Resource sharing and payoff allocation in a three-stage system: Integrating network DEA with the Shapley value method

Resource sharing exists not only among multiple entities but also among various stages of a single network structure system. Previous studies focused on how to allocate total given sharable resources to stages to maximize the efficiency of the network structure system, and a few discussed the fair allocation of potential gains obtained from resource sharing. In this study, we explore a new case in which the common inputs (or shared resources) of all stages are known. By constructing a game that regards each stage as a player, we integrate cooperative game theory with network data envelopment analysis (DEA) to explore the payoff allocation problem in a three-stage system. We build network DEA models to calculate the optimal profits of the system before and after resource sharing (i.e., pre- and post-collaboration optimal profits), and then apply the Shapley value method to allocate the increased profits of the system to its stages. Results indicate that the game among stages in a three-stage system is superadditive. A numerical example is provided to illustrate our method.