Clubs, Commonality, and the Core: An Integration of Game Theory and the Theory of Public Goods

One of the proposed conditions for equilibrium in a non-zero-stum, many-person game is that the outcomes lie in "the core". The concept of the core seems to have numerous applicationis to economics, but up to the present almost its sole application has been the demonstration that competitive market equilibrium is in the core.2 This article will attempt to apply the same concept to the theory of clubs-voluntary organizations for mutual benefit-of which a special case is the provision of public goods. The results inidicate further possibilities for development of this concept in political economy. This should not suggest, however, that the core concept is a silmiple one and its application without difficulty. At best the notion is complex and at times the analysis is difficult to manipulate.3 In the most general terms, a position in the core is supposed to represent a stable allocation or outcome of a game, one from which there is no tendency to movement caused by coalition formationi and dissolution. In specific terms, the core is the set of undominated imputations; an allocation in the core is one for which there exists no blocking coalition. A mathematical definition of the core, due to Luce and Raiffa, may be given: If the set of payments to the n players in a game is denoted as x=(x1, x2, ..., x.), then an imnputation belongs to the core if it satisfies this requiremenit: