WARP implies choice function is raionalizable.

Say we have a choice function $C(B)$, $B$ is a closed convex compact set. I am looking for a intuitive example of $C$. The $C$ is economic meaningful, and it violates transitivity (I'll explain what transitivity means).

Let the relation $\succsim$ be defined as such: $x=C(B)$ and $y\in B$ implies $x\succsim y$. $x\succsim y$ and $y\not\succsim x$ implies $x\succ y$. The transitivity is transitivity of $\succsim$.

By economic meaningful, here is what I mean: the choice function is observed or is meant to be observed or is plausible in some real-life economic situations. For one example, one constraint is that, let's consider the $R^2$ set, $B$ is a close convex compact set. The choice function needs to be monotonic, such that if $x\geq y$ imply $x\succsim y$.

For another example, requiring $B$ to be convex compact is economically intuitive, as, if you can buy two apple or two banana, then you should have enough money to buy one apple and one banana most of the times.



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