Abstract Consumers’ preferences and choices are traditionally described by appealing to two classical tenets of rationality: transitivity and completeness. In 1971, Schmeidler proved a striking result on the interplay between these properties: On a connected topological space, a nontrivial bi-semicontinuous preorder is complete. Here we reformulate and extend this well-known theorem. First, we show that the topology is not independent of the preorder, contrary to what the original statement suggests. In fact, Schmeidler’s theorem can be restated as follows: A nontrivial preorder with a connected order-section topology is complete. Successively, we extend it to comonotonic bi-preferences: these are pairs of relations such that the first is a preorder, and the second consistently enlarges the first. In particular, a NaP -preference (necessary and possible preference, Giarlotta and Greco, 2013) is a comonotonic bi-preference with a complete second component. We prove two complementary results of the following kind: Special comonotonic bi-preferences with a connected order-section topology are NaP -preferences. Schmeidler’s theorem is a particular case.

中文翻译：

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传递性和完整性之间的双偏好相互作用：重新制定和扩展 Schmeidler 定理

摘要 消费者的偏好和选择传统上是通过诉诸理性的两个经典原则来描述的：传递性和完整性。1971 年，Schmeidler 证明了这些性质之间相互作用的惊人结果：在连接的拓扑空间上，一个非平凡的双半连续预序是完整的。在这里，我们重新表述并扩展了这个众所周知的定理。首先，我们证明拓扑并不独立于前序，这与原始陈述所暗示的相反。事实上，Schmeidler 定理可以重述如下：具有连通序截面拓扑的非平凡预序是完备的。随后，我们将其扩展到共调双偏好：这些是成对的关系，其中第一个是前序，第二个始终扩大第一个。特别是，NaP 偏好（必要和可能的偏好，Giarlotta 和 Greco，2013 年）是一个具有完整第二分量的共调双偏好。我们证明了以下两种互补结果：具有连接顺序截面拓扑的特殊共调双偏好是 NaP 偏好。施梅德勒定理是一个特例。