Abstract:

A cooperative game with transferable utility， shortly TU-game， is a pair consisting of a finite set of players and a characteristic function assigning a worth to each coalition of players. However， in many situations cooperation is restricted by cooperation structures such as hierarchical， technical or communicational structure. Aumann and Drèze （1974） first considered the situation in which there exists an exogenous coalition structure. Later， Myerson （1977） introduced TU-games with restricted cooperation possibilities represented by an undirected graph.In practice， there are some situations of cooperative decision making in which the cooperation between agents is restricted by both a communication structure and an organizational structure. van den Brink et al. （2015） considered the partition-graph value for TU-games with coalition and graph structures. The partition-graph value reflects not only the coalition structure but also the graph structure of the game will influence the allocation process， but disregards the uncertainty of cooperation structure.In this paper the TU-games with coalition and probabilistic graph structure are proposed and the partition-probabilistic graph value for the game model in introduced by considering the uncertainty of graph structure. It is shown that the partition-probabilistic graph value is uniquely determined by probabilistic partition component efficiency and probabilistic fairness. Finally， the partition-probabilistic graph value is illustrated by an example of transnational material transportation and compare it with other values.

Key words: TU-game, allocation rule, coalition structure, probabilistic graph, partition-probabilistic graph value