Cost-sharing in Parking Games

TL;DR

This paper introduces parking games derived from total displacement statistics, proves their supermodularity, and provides a polynomial-time algorithm for computing the Shapley value, advancing cost-sharing methods in cooperative game theory.

Contribution

It presents the first polynomial-time algorithm for Shapley value computation in parking games, a new class of supermodular cost-sharing games based on parking functions.

Findings

Parking games are supermodular with an empty core.

The Shapley value can be computed efficiently using a new algorithm.

The algorithm exploits permutation-invariance and dynamic programming techniques.

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.