# Weak axiom of Revealed Preference application

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?

Just because $b$ is chosen from the set $\{a,b\}$ doesn't mean that $b$ should always be chosen, e.g. from the set $\{b,c\}$. It just means that $b$ is revealed-preferred to $a$, but it doesn't mean that $b$ is preferred to all other possible alternatives (in particular, $c$). For example, the preference $c\succ b\succ a$ is perfectly consistent with the revealed choice pattern of the firm.
Write out the two conditions explicitly: \begin{align} C(\color{red}{\varnothing},\color{red}{\{a\}},\{b\},\color{red}{\{a,b\}})&=\{b\}\tag{1}\\ C(\color{red}{\varnothing},\color{red}{\{b\}},\{c\},\color{red}{\{b,c\}})&=\{c\}\tag{2} \end{align} $(1)$ just says that whenever the four choices --- hire no one, hire $a$ only, hire $b$ only, hire both --- are present, the firm hires $b$ only. Likewise, $(2)$ says whenever the four choices --- hire no one, hire $b$ only, hire $c$ only, hire both --- are present, the firm hires $c$ only. These two are consistent with WARP (for $(1)$, the x in your quoted definition is $a$ and y is $b$; for $(2)$, x is $b$ and y is $c$).
For restrictions on $C(2^{\{a,b,c\}})$, writing it out explicitly, we get $$C(\color{red}{\varnothing},\color{red}{\{a\}},\color{red}{\{b\}},\color{black}{\{c\}},\color{red}{\{a,b\}},\color{black}{\{a,c\}},\color{red}{\{b,c\}},\color{black}{\{a,b,c\}})=\{\{a\},\{a,c\},\{a,b,c\}\}$$ where the elements colored in red are revealed to be inferior according to $(1)$ and $(2)$.