Cournot–Nash equilibria in limit exchange economies with complete markets and consistent prices

n this paper, we analyse a model of non-cooperative exchange “à la Cournot–Nash”, proposed by Lloyd S. Shapley, in limit exchange economies. In contrast with the case with a finite number of traders, analysed by Sahi and Yao [Sahi, S., Yao, S., 1989, The non-cooperative equilibria of a trading economy with complete markets and consistent prices, Journal of Mathematical Economics 18, 325–346], we show that the non-uniqueness of market clearing prices induces an indeterminacy in traders' payoffs for individual deviations. In order to overcome this difficulty, we define a Cournot–Nash equilibrium concept by considering as possible equilibria only the strategy selections for which the aggregate bid matrix is irreducible. Then, we show an equivalence “à la Aumann” between the set of Cournot–Nash equilibrium allocations and the set of Walras equilibrium allocations under the assumption that the set of commodities in the economy is a net. Finally, we show the existence of a Cournot–Nash equilibrium as an easy corollary of the equivalence theorem