Abstract
Classically, a cooperative game is given by a normalized real-valued function v on the collection of all subsets of the set N of players. Shapley has observed that the core of the game is non-empty if v is a non-negative convex (a.k.a. supermodular) set function. In particular, the Shapley value of a convex game is a member of the core. We generalize the classical model of games such that not all subsets of N need to form feasible coalitions. We introduce a model for ranking individual players which yields natural notions of Weber sets and Shapley values in a very general context. We establish Shapley’s theorem on the nonemptyness of the core of monotone convex games in this framework. The proof follows from a greedy algorithm that, in particular, generalizes Edmonds’ polymatroid greedy algorithm.
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Faigle, U., Peis, B. (2008). A Hierarchical Model for Cooperative Games. In: Monien, B., Schroeder, UP. (eds) Algorithmic Game Theory. SAGT 2008. Lecture Notes in Computer Science, vol 4997. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79309-0_21
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DOI: https://doi.org/10.1007/978-3-540-79309-0_21
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