Extracting individual race results from Mario Kart final scores

Question

In Mario Kart, one "cup" involves 4 races, and after every race each racer gets points awarded based on what place they came in (better rank means more points). After playing it enough I grew curious about the mathematical problem of trying to reason backwards from the final scores to the individual race results - when is it possible and how might one do it?

Formally, the problem goes like this: suppose we have a scoring vector $\mathbf{v}\in\mathbb{Z}^n$ with components that are strictly decreasing with index. There are $r$ permutation matrices $P_1,P_2,\dots,P_r$ that we don't have, but we do have the final score vector $\mathbf{f}=(P_1+P_2+\cdots+P_r)\mathbf{v}$. What conditions on our information - $\mathbf{v},r,\mathbf{f}$ - allow there to be a unique solution, and how might we go about computing them? I imagine if $r$ is small enough compared to $n$ and the scoring vector $\mathbf{v}$ is "sparse" or the components "independent" enough (a fuzzy intuition), there will probably be a unique solution, but it doesn't look at all amenable to linear algebra methods that I know of.

More generally, we could understand the scoring vector to hail from a different type of vector space, or understand the score-distributing matrices $P_i$ to be from a different candidate set than permutation matrices (perhaps from a group representation of something other than the symmetric group?).

Answer 1

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.