Asymptotic equivalence between Cournot–Nash and Walras equilibria in exchange economies with atoms and an atomless part

In this paper, we consider an exchange economy à la Shitovitz (Econometrica 41:467–501, 1973), with atoms and an atomless set. We associate with it a strategic market game of the kind first proposed by Lloyd S. Shapley, known as the Shapley window model. We analyze the relationship between the set of the Cournot–Nash allocations of the strategic market game and the Walras allocations of the exchange economy with which it is associated. We show, with an example, that even when atoms are countably infinite, any Cournot–Nash allocation of the game is not a Walras allocation of the underlying exchange economy. Accordingly, in the original spirit of Cournot (Recherches sur les principes mathématiques de la théorie des richesses. Hachette, Paris, 1838), we partially replicate the mixed exchange economy by increasing the number of atoms, without affecting the atomless part, and ensuring that the measure space of agents remains finite. Our main theorem shows that any sequence of Cournot–Nash allocations of the strategic market games associated with the partial replications of the exchange economy has a limit point for each trader and that the assignment determined by these limit points is a Walrasian allocation of the original economy.