We consider a bilateral oligopoly version of the Shapley window model with large traders, represented as atoms, and small traders, represented by an atomless part. For this model, we provide a general existence proof of a Cournot-Nash equilibrium that allows one of the two commodities to be held only by atoms. Then, we show, using a corollary proved by Shitovitz (1973), that a Cournot-Nash allocation is Pareto optimal if and only if it is a Walras allocation.

中文翻译：

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具有原子和无原子部分的双边寡头垄断中古诺-纳什均衡的存在性和最优性

我们考虑 Shapley 窗口模型的双边寡头垄断版本，其中大交易者表示为原子，小交易者表示为无原子部分。对于这个模型，我们提供了古诺-纳什均衡的一般存在证明，该均衡允许两种商品之一仅由原子持有。然后，我们使用 Shitovitz (1973) 证明的推论表明，古诺-纳什分配是帕累托最优的当且仅当它是瓦尔拉斯分配。