Extracting individual race results from Mario Kart final scores

It never occurred to me to play Mario Kart in this way :)

One sufficient condition would be to have "maximally independent" values of $v_i$, for example that they are all logarithms of different primes; that way it is easy to reconstruct the full set scores for each player. However this would allow only to reconstruct the sum of the matrices, and not the set of individual matrices $P_i$.

To have unique $P_i$ one obvious requirement is that for any $n \geq 2$ no set of $n$ players can have a set of $n$ scores in common -- for instance if players 3 and 4 both have the scores x and y, then there are at least two ways to construct the $P_i$. In matrix terms, this means that no submatrix of the "sum matrix" can be strictly positive. Perhaps Perron–Frobenius-like results can help from there.