  In parking problems, a given number of cars enter a one-way street
sequentially, and try to park according to a specified preferred spot in the
street. Various models are possible depending on the chosen rule for
collisions, when two cars have the same preferred spot. We study a model
introduced by Harris, Kamau, Mori, and Tian in recent work, called the MVP
parking problem. In this model, priority is given to the cars arriving later in
the sequence. When a car finds its preferred spot occupied by a previous car,
it "bumps" that car out of the spot and parks there. The earlier car then has
to drive on, and parks in the first available spot it can find. If all cars
manage to park through this procedure, we say that the list of preferences is
an MVP parking function. We study the outcome map of MVP parking functions,
which describes in what order the cars end up. In particular, we link the
fibres of the outcome map to certain subgraphs of the inversion graph of the
outcome permutation. This allows us to reinterpret and improve bounds from
Harris et al. on the fibre sizes. We then focus on a subset of parking
functions, called Motzkin parking functions, where every spot is preferred by
at most two cars. We generalise results from Harris et al., and exhibit rich
connections to Motzkin paths. We also give a closed enumerative formula for the
number of MVP parking functions whose outcome is the complete bipartite
permutation. Finally, we give a new interpretation of the MVP outcome map in
terms of an algorithmic process on recurrent configurations of the Abelian
sandpile model.
