Form of Ballot to be used in Tally
Let a ballot (to be called an executable ballot) consist of an ordered list of rules.
A rule consists of two parts: head and tail.
A rule head consists of a Boolean combination of constraints concerning the question of which candidates remain in the running or, equivalently, which candidates have been eliminated. Note that one of the possible Boolean combinations is the tautology. (The contradiction would also be legal, but useless).
A rule tail consists of a Score ballot.
Tallying proceeds in rounds. A round eliminates one or more candidates. The last candidate standing wins.
For the case where the last round results in a tie, any number of rules (generic sense, not technical sense of rules on a ballot) could be chosen from among for resolving the tie, e. g. random selection.
In a given round, the tally examines the ballots. For each ballot, the algorithm must look for a single rule to apply in the current round. Recall that the rules are ordered on the ballot, and that the order signifies. The algorithm must choose the first rule in the order, whose head evaluates to true under the conditions of the current round. We may say that this rule fires for the round. If no rule fires, the ballot is exhausted and must be thrown out.
During the round of tallying, the tallying algorithm must accumulate for each candidate, the sum over the scores given to that candidate by the tails of the rules that fired from across the ballots.
The candidate receiving the lowest total score is eliminated. If more than one candidate ties for the lowest score, all who so tie are eliminated.
Some Properties of This System
Unless I am missing something, IRV ballots can be mapped into executable ballots in such a way that if all voters vote in IRV style, the result of the system will be the same as in IRV. If this is not the case, I want to revise the system. IRV stops on finding a majority, but does that rule affect the outcome?
Score ballots can be mapped into executable ballots in such a way that if all voters vote in Score style, the result of the system will be the same as in Score.
The system is as democratic as any other winner-take-all system. A vote’s antivote includes the same rule heads in the same order, but negated tails. Thus the antivote tracks the vote through all the rounds and opposes it fully and exactly in each round.
If the above system seems interesting from the point of view of how it distributes power among the voters and from the point of view of applicability to the fight against two-party dominance (2PD), but is objected to on the grounds that the ballots are too complex, then perhaps a solution would be to design some simplified ballot forms and prescribe a mapping from the simplified ballots to executable ballots.