  In this paper, we study the total displacement statistic of parking functions
from the perspective of cooperative game theory. We introduce parking games,
which are coalitional cost-sharing games in characteristic function form
derived from the total displacement statistic. We show that parking games are
supermodular cost-sharing games, indicating that cooperation is difficult
(i.e., their core is empty). Next, we study their Shapley value, which
formalizes a notion of "fair" cost-sharing and amounts to charging each car for
its expected marginal displacement under a random arrival order. Our main
contribution is a polynomial-time algorithm to compute the Shapley value of
parking games, in contrast with known hardness results on computing the Shapley
value of arbitrary games. The algorithm leverages the permutation-invariance of
total displacement, combinatorial enumeration, and dynamic programming. We
conclude with open questions around alternative solution concepts for
supermodular cost-sharing games and connections to other areas in
combinatorics.
