Score Voting is the solution -- give candidates a score from 0-99 and take the highest average score. Works great to decide Olympian Gold/Silver/Bronze medalists. Plus, no vote is wasted, and bicameralism is transcended.
It works in situations where there are few voters and their votes are all publicly visible.
Voters do not as a rule have a mathematical score for a candidate, excluding the case where they score equivalently to Approval.
Take a candidate which you did not vote for but do not hate, is your score for the candidate 1, 2 or 3.141529? Can you give an explanation for your score and why it is that exact value and not a bit higher or lower?
Asking people for information they do not have (a mathematical score for each candidate) leads to GIGO - unless just about everyone does Approval, in which case Score's just a more complicated, worse version of Approval.
Score's tactical logic leads to most voters giving the extreme scores to every candidate, and psychology means all these voters will end up justifying it to themselves - leading to extreme polarization when the losing side is convinced the winner has no redeeming qualities whatsoever. The last presidential term showed what happens than, now repeat that for all levels of government all the time.
Worst of all, some very few people will actually give partial scores. Why is that bad? Well, what's the most common way to calculate scores today? There are various single-issue and ideological organizations which put out scores measuring representatives. Just about all these organizations represent absolutist POVs regarding their single issue or ideology (e.g. the NRA has a score for every representative). So to get a high score, representatives are pushed to extremes.
In the general election, using plain score voting is a bit dangerous; if you consider a Trump vs Clinton race where all the Trump voters give him a ten out of ten, and most of the Clinton voters give her a six or seven, then Trump wins by a bigger margin than he did in 2016. (STAR fixes this with an immediate runoff of the top two after maximalizing everyone's vote.) But in a primary election, it might be good to get the candidate with the most enthusiasm behind them.
One difficulty in implementing something like this for the Democratic primary is that it isn't very compatible with the delegate system. Maybe that's not entirely a bad thing and delegates should be done away with, but sometimes it's good to have a group of people at the convention who can make decisions if something unexpected happens to one of the candidates.
You could have individual states adopt different voting systems for their primaries and just use them to allocate delegates. For instance, instead of allocating delegates proportionally to anyone with more than 15% of the vote using first-past-the-post ballots like we do now, allocate delegates proportional to their candidate's total score, perhaps with some cut-off (like, no delegates awarded if your score average is less than 4 out of 10) or a rule that only the top N candidates are awarded delegates.
(One of the reasons I prefer approval voting over score voting is that there's less of a gap between strategic voting and honest voting. Everyone is forced to give a maximalized yes/no vote for each candidate rather than casting a lukewarm vote.)
You're right that score voting does better in that simulation when everyone votes honestly. But if everyone votes strategically, then score voting and approval voting are the same. (This makes sense because tactical score voting is mostly just score voting with maximalized votes, which is basically the same as approval voting.)
Using approval voting basically forces everyone to use tactical voting. Which might be less good in some metrics but it ensures that all voters have basically the same amount of influence, as opposed to tactical voters having more influence than honest voters. I think having equal influence is pretty important.
I'm not sure if this simulation that everyone likes to cite has been run more recently with STAR voting. That would be interesting.
What are the incentives for someone not to vote 99 for the candidate he wants to win, and 0 for all others ?
I think we’ve proved that FPTP works badly a lot of the time, time to try something else