Is possible that, in a choice structure $(B,C)$, $C(b) = \emptyset$, for some $b \in B$

Question

It is a very simple question but with some implications. I just start reading Mas-Colell and it's not clear for me if it is possible a choice structure, $(B,C(.))$, where $\exists b \in B, C(b)= \emptyset$.

For example: $B=\{\{x,y,z\},\{x,y\},\{x,z\},\{y,z\},\{x\},\{y\},\{z\}\}$

$C(\{x,y,z\}) = \{z\}$

$C(\{x,y\}) = \emptyset$

$C(\{y,z\}) = \{z\}$

$C(\{x,z\}) = \{z\}$

$C(\{x\}) = \emptyset$

$C(\{y\}) = \emptyset$

$C(\{z\}) = \{z\}$

is a valid choice structure?

Mas-Colell defines choice structure and choice rules in section 1.C, page 10 on oxford press 1995 edition.

I first though about it because I think it would have some interesting implications, but it was false. Anyway, the question remains. Thanks!

Answer 1

One usually rules this out by assumption, as Mas-Colell, Whinston, and Green do in their textbook (). The goal is often to try to characterize preferences that guarantee (at least for finite choice sets) the existence of choices from a revealed preference point of view. There is also the methodological point that if you confront a decision-maker with a choice problem and chooses none of your options, then you haven't really included all options in the first place.

That being said, one can still study the mathematics of possibly empty-valued choice rules, and this has been done. One can find some material within this more general context in the book [Aleskerov, Fuad, Denis Bouyssou, and Bernard Monjardet. Utility maximization, choice and preference. Vol. 16. Springer Science & Business Media, 2007.]