The following is a problem I am dealing with related to Weak Axiom of Revealed Preference. I have given my solution below to the situation. What I am not getting is how is WARP not violated?
A law firm looking to hire to fill three positions gets applications from Andrew, Barbara and Celia.
The law firm's set of alternatives is the set of possible hiring decisions:
$ X = \{ \phi, \{a\}, \{b\}, \{c\}, \{a,b\}, \{b,c\}, \{a,c\}, \{a,b,c\} \} $
For any $Y \subset \{a,b,c\}$, define the power set of Y as
$ 2^{Y} \equiv \{Z | Z \subset Y \}$.
$ 2^{Y}$ is the set of hiring decisions that the firm can make when it receives applications from the lawyers in Y.
The law firm's budget sets $ B \in \mathcal{B} $ are the sets of hiring decisions it can make after receiving applications from some combination of Andrew, Barbara and Celia :
$\mathcal{B} = \{2^{Y} |Y \subset \{a, b, c \} \}$
1) When it receives applications from Andrew and Barbara, it will choose to hire Barbara (and not Andrew):-
$ C(2^{\{a,b\}}) = \{b\} $
2) When it receives applications from Barbara and Celia, it will choose to hire Celia (and not Barbara):
$ C(2^{\{b,c\}}) = \{c\} $
The following is the problem I am confused in : Q) What restrictions does the weak axiom place on the firm's hiring decision $C(2^{\{a,b,c\}})$ when it receives applications from Andrew, Barbara and Celia?
My Solution:
According to Mas-Colell et al (Definition 1.C.1) , the Weak Axiom of Revealed Preference says that if x is ever chosen when y is available then there can be no budget set containing both alternatives for which y is chosen and x is not.
So based on my understanding of WARP, in my situation above, when Andrew and Barbara apply the firm chooses Barbara , i.e. $ Barbara \succsim_R Andrew$ and when Barbara and Celia apply, the firm chooses Celia i.e. : $ Celia \succsim_R Barbara $
Here we see that since Barbara is not chosen over Celia , WARP is violated. Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set. So when Andrew , Barbara and Celia apply, and WARP violates the above relation given , the firm would hire only Andrew.
What I am not getting is how is WARP not violated?