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arXiv · 2309.12265

Cost-sharing in Parking Games

Jennifer Elder·Pamela E. Harris·Jan Kretschmann·J. Carlos Martínez Mori
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Abstract

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 an alternative solution concept for supermodular cost-sharing games and connections to other areas in combinatorics.

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Jennifer Elder, Pamela E. Harris, Jan Kretschmann, J. Carlos Martínez Mori. 2023-09-21. Cost-sharing in Parking Games. https://doi.org/10.46298/dmtcs.13113

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