(If you're not familiar, approval voting is a special case of score voting where you are giving either a 0 or 1 to each candidate; in other words the range is binary. It's not quite as expressive for voters as score voting, but still a huge improvement over what we use now and pretty low on "voter regret" since it avoids having to vote strategically. And it's arguably easier to use and understand than score voting. I would be so happy with either score or approval.)
One would work where you have 1 vote per ballot item. If there were 24 ballot items, you get 24 votes. You can allocate your votes across the ballot however you saw fit. If you thought president and senate were more important than city council and an infrastructure bond; you could put 18 votes for president, 6 votes for senate, and nothing for local issues. No idea how this would be normalized but it’s a novel concept.
Another idea is to rank options in each issue, and if my first option loses; my votes go to the second option. Say I wanted to vote libertarian for president in the US but was afraid it was a protest vote that wouldn’t win so I voted Democrat because I didn’t like the GOP candidate. Under this system I could say I want the libertarian candidate first, democrat second, and not pick the gop candidate.
After the votes were counted and the libertarian lost, my vote would be reallocated to the democrat; and the votes would be re-tallied. This would encourage people to vote for a non binary political candidate since they knew their vote wouldn’t be “thrown away”.
interesting concept indeed, but i think the normalization function would be a serious vulnerability. most state and national level elections in the US tend to be pretty close, so the normalization function would pretty much end up deciding the winner. people who realize this would fight hard to get a biased function implemented, and you would be stuck trying to explain statistics to the general public. basically the gerrymandering situation, but even harder for the public to understand.
That seems extremely problematic to have votes determined by ballot items. It's impossible to normalize it. Even if you just determined that "each voter nationwide gets X voting points" you'd have politicians that would try to dilute or concentrate their ballots in order to game things.
Yeeeyaw
The voting system sounds reasonable, except local, state and national elections are much more independent than you're describing. The Constitution specifically and purposefully separates these entities, and trying to sew the legislation together would be more of a nightmare than it's worth.
For instance if I want Candidate A to win, I'll give him a 10. But if I'm only half bought into candidate B, I can give them 5, and could give Candidate C nothing because I really don't like them. Now if the rest of the country doesn't like A, and the battle is between B and C, then I'd feel good that I could contribute to the goal.
The actual process is up for remix of course, but that was the idea.