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Can the choice function of a non-empty and finite set be the empty set? Or is this by definition of the choice function impossible? Does there need to be always at least one winner if we evaluate non-empty and finite sets?

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    $\begingroup$ Can a choice function point to an empty set/have an empty set as value? Is this what you meant to ask? $\endgroup$
    – Giskard
    Commented Sep 28, 2022 at 16:49
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    $\begingroup$ Also, is there any context for this question? What framework/model/book are you studying? $\endgroup$
    – Giskard
    Commented Sep 28, 2022 at 16:50
  • $\begingroup$ Well, I am reading a book about behavioral theory and I am currently studying preferences, choice sets and utility function representation. Imagine you have any set A which is finite and is non-empty. For example A={x,y,z}. I am wondering if it can be the case that C({x,y,z}={}. This means: Can it be the case that the choice function assigns to any non-empty and finite set the empty set? Or is there always at least one element that needs to be the "output" of the choice function if the mentioned conditions are met? I hope this makes it clearer. $\endgroup$
    – dewewdew
    Commented Sep 28, 2022 at 17:18
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    $\begingroup$ Is there a particular reason for you not to reference the book? (by title and author) $\endgroup$
    – Giskard
    Commented Sep 28, 2022 at 18:21
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    $\begingroup$ economics.stackexchange.com/questions/42094/… $\endgroup$ Commented Sep 29, 2022 at 12:48

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The choice correspondence is typically assumed to be nonempty-valued. As these notes by John Nachbar explain:

[The requirement of nonemptiness of the choice correspondence] eliminates the possibility of Buridan’s Ass (a donkey that starves because it cannot make up its mind as to which of two bails of hay to eat). If I want to allow the possibility of “no choice,” then I have to include “no choice” as an element of [the set of alternatives].

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