Robust Cost Allocation in Stochastic Cooperative Games: An Application in Last-mile Delivery

Abstract

Problem definition: This paper considers a class of stochastic cooperative games from a platform perspective, where restricted data sharing among players poses significant challenges. The platform aggregates data from all participants but cannot share information directly. A key challenge for the platform is to propose an attractive and fair cost allocation solution that incentivizes all players to join the grand coalition. 

Methodology/results: We develop the robust optimal cost allocation problem (ROCAP) framework, which incorporates two key constraints: the enhanced coalitional stability constraint that ensures provided allocation is more advantageous than the best-case cost of any subcoalition, and the worst-case budget balance constraint, which guarantees the grand coalition cost is fully covered under the worst-case scenario. These constraints collectively define the "enhanced robust core", which extends the traditional core and robust core concepts to uncertain scenarios where data sharing is restricted and the platform aggregates information to facilitate cooperation.  Practically, we develop a cooperative delivery game with uncertain demand locations as an application to illustrate how the robust cooperative game operates. Technically, we derive tractable reformulations for the proposed ROCAP model and design an algorithmic framework to calculate the enhanced robust core efficiently. Extensive numerical experiments demonstrate that our enhanced robust core concept significantly outperforms traditional cost allocation methods, such as the Shapley value and nucleolus, in the robustness performance, with an improvement from approximately 40% to 99%.  

Managerial implications: An important insight from our research is that the enhanced robust core offers greater advantages for larger companies compared to traditional cost allocation methods, such as the Shapley value and nucleolus. In practice, this is particularly beneficial for forming grand coalitions, as larger companies often exhibit reluctance to collaborate with smaller companies to safeguard their market share and confidential information.

Keywords: cooperative games, cost allocation problem, core, distributionally robust optimization, data-driven, delivery scenario, TSP approximation model, Wasserstein distance