Different ways of measuring expressiveness for rated-type ballots

Image from Critique #3 in

I think it’s worth pointing out that the number for Score-type ballots is somewhat inflated, depending on the measure of expressiveness you’re using. If we ignore write-in candidates, and are talking about an election with 3 candidates, with a scale of 0 to 3, then we can observe that the following 2 rated ballots are equivalent in terms of margins:

A:3 B:2 C:2

A:2 B:1 C:1

Both of these can be written in a pairwise table (if counting how many more, if any, points the voter gives the row candidate over the column candidate) as:

||| A B C
A — 1 1
B 0 — 0
C 0 0 —

For 3 candidates, the possible number of permutations under this “results-based” measure of expressiveness (i.e. when we measure expressiveness in terms of how a vote can change the result in the context of single-winner and Bloc Score) is 36, rather than 62 or 64. Details:

Summary

A:1 B:0 C:0 (= 2 other votes; if we add 1 point to every candidate, it becomes A:2 B:1 C:1, and add another point, and it’s A:3 B:2 C:2. We can’t add yet another point, since that would put A above the max score)
A:0 B:1 C:0 (=2)
A:0 B:0 C:1 (=2)
A:1 B:1 C:0 (=2)
A:0 B:1 C:1 (=2)
A:1 B:0 C:1 (=2)

A:3 B:2 C:1 (=1, since remove 1 point and it becomes A:2 B:1 C:0)
A:3 B:1 C:2 (=1)
B:3 A:2 C:1 (=1)
B:3 A:1 C:2 (=1)
C:3 A:2 B:1 (=1)
C:3 A:1 B:2 (=1)

A:3 B:1 C:1 (=1)
B:3 A:1 C:1 (=1)
C:3 A:1 B:1 (=1)
A:3 B:3 C:1 (=1)
A:3 B:1 C:3 (=1)
A:1 B:3 C:3 (=1)

A:0 B:0 C:0 (=3, because we can add 1 point to every candidate 3 times before reaching the max score. However, ignore 1 of these, because the possibility of max-scoring every candidate was already subtracted out in the image above)

So there are 26 votes that can be removed as equivalent to these ones in the result-based expressiveness measurement. Essentially we get these by taking votes that aren’t normalized, and bumping them up or down x points until we hit both ends of the scale.

Just to be comprehensive, I’ll also show how many votes can be removed if you consider non-normalized votes as invalid expression (i.e. because they weaken your vote), even though the case for doing this isn’t quite as strong: there are 20 votes mentioned in the above analysis, and all of them are non-normalized votes, since in order for two or more votes to be equivalent in terms of margins, it has to be possible to add/subtract a constant to their score for every candidate, and that’s not possible if in any of the votes, one candidate is at the max score, and one candidate is at the min score. Ignoring the 2 votes where every candidate is scored the same means that there would be 18 votes to consider from the above analysis, so that’s 36-18=18 possible normalized votes that differ in terms of pairwise margins between the candidates.