<p>A choice structure can be whatever it wants to be. A choice structure that is rationalizable by a preference relation must (among other things) satisfy Sen's $\alpha$. </p>

<p>So if you want to show that a rationalizable choice structure satisfies $\alpha$ then you can show that any that satisfies WARP will also satisfy $
\alpha$. </p>

<p>I recommend Kreps or the UPitt notes for more info on the relationship between choice structures and preference. </p>

<p><a href="http://www.pitt.edu/~luca/ECON2100/lecture_02.pdf" rel="nofollow">http://www.pitt.edu/~luca/ECON2100/lecture_02.pdf</a></p>

CC BY-SA 3.0
2016-09-11T17:59:58.660Z
So do you mean a choice structure satisfies the $\alpha$ condition only if we can find a rational preference $R$ s.t. $c_R=c$?
ask
CC BY-SA 3.0
2016-09-11T18:03:34.153Z
Wait. "only if" means sufficient right? Then no as $\alpha$ is necessary, but not sufficient.
VCG
CC BY-SA 3.0
2016-09-11T16:48:29.433Z
2016-09-11T17:06:27.000Z
2016-09-11T17:06:27.000Z
VCG
<p>I am trying to prove a statement that if $c_R$ is a choice structure, then it must satisfy Sen's $\alpha$ condition.</p>

<p>However, if $X=\{a,b,c \}$, I can construct some choice structure like $c(\{a\})=a,c(\{b\})=b, c(\{c\})=c, c(\{a,b\})=a,c(\{b,c\})=b,c(\{c,a\})=c,c(\{a,b,c\})=a$.</p>

<p>Such choice structure clearly violates the $\alpha$ (shrinking) condition.</p>

<p>So can someone point out where do I make a mistake? Or is the statement assuming the relation $R$ to be rational?</p>

<p>Thanks!</p>

<microeconomics><academic-graduate>
$\alpha$ condition and choice structure
ask
# $\alpha$ condition and choice structure

(Asked by: username_0 on 2016-09-11)

I am trying to prove a statement that if $c\_R$ is a choice structure, then it must satisfy Sen's $\alpha$ condition.

However, if $X=\{a,b,c \}$, I can construct some choice structure like $c(\{a\})=a,c(\{b\})=b, c(\{c\})=c, c(\{a,b\})=a,c(\{b,c\})=b,c(\{c,a\})=c,c(\{a,b,c\})=a$.

Such choice structure clearly violates the $\alpha$ (shrinking) condition.

So can someone point out where do I make a mistake? Or is the statement assuming the relation $R$ to be rational?

Thanks!

**Tags:** microeconomics, academic-graduate

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## 1 answers

### Answer by username_1 on 2016-09-11. Score: 1

A choice structure can be whatever it wants to be. A choice structure that is rationalizable by a preference relation must (among other things) satisfy Sen's $\alpha$.

So if you want to show that a rationalizable choice structure satisfies $\alpha$ then you can show that any that satisfies WARP will also satisfy $
\alpha$.

I recommend Kreps or the UPitt notes for more info on the relationship between choice structures and preference.

<http://www.pitt.edu/~luca/ECON2100/lecture_02.pdf>

> **Comments:**
> **username_0**: So do you mean a choice structure satisfies the $\alpha$ condition only if we can find a rational preference $R$ s.t. $c_R=c$?  
> **username_1**: Wait. "only if" means sufficient right? Then no as $\alpha$ is necessary, but not sufficient.  

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