Abstract:

A cooperative game describes real-world allocation problems by specifying the worth of all potential coalitions of players. A solution of cooperative games aims to distribute the worth of the grand coalition among all players fairly. The Shapley value is one of the most important solution concepts in cooperative game theory. The paper focuses on the Shapley values for cooperative games with a coalition structure or a communication network. A coalition structure is a partition of the player set， where each component acts as an intermediary node during the allocation process. The Aumann-Drèze value， the Owen value， and the two-step Shapley value are well-known extensions of the Shapley value to cooperative games with a coalition structure. A communication network consists of nodes and edges， where nodes and edges represent players and communication relationships between players， respectively. Only connected players in the network can communicate with each other. The Myerson value， the position value， and the average tree solution are well-known extensions of the Shapley value to cooperative games with a communication graph. The underlying paper aims to review these extensions. For coalition games， both the traditional coalition structure and the extended forms of coalition configurations and level structure are concerned. For network games， the extended form of hyper-network is also studied. Moreover， the extensions of the Shapley value to cooperative games with both a coalition structure and a communication network are also reviewed， where two cases are considered： 1） coalition network games in which the coalition structure and the communication network are independent； 2） two-layer network games in which the two are interrelated. According to these reviews， the relationship between different Shapley value extensions are more clear， which may support decision making in reality.

Key words: Cooperative game, coalition structure, communication network, Shapley value