Atomic Cournotian traders may be Walrasian

In a bilateral oligopoly, with large traders, represented as atoms, and small traders, represented by an atomless part, when is there a nonempty intersection between the sets of Walras and Cournot-Nash allocations? Using a two-commodity version of the Shapley window model, we show that a necessary and sufficient condition for a Cournot-Nash allocation to be a Walras allocation is that all atoms demand a null amount of one of the two commodities. We provide four examples which show that this characterization holds non-vacuously. When our condition fails to hold, we also confirm, through some examples, the result obtained by Okuno et al. (1980): small traders always have a negligible influence on prices, while the large traders keep their strategic power even when their behavior turns out to be Walrasian in the cooperative framework considered by Gabszewicz and Mertens (1971) and Shitovitz (1973).