Abstract
This paper studies a class of cooperative games, called graphical cooperative games, where the internal topology of the coalition depends on a prescribed communication graph among players. First, using the semitensor product of matrices, the value function of graphical cooperative games can be expressed as a pseudo-Boolean function. Then, a simple matrix formula is provided to calculate the Shapley value of graphical cooperative games. Finally, some practical examples are presented to illustrate the application of graphical cooperative games in communication-based coalitions and establish the significance of the Shapley value in different communication networks.
Highlights
Game theory provides a formal mathematical formation to describe the complex interactions among rational players [1]
The fundamental unit in cooperative game theory is the set of players or coalition, and the theory studies the behaviour of rational players when they cooperate. e fundamental problem in cooperative game theory is how to allocate the profit or value of a coalition to its individual players in such a way that players are encouraged to cooperate
For modelling the communication graph, cooperative games in the graph form, called graphical cooperative games or graphical coalitional games, are introduced by Myerson in [3]. ese are games where the internal structure of the coalition is described by a network
Summary
Game theory provides a formal mathematical formation to describe the complex interactions among rational players [1]. Is game model is used to study biologic networks [8] and so on It is pointed out in [3] that the Shapley value is the only possible function that provides a fair allocation in graphical coalitional games. Its computational complexity becomes an obstacle both in practical applications and Mathematical Problems in Engineering theoretical deductions [9]; a remark is presented in [6] to state that “the Shaple value is computationally expensive, but for fairly large structures one time computation is still affordable For such a graph that even one time computation is not affordable, approximations can be used for its computation.”.
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