SFB 303 Discussion Paper No. B-151

Author: Holtz Wooders, Myrna
Title: Inessentiality of Large Coalitions and the Approximate Core Property: Two Equivalence Theorems
Abstract: We show that large games, ones with many players, satisfy an economically intuitive condition that almost all gains to coalition formation can be realized by coalitions bounded in absolute size if and only if they have the approximate core property (all sufficiently large games have nonempty approximate cores) and satisfy boundedness of contributions to average payoffs. This extends some of the results in the literature on nonemptiness of approximate cores. The main point here is the equivalence of the inessentiality of large coalitions condition and the approximate core property. We also consider a less restrictive inessentiality-of-large-coalitions condition, where the size of the game required for realization of almost all gains to coalition formation can depend on the attributes of the players of the game (the uniform bound on the size of coalitions in the preceeding paragraph is dropped). We show that large games satisfy this condition if and only if they satisfy a weak form of the approximate core property. If we adopt the standard that a necessary condition for a large game to be analogous to a competitive economy is nonemptiness of an (approximate) core or existence of an approximate equilibrium, then our results indicate that inessentiality of large coalitions is necessary for the competitiveness of large games.
Creation-Date: June 1990
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