A bi-preference interplay between transitivity and completeness: Reformulating and extending Schmeidler's theorem

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.