IRV is automatic exhaustive single-choice. Baldwin is automatic exhaustive Borda. Automatic exhaustive score would entail interpolating each voter’s remaining scores to fit the original range after each elimination. Automatic exhaustive approval would have the voter approve each candidate he scored higher than the average of his remaining scores after each elimination. Of these, I have no doubt that automatic exhaustive approval would best incentivize honest voting.
But exhaustive balloting is only one type of repeated balloting. Rather than eliminating a candidate each round, which risks the elimination of the rightful winner, why not update the expected winner and runner-up each round, having each voter vote strategically as if the winner and runner-up of the previous round are the expected winner and runner-up, respectively? This would not be an improvement in the single-choice case, where it would be equivalent to the contingent vote, but it might be an improvement in the case of approval.
In the first round, each voter would approve all candidates he scored above the midrange. In subsequent rounds, each voter would approve each candidate he prefers to the previous round’s winner and the previous round’s winner if he prefers him to the previous round’s runner-up. The final round is the first round with the same winner and runner-up as a previous round.
A Condorcet pairwise comparison can be thought of as a simulated strategic single-choice election with the given candidates as the two frontrunners. The Smith set can be constructed by repeatedly eliminating those elections that include at least one frontrunner that isn’t the winner of at least one remaining election. The following is an approval-based alternative that results in a subset of the Smith set:
Construct a pairwise comparison matrix. In each cell, place the net preferences for the row candidate over the column candidate (place nothing in the self-comparison cells). Construct another candidate matrix. In each cell, place an ordered pair: the winner, followed by the runner-up, of an approval election in which the row candidate is the expected winner and the column candidate is the expected runner-up (they can be found by finding the highest and second highest values of those in the corresponding cell of the first matrix and the row candidate’s column in the first matrix).
Black out all cells whose row and column candidates are not represented by an ordered pair (row then column) in some remaining cell. Repeat until repeating ceases to black out any more cells. The candidates first in at least one remaining cell are the candidates capable of winning an approval election given some plausible expected winner and runner-up (or, alternatively, those capable of winning an infinitieth round of infinitely repeated approval voting given some initial expected winner and runner-up).