Can scoring be made biased to the lower end of the range?

Is there any way to make it more valuable for candidates to take voters from 0/10 to 5/10 than from 5/10 to 10/10 in utility? It seems that an increase in utility at the bottom end of the scale means more overall than at the top, and in addition, this would bias candidates towards reaching out to voters who don’t like them. However, whatever solution is arrived at shouldn’t bias the scale too much, otherwise a lowest common denominator-type candidate would win.
The simplest way I can think of is to make the lower end of the scale (0, 1.5, 2.25, 3.25, 4…) since now a voter who partially likes the candidate must give more support to have it registered.

Why should that be? In any case, you could weight ballots when computing a candidate’s total score so that higher weight is given to lower percentiles. For example, ordering candidates by 4×Total (sum) score on lowest quartile of ballots+3×Total score on 2nd quartile of ballots+2×Total score on 3rd quartile of ballots+Total score on highest quartile of ballots. Though that specific formula may be more extreme than what you intended.

My sequential Vote unitarity system has a linear map from score given to the score spent. This could be done in any way really. I honestly do not see why this would be good. It distorts the mapping of score to ultility.

Another option is how you add score. Giving two winners a score of 1 one results in a score of 2. We could distort the addition metric too. ie 2 + 2 = 3 do you spend less. This is independant from the first possible alteration but also I do not see the point of it. Warren and I made a generalized optimal version of vote unitarity and we left this as an unspecified function.

To make an analogy, it’s better to rescue some drowning people and hand them good clothing than it is to put dry, well-clothed people in golden mansions.
Another way of looking at it is that we want to take people from 0 to approving the candidate, rather than from approving the candidate to loving the candidate. It should be marginally in favor of the former.

Interestingly enough, we might want to go the opposite way for PR, so candidates are more incentivized to be a voter’s favorite than to reach out to more voters.

Interesting option: Do STAR, but your ballot can only go to someone in the runoff if you score them 5-9. (Otherwise, you like neither so your vote is not given to either. If >50% of votes are wasted in this way we call for new elections.)

That’s rather fascinating, though I think it can be improved: only let candidates enter the runoff based on scores that are 5-9. So there is no way for a candidate to enter a runoff without substantial approval, though I would make the 0-4 votes count in the runoff in this version.

This approach does have it’s flaws. Any method of this type should fail IIB. You also have to be carefull with this approach as percentile function based methods can even fail monotonicity (in that thread for whatever reason I didn’t notice that the rating participation criterion I made up was equivalent to the monotonicity criterion for rated methods that assign candidates a score and pick the candidate with the highest score) though that shouldn’t be something you have to worry about if the function you use is always increasing or always decreasing i.e. monotonic but it would still likely fail participation.

There is an alternative way to go about this: just plug the scores into a function before averaging them. For example, instead of picking the candidate with the highest average score, you can pick the candidate with the highest average square root score.

Oddly enough, there is one method that does this (with the digamma function instead of the square root function): Warren’s PSI voting. While his PSI voting was intended for proportional multi-winner elections, it’s single winner case doesn’t reduce to normal score voting like his other multi-winner methods but rather a method that does what you want it to do: the amount a candidate’s quality increases when you change their score from 0/10 to 5/10 is a lot greater then when you change it from 5/10 to 10/10.

This is similar to what I described, except that the scale on the ballot would still be labeled 0, 1, 2, 3, 4, … instead of f(0), f(1), f(2), f(3), f(4).

There is the problem of strategic voting where voters can adjust their score to by picking the score value that when you plug it in to f(x) is closer to their actual score, but then again voters can be maximally strategic by giving min/max scores anyways.

Can the amount be tuned? I’d say about 55% of the quality should come from being 5/10 and 45% from 10/10, just to provide some incentive for candidates to reach out without irritating voters.

Yes. PSI does have a tuning parameter Δ. When PSI is used for multi-winner elections, it is only proportional when Δ is between 1 and 1/2. When Δ approaches infinity, PSI voting becomes score voting and when it approaches 0, single winner PSI becomes approval voting where you approve of all the candidates you don’t give a 0 to. You can also do something similar with the square root approach by raising the scores to the power of P and changing P to values other then 1/2. When P is 1 you get normal score voting and when it approaches 0 you again get the method in which you approve of all the candidates you don’t give a 0 to.