DEA game for internal cooperation between an upper-level process and multiple lower-level processes

Cooperation is an important strategy to improve companies' profitability. As a data-driven tool for performance evaluation, data envelopment analysis (DEA) is often used to measure the benefits from cooperation between independent decision making units (DMUs) or systems, but few involve the internal cooperation in a network structure system. For a double-level system that consists of an upper- and multiple lower-level processes, prior to cooperation, its internal processes are independent and fail to yield profit. Once the upper-level process cooperates with at least one lower-level process, the system can operate to obtain profit. At present, no research explores the internal cooperation in a double-level system from the perspective of relative performance, considering such dependency between upper- and lower-level processes. To fill this gap, we propose a double-level DEA game by integrating network DEA and cooperative game theory, and prove this new game is monotone, super-additive, and has non-empty core. We use an example of 15 supply chains including one supplier and three retailers to validate our approach and provide an example as a real-life application. Results show the Shapley value and nucleolus have the best performance in allocating the total profit in terms of fairness and stability, respectively.