[This is my first version of prescription of Balanced Elimination Voting. It is superseded by the second version, which is worded very differently but implies the same semantics.]
This proposes a form of election for a single winner.
It is an attempt to keep the best characteristic of Approval Voting while still appealing to advocates of Instant Runoff Voting.
In this form of election, a voter who opposes a candidate and a voter who supports the candidate have equal power, in the sense that adding a million supporters to the election would be balanced by adding a million who oppose that candidate. This is in contrast to for example the choose-one plurality form of election where for example if there are three candidates, it takes twice as many voters who oppose candidate A to balance the effect of a given number of supporters. This happens because the anti-A vote will be split between B and C. It may not be obvious why IRV also supports such vote splitting, but I believe IRV advocates cannot bring a proof that IRV gives equal power to all the voters. However, like Approval, the present proposal can be proved to provide equality simply by pointing out that every possible vote has an antivote, that is, a vote that would balance the effect of the first vote.
This form of election cannot be tallied by the precincts. However, I promote the opinion that the costs of elections are swamped by the cost of bad government resulting from electing a wrong candidate.
The voter has the option of including so many sections in her ballot as she chooses.
The sections are ordered on the ballot; their order matters.
A section in a ballot can mention any of the candidates the voter chooses to, up to all of the candidates.
When a candidate is mentioned in a section, the voter rates the candidate up or down for that section. A voter can vote a given candidate up in one section and down in a different section.
When a candidate is mentioned in a section, the voter marks the candidate as key or not key for that section.
The voter can include in a section, an indication of how to treat candidates whom the voter does not specifically mention by name in the section. I. e. a default treatment. This may be particularly relevant if write-in votes are allowed.
The voter chooses a conjunction for each section, either “and” or “or”.
Tallying proceeds in rounds, where the goal of each round is to eliminate a candidate. The last candidate remaining wins.
A round accumulates a total score for each candidate. The round eliminates the candidate with the lowest total score.
From each ballot, one section (at most) contributes to the scoring in a round.
The determination of what section of the ballot to apply in the round begins by determining which sections light up for the round, based on the following rules.
If the conjunction word of a section is “and”, the section lights up iff all its key candidates are still in the running for the round. “Still in the running for the round” means not having been eliminated in any prior round.
If the conjunction word of a section is “or”, the section lights up iff any of its key candidates is still in the running for the round.
Perhaps the last section on the ballot should be regarded to light up regardless of the above rules. Otherwise, a ballot could be exhausted even if it has at least one section. This would probably not be the voter’s intent.
The section that applies to the round is the earliest (i. e. nearest to the beginning of the ballot) section that lights up.
When the ballot is counted for the round, the score of each candidate voted up in the applied section is incremented and the score of each candidate voted down in the applied section is decremented.
Each candidate starts with a total of zero at the beginning of the round; that is to say, the tally calculates the total for that candidate for that round independently of what happens during the other rounds.
The election could offer voters the options of choosing simplified ballots. The election machinery would then in effect translate the simplified ballots into ballots having the form prescribed in the above section Executable Ballots, for the tally.
Approval-style Simplified Ballots
A voter who wishes to cast an Approval-style vote can simply supply a single Section whose conjunction is “and” and on which none of the candidates is key. The up or down ratings with respect to the candidates indicate approval or disapproval.
If all voters cast such Approval-style ballots, the outcome of the election will be the same as that of an Approval election.
Ranking-style Simplified Ballots, Type “A”
A voter who chooses this type of simplified ballot can rank the candidates. Equal ranking is permitted.
Translation to an executable ballot starts with the least favored candidate(s) i. e. the most hated (by this voter) candidate(s) as indicated on the simplified ballot. The first section on the executable ballot to be put out by the translation process, marks these hated candidates as key, rates them down, and rates all better candidates up. The conjunction is “or”.
Suppose the voter ranks a single candidate as most hated. Then under this translation scheme, the rounds of tallying in which that candidate is not yet eliminated will throw the voter’s full political weight toward eliminating that candidate. If every voter were to use this type of simplified ballot, then the first round would eliminate the candidate most hated by the most count of voters.
The translation continues with the second-most-hated (by this voter) rank of candidates from the simplified ballot, generating on the output executable ballot, another section. This section names the second most hated (by this voter) candidates as key, rates them down, rates up the better candidates, and sports “or” as its conjunction.
This pattern of translation continues through the ranks provided on the simplified ballot up to but excluding the most favored (by this voter) rank of candidates (perhaps naming a single candidate, the voter’s absolute favorite). For these candidates or this candidate, the translation generates the final section of the executable ballot. This last section sports “and” as its conjunction, marks no candidates key, up-votes the favorite candidate(s), and down-votes the lesser candidates.
Score Voting Variant
Instead of just allowing an up or down vote on each candidate in a section, we could allow scoring them -50, -49, [-40], , , 49, or 50. The score the voter gives to a candidate would be added to that candidate’s tallies in the rounds to which the section applies. Using a range of only four or five or seven possible score values may keep the election practical to tally by hand (although there would have to be back-and-forth communication between the precincts and the center as the tally proceeds from round to round).
Adaptation to Multi-winner Elections
I don’t see any way I would want to adapt this idea to the multi-winner case, and I do not suggest to others to try to think in that direction. (For legislatures, I suggest Choice of Representation, a scheme in which every citizen chooses her representative, and the representatives vote in the legislature with the proxy power of their constituents. If this would lead to overcrowding of the legislative chamber, the less-chosen representatives could participate in discussion and voting via remote communication, and might not be supplied at public expense with offices and staffs. Jim Mueller of Wisconsin advocates Choice of Representation by that name.)