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Ensuring the boundedness of the core of games with restricted cooperation

  • Michel Grabisch

    ()

    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne, EEP-PSE - Ecole d'Économie de Paris - Paris School of Economics - Ecole d'Économie de Paris)

The core of a cooperative game on a set of players $N$ is one of the most popular concepts of solution. When cooperation is restricted (feasible coalitions form a subcollection $\cF$ of $2^N$), the core may become unbounded, which makes its usage questionable in practice. Our proposal is to make the core bounded by turning some of the inequalities defining the core into equalities (additional efficiency constraints). We address the following mathematical problem: can we find a minimal set of inequalities in the core such that, if turned into equalities, the core becomes bounded? The new core obtained is called the restricted core. We completely solve the question when $\cF$ is a distributive lattice, introducing also the notion of restricted Weber set. We show that the case of regular set systems amounts more or less to the case of distributive lattices. We also study the case of weakly union-closed systems and give some results for the general case.

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Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number hal-00650964.

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Date of creation: 2011
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Publication status: Published, Annals of Operations Research, 2011, 137-154
Handle: RePEc:hal:cesptp:hal-00650964
Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00650964
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