Give an example that the following condition does not imply WARP

Question

I know how to prove that Weak Axiom of Revealed Preference (WARP) implies the following condition: if $a\in B_1, B_1 \subseteq B_2, a\in C(B_2)$, then $a\in C(B_1)$. $C$ here is a notation for choice correspondence. Could someone provide an example that the above condition does not imply WARP? Thanks.

Answer 1

Let $B_1 = \{w,x,y\}$ and $B_2 =\{x,y,z\}$. Let $C(B_1)=\{x\}$ and $C(B_2)=\{y\}$. This example satisfies your property (it holds vacuously), but fails WARP.