This is an unnecessary complication but I get what you're saying. We could instead label a ballot numerically as: strongly disapprove (1), disapprove (2), neutral (3), approve (4), strongly approve (5). Or rather, both. What I'm getting at here is that nothing in the math changes so it isn't correct to say that these modifications make a system more balanced or expressive. I'll make no arguments about popularity or satisfaction since this is opinionated.

A benefit of cardinal systems that you are actually pointing out is that not only can we determine winners, but we can also get a lot of information about voter preference. This is quite useful to politicians and researchers (understanding how represented people feel). But I will also note that a strictly negative ballot is likely to be rare as it is purely a protest ballot.

Even though it may be mathematically equivalent, I think the psychological appeal is important to motivate people to vote at all, if not think clearly about how they vote, and do it honestly and wisely.

Human factors are important -- remember the Butterfly Ballots of the 2000 elections!

https://en.wikipedia.org/wiki/2000_United_States_presidentia...

I think that 0 being "don't care" and negative numbers being negative percentages is easier to grasp and more balanced and granular than (3) being neutral. Even -3 to 3 would be better than 1 to 7.

Voting "A: -100%, B: -100%, C: -100%" weights each candidate the same as "A: 0%, B: 0%, C: 0%" the same as "A: 100%, B: 100%, C: 100%", but as you said, it's useful to get an overall count of the voters' absolute satisfaction with each candidate.

A popular positively winning candidate would have a more legitimate claim to a mandate, and the lesser of two evils candidates would be branded as such.

> Even though it may be mathematically equivalent, I think the psychological appeal is important to motivate people to vote at all, if not think clearly about how they vote, and do it honestly and wisely.

I agree with you here, maybe I did not articulate my previous comment correctly to reflect this.

The biggest issue with a proposal like this is that a lack of any preference given whatsoever should not count positively towards a candidate. If someone labels candidate A as +3, candidate B as -3, and doesn't label candidate C, candidate C should not be given more preference than candidate B. People will either not understand the ramifications (they might not know someone and think that "no preference" is truly no preference when in reality it's positive preference), will forget, be lazy and not fill in some candidates, etc. I don't think that's a desirable trait since it's essentially ascribing a vote which the voter did not want. Of course if you want to make -3 (on a -3 to +3 scale) the default value assigned to "no preference" votes, perhaps that would be okay.