidates, things can get complicated (at least for pairwise counting methods). So some criteria could probably be invented regulating how easy it ought to be for a vote-counting method to handle write-ins.
• There probably already are criteria or math describing how much data needs to be recorded at each step of vote-counting, but it might be interesting to see more study on this.
• The vote-counting methods for PR methods probably have some fascinating properties of their own that could be studied.
  • I actually had described in the old voting-theory forum an alternative way to do the vote-counting for certain sequential cardinal PR methods. For example, with SPAV, the normal way of vote-counting is to do several rounds of vote-counting, with the data taken from voters' ballots in one round completely ignored in future rounds; I envisioned an alternative vote-counting method (partially inspired by my negative pairwise counting idea) in which, for the first round of vote-counting (i.e. the round of vote-counting used to decide the first winner), the vote-counting is done as usual, but then in successive rounds, we simply subtract from the support recorded on a voter's ballot for each of their approved candidates as that voter begins to see some of their approved candidates get elected, rather than having to entirely re-count all ballots.
    • An example of this would be if you had 30 voters in SPAV who approved AB and disapproved CD, while the other 70 voters in the election all have different preferences and had all disapproved 'candidate A'; supposing that in the first round, A is elected, you would now reweight the 30 AB ballots to 1/2 weight (i.e. they are now treated as having the power of only 15 ballots), subtracting 15 votes from the vote tally for candidate B (since the 30 ballots that had supported B are now treated as having half-weight, or having lost the equivalent of 15 ballots' worth of weight), while not having to count any other ballots or change any other aspects of the tally.
    • The advantage of this approach is that only those ballots which have had one of their candidates elected in the immediately previous round need be (re-)counted in the current round, rather than having to re-count all the ballots.