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The Nakamura theorem for coalition structures of quota games. (English) Zbl 0846.90132

Summary: This paper considers a model of society \(\mathcal S\) with a finite number of individuals, \(n\), a finite set of alternatives, \(\Omega\), effective coalitions that must contain an a priori given number \(q\) of individual. Its purpose is to extend the Nakamura Theorem (1979) to the quota games where individuals are allowed to form groups of size \(q\) which are smaller than the grand coalition. Our main result determines the upper bound on the number of alternatives which would guarantee, for a given \(n\) and \(q\), the existence of a stable coalition structure for any profile of complete transitive preference relations. Our notion of stability, \({\mathcal S}\)-equilibrium, introduced by Greenberg-Weber (1993), combines both free entry and free mobility and represents the natural extension of the core to improper or non-superadditive games where coalition structures, and not only the grand coalition, are allowed to form.

MSC:

91A12 Cooperative games
91D10 Models of societies, social and urban evolution
91A40 Other game-theoretic models
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