The Nakamura Theorem for coalition structures of quota games

This paper considers a model of society $S$ with a finite number of individuals,n, a finite set off alternatives, Ω effective coalitions that must contain ana priori given numberq of individuals. Its purpose is to extend the Nakamura Theorem (1979) to the quota games where individuals are allowed to form groups of sizeq 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 e andq, the existence of a stable coalition structure for any profile of complete transitive preference relations. Our notion of stability, $S$ -equilibrium, introduced by Greenberg-Weber (1993), combines bothfree entry andfree 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.