Abstract
The traveling salesman game (TSG) consists of dividing the cost of a round trip among several customers. One of the most significant solution concepts in cooperative game theory is the Shapley value, which provides a fair division of the costs for a variety of games including the TSG, based on the marginal costs attributed with each customer. In this paper, we consider efficient methods for computing the Shapley value for the TSG. There exist two major variants of the TSG. In the first variant, there exists a fixed order in which the customers are serviced. We show a method for efficient computation of the Shapley value in this setting. Our result is also applicable for efficient computation of the Shapley value in ride-sharing settings, when a number of passengers would like to fairly split their ride cost. In the second variant, there is no predetermined fixed order. We show that the Shapley value cannot be efficiently computed in this setting. However, extensive simulations reveal that our approach for the first variant can serve as an excellent proxy for the second variant, outperforming the state-of-the-art methods.
Highlights
An important combinatorial optimization problem is the Vehicle Routing Problem (VRP), where there is a set of customers situated in different locations on a map, and the goal is to find an optimal route for a vehicle to deliver or pick up some goods to the customers [1]
2) We show that, while there exist no polynomial algorithm for computing the Shapley value of the general traveling salesman game (TSG), the Shapley value computed for routing games can be used as an excellent proxy for the Shapley value in a TSG, establishing a new stateof-the-art
We develop a proxy for the Shapley value for the general TSG problem, which is based on the Shapley value for routing games, and show that it outperforms the current state-of-the-art proxies without requiring extensive computation
Summary
An important combinatorial optimization problem is the Vehicle Routing Problem (VRP), where there is a set of customers situated in different locations on a map, and the goal is to find an optimal route for a vehicle to deliver or pick up some goods to the customers [1]. As stated by Özener and Ergun [8], “In general, explicitly calculating the Shapley value requires exponential time It is an impractical cost-allocation method unless an implicit technique given the particular structure of the game can be found”. In the context of supplies delivery, the order may be attributed to the order in which the requests were received, the urgency of the service, or the frequency of customer usage of the service In such cases, the order must be preserved when determining the travel cost of the tour with a subset of the customers. The contributions of this paper are two-fold: 1) We show an efficient method for computing the Shapley value of each customer for routing games, which is in contrast to a previous conjecture made in the literature [9].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
