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We study cooperative games in which some of the coalitions are not viable and in addition, there are externalities among the feasible coalitions. These games are called here restricted partition function form games. For this class of games, two extensions of the Shapley value are proposed and characterized. [ABSTRACT FROM AUTHOR]

In the classical approach of von Neumann and Morgenstern to cooperative games, it was assumed that the worth of all coalitions must be given. However, in real-world problems, the worth of some coalitions may be unknown. Therefore, in this study, we consider the Shapley value for convex partially defined games using two approaches. Firstly, we introduce a polytope that includes the set of Shapley values that can be obtained from a given convex partially defined game and select one rational value in some sense from the set. The elements of this polytope are said to be the Shapley payoff vectors. Secondly, we obtain the set of Shapley values that can be obtained from a given convex partially defined game and select one rational value in some sense from the set. Moreover, we axiomatize the proposed two values. [ABSTRACT FROM AUTHOR]

The object of this paper is to study restricted cooperative games, that is, cooperative games for which the worth of some coalitions is unknown. We consider a value for these restricted cooperative games whose definition is based on the Harsanyi's dividends approach, and can therefore be seen as an extension of the Shapley value. We provide a characterization of this value with three axioms: Carrier, Symmetric-partnership and Additivity, which are similar to those proposed by Shapley (in: Kuhn and Tucker (eds) Contributions to the theory of games, Princeton University Press, Princeton, 1953). In addition, we characterize this value on the subclass of restricted cooperative simple games. Finally, we apply this value for restricted cooperative games to analyze the power distribution of the Catalonian Parliament in 1980 and compare the results with those of the coalitional value in Carreras and Owen (Math Soc Sci 15:87–92, 1988). [ABSTRACT FROM AUTHOR]

In the classical approach to cooperative games it is assumed that the worth of every coalition is known. However, in the real world problems there may be situations in which the amount of information is limited and consequently the worth of some coalitions are unknown. The games corresponding to those problems are called partially defined cooperative games (PDGs in short). The Shapley values for PDGs proposed so far need so much time to calculate them. Since the Shapley value plays an important role in applications of cooperative game theory to real world problems, we should consider methods to obtain the Shapley value or its approximation efficiently. We propose an approximated Shapley value which can be obtained efficiently. An axiom system of the proposed value is also shown. [ABSTRACT FROM AUTHOR]