I struggled to name what I'm asking about here. I'm talking about the kind of voting that happens on many websites, including stackexchange: you can upvote something (a question), or you can downvote it. In other words, you can cast a positive vote or a negative vote on anything.
Note: I don't mean simply where you give each candidate (or whatever) a positive score from (say) 1 to 5. Youtube used to use this system for its videos, but switched to what I'm calling negative voting: thumbs up or down.
Has this ever been tried for real-world elections? You could for example vote:
Socialists: 0 (no vote)
Communists: 0 (no vote)
National Front: -1 (negative vote)
To tally the results the returning officer would subtract a candidate's negative votes from their positive votes, and award the victory in this district to the candidate or party with the highest net score.
Gaullist, 2000 up, 1500 down.
Socialist, 1500 up, 500 down.
Communist, 100 up, 75 down.
National Front, 300 up, 400 down.
The winner in this seat would be the Socialist candidate.
Has this ever been used for anything other than internet purposes? I'm pretty sure it's never been used for government elections at any level, or major political parties, but it would be interesting to know.
Has any prominent interest group or academic ever proposed using it? What is the thinking on whether it would be possible to design a negative voting system which satisfied all the requirements of a fair voting system described by Arrow?
Edit: It's gratifying that this question has received so much attention. I'd like to point out I'm not proposing this system or even trying to predict what its effects would be on elections, politics or governments. This isn't the place to make proposals, or to knock the imagined proposals of others.