It's adds a good bit of complexity, with potentially small gains. It encourages voters, who already generally know very little about the candidates to have to make even more candidate level decisions. The most likely result is essentially the same as the party line voting we get today, with the only potential location where it results in a "new" result compared with a traditional vote is a jungle primary, but allowing for a "party" vote rather than a candidate vote.

I'd think, if you were going to do it, you should use a system like a poll where there are more points for a first place vote, but still points on a second place vote. Imagine 4 candidates with the following votes:

Alice gets 48% of the first place votes, Bert gets 32%, Carrie gets 15% and Dan gets 5%.

After the first round, Dan is out. So we open the second place votes of Dan's voters, they are proportional to the first round voting for the other 3 candidates and Alice wins with the extra bump of just under 2.5%.

A researcher then looks at the remainder of the votes, and while it turns out Dan got 5% and his voters otherwise split proportionally, Dan himself would have gotten 95% of the second place votes if they had been looked at, as he was everyone's second choice except for his own voters. Alice didn't appear as a 3rd or 4th choice for any of those voters, but those that voted 2nd for Alice gave Bert their 3rd place votes.

We dig further and we find that other than Alice's own voters and the pro rata portion of Dan's Alice wasn't anyone's 3rd or 4th choice. In fact, Carrie's and Alice's voters went 100% for Bert as a third choice, and Bert's went 100% for Carrie as a third choice.

Did we really get the optimal candidate? Or should it have been Dan, the overwhelming 2nd place favorite? Or even Bert who, unlike Annie showed as a 1st, 2nd or 3rd place on every ballot?

The downside here is that it essentially just encourages even more braindead party line voting, and that is just easily resolved by allowing a party line vote on the final candidates from each party.