@parker_friedland gave an explanation for Jefferson party list reweigting that got me thinking.
Essentially you are taking a conserved vote power and splitting it between previous winners and potential winners. Whichever potential winner has the most distributed vote power available to them wins. That is a good and compelling explanation. It also fits well with the concept of Vote unitarity. I think it also carries over to SPAV each voter has their vote split between all their approved winners and the next potential one. The formula for their ballot weight is 1/(1+W) but that is deceptive. The 1 in the denominator is the actually the check if they approve of the other candidate or not. If we instead wanted to reweight the approval matrix, A, for voters, v, candidates, c, and current winners, w. We can express SPAV as
The approval matrix in the numerator comes from the distributive law and the one in the denominator come from the fact that it is 0 or 1 but only 1 when the numerator is non-zero. Clearly this is correct but it is not a simplification. It does however show that what Parker was saying is true. You choose the winner who would win when everybody had to spread their vote weight over all winners and the next potential winner.
The Idea I had today is to just to the same update on each round giving a new score matrix derived from the original scores. That would give.
Recall that score matrices are always normalized to [0,1].
This means you spread your vote power over each to be elected candidate by the ratio of the score you gave to them over the total amount of score you would have spent if they were elected. This totally seems like a reasonable system to me. It is a bit like Phragmen but with voter weight not candidate weight.
The interesting thing is that we were trying to motivate RRV theoretically. This is not RRV. RRV does not have the extra score matrix in the denominator. Since it is a different system I am going to call it Sequential Thiele Voting to conform with Sequential Monroe Voting and Sequential Phragmen Voting. Also Distributed Score Voting has already been claimed by @Essenzia.
I like the idea of adjusting the score matrix better than the ballot weight. It is more fine grained. This is where it gets the alignment with Vote Unitrity. SSS updates the score matrix too. So maybe this is the implication for adding Vote Unitarity to different philosophies. For a while I have been trying to determine if Vote Unitarity is a different philosophy to Monroe but if what I have said above is accurate then it can equally well be applied to Thiele. This makes it really more like an additional constraint.
Relative to RRV it down weights less than the Jefferson variant. This is bad since the Jefferson variant seems to already not down-weight enough. We could have a Webster variant by putting a 2 in front of the sum. It would then be somewhere between Webster RRV and Jefferson RRV. It seems hackish to just say already won candidates are worth twice what potential candidates are worth but that seems to be what Webster does.
I could also be totally wrong and this is just a terrible system. I only came up with it today while trying to sort out how to justify RRV with words alone. I am sure you will all tell me if it is terrible.