Exhaustive Approval, Automatic Repeated Ballot, and Approval-based Condorcet

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).

I don’t quite understand the idea. That being said, I think that this assumption

may not be a good way to model approvals. Could the idea be strengthened by having voters use an approval threshold with their scores to indicate who they want to approve to start off with?

The idea is to adapt exhaustive ballot, repeated ballot and Condorcet to approval (they’re all effectively single-choice-based). The same could be done for STAR: make the first round approval, and the incentive to falsely min-max favorites disappears; make the second round approval, with the first round’s winner and runner-up serving as the primary (i.e. for all candidates other than the winner) and secondary (i.e for the winner) approval thresholds, and the risk of early elimination of the rightful winner disappears.

Maybe. I did consider a voter-chosen approval threshold but decided that would be just one more mark for the voter to make. On the other hand, it would reduce the necessary range size (if only integers are allowed) from 2n-2 to n, n being the number of candidates. To be clear, the choice of midrange is not an assumption or based on an assumption. I don’t expect scores to be proportional to utilities anyway (there’s no strategic reason for them to be). I chose midrange instead because it’s neutral and easy to spot and minimizes the necessary range size.

Perhaps you could do both; use the midrange as the default threshold, but allow the voter to choose their own threshold if they want.

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I like that. Maybe the voter could opt out of having their vote change from round to round as well.

If I’m understanding correctly, is this basically “Repeatedly eliminate the Condorcet loser”? Because keep in mind that cycles can occur at the bottom of the Condorcet ranking too.

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You’re right, my bad. I think my approval-based set is still a non-empty subset of the Smith set, though. In fact, it can be smaller than Schulz and Landau as well. For example, suppose there is a cycle of 3 sets of candidates S>T>U>S, all members of S beating all members of T beating all members of U beating all members of S. Suppose that, within each set, social preference is transitive. All candidates are in Schultz and Landau, whereas only the top candidate in S, T and U is in my set.

If we clone each candidate, the winner and runner-up are always some candidate and his clone. With his clone as the approval threshold, a candidate is approved by half the voters (or with .5 probability) in approval or neither approved nor disapproved in combined approval. This simplifies the definition of my set: the set of candidates x for which there is some member y (which may or may not be x) that x is most preferred to. Formally, the Most Preferred (?) set S is the set of candidates x such that:

∃ y∈S, ∀ z, Vxy-Vyx ≥ Vzy-Vyz,

where Vij is the number of voters who prefer i to j.