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.