Title: On Large Games and Competitive Markets 1: Theory
Abstract: This paper discusses a body of research directed towards establishing that large economies where almost all gains to collective activity can be captured by small groups of participants are competetive. Specifically, such economies are game-theoretically equivalent to competitive exchange economies. The ecomomies include ones with clubs, coalition production, local public goods, and also exchange economies with nonconvexities and indivisibilities. A framework of games in characteristic form is introduced. A large game is one with a "large" but finite number of players. The framework has the substitution property, the property that in any large game most players have many close substitutes. The games satisfy the conditions of effectiveness of small groups if almost all gains to cooperative activities can be realized by cooperation within bounded-sized coalitions in partitions of the total player set. Note that this does not rule out arbitrarily large coalitions; it only means that large coalitions are inessential for the realization of almost all gains to group formation. Roughly, I establish that effectiveness of small groups is necessary and sufficient for: (a) the market-game property, that all sufficiently large games are approximately market games (ones derived from exchange economies where all agents have continuous, concave utility functions); (b) the core-convergence property, that approximate cores of large games converge to Walrasian payoffs of markets; (c) continuum representability, the property that small groups cannot have significant per-capita effects on large groups (i.e., that the games can be adequately represented as ones with a continuum of players); and (d) the approximate core property, that all sufficiently large games have nonempty approximate cores. I emphasize that, with some small qualifications, effectiveness of small groups conditions are necessary and sufficient for these properties, and in particular, for core convergence. This contrasts with earlier results showing sufficient conditions. In addition, I establish that for large games the approximate core is typically "small" and converges to a single payoff. This is an immediate consequence of the properties that large games are approximated by large markets where all agents have the same continuous and concave utility function and (b) above. In "Games and Competitive Markets 2: Applications" the application of the method and results to a number of economic problems is discussed. I apply my approach to economies with public goods with satiation and/or congestion. I show that when collective consumption by the group of the whole is optimal, effectiveness of small groups is necessary and sufficient for convergence of the core to the Lindahl payoffs. This application is chosen to present a boundary of the applicability of my framework. The application of the results to collectively consumed and/or produced commodities, where it may be optimal to have many distinct groups, is discussed, and related to other research. The results of this paper are applied to show that large economies satisfying substitution and effectiveness can be approximated by exchange economies where the goods are the attributes of players and where hedonic prices emerge as Walrasian prices. A number of other applications are suggested.
Creation-Date: November 1991
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