<p>Yes, they are equivalent. Here is a formal proof by contradiction.</p>

<p><strong>WARP $\Rightarrow$ Choice coherence</strong></p>

<p>Suppose that WARP holds but that choice coherence is not true. There exists $B,B'$, $x,y \in B \cap B'$ such that $x \in c(B)$, $y \notin c(B)$ and $y \in c(B')$. </p>

<p>But WARP applied to the conditions $y \in c(B')$ and $x \in c(B)$ implies $y \in c(B)$. This is a contradiction.</p>

<p><strong>Choice coherence $\Rightarrow$ WARP</strong></p>

<p>Suppose now that choice coherence is true but that WARP is falsified. There exists $B,B'$, $x,y \in B \cap B'$ such that $x \in c(B), y \in c(B')$ and $x \notin c(B')$. </p>

<p>But choice coherence applied to the conditions $y \in c(B')$, $x \notin c(B')$ yields $x \notin c(B)$. This is a contradiction.</p>

CC BY-SA 3.0
2015-06-01T07:57:36.393Z
This seems to have been a homework question.
Giskard
CC BY-SA 3.0
2015-06-02T16:36:32.257Z
No, This was due to the different versions of the definitions in MWG and Kreps. Former mentions it as weak axiom of revealed preference while Kreps calls it choice coherence. I just wanted to be sure of my understanding that economically they both mean the same. Doing self study for now, would be nice if people can point to some good sources to actually practice and build up on these concepts.
subrat
CC BY-SA 3.0
2015-06-01T07:02:38.843Z
2015-06-01T07:02:38.843Z
Oliv
<p>The new Kreps is clearer about it! Choice coherence and WARP are contrapositives. </p>

<p>Starting from choice coherence, write the contrapositive as:</p>

<p>For all x,y in B and B', </p>

<p>y in C(B') => not { [x in C(B)] and [y not in C(B)]} </p>

<p>which is equivalent to </p>

<p>y in C(B') => [x not in C(B)] or [y in C(B)]</p>

<p>So if x is in C(B), then y must be in C(B). </p>

CC BY-SA 3.0
2015-06-17T12:12:45.983Z
2015-06-17T12:12:45.983Z
is it normal
<p>$B$ and $B'$ are elements of the family of subsets of $X$</p>

<p>WARP<br>
For every pair $x,y \in B \cap B' $ and if $x \in c(B)$ , then if $y \in c(B'), x$ must $\in c(B').$</p>

<p>Choice Coherence<br>
For very pair $x,y \in B \cap B'$ and if $x \in c(B)$ and $y  \notin c(B)$, then $y$ must $\notin c(B').$</p>

<p>Are these two equivalent. If yes, how can we prove it?</p>

CC BY-SA 3.0
2015-06-01T13:50:14.703Z
you should attempt to prove it by your own means or at least show effort, this seems to be a homework question.
user157623
<microeconomics><choice-theory>
Weak axiom of revealed preference and choice coherence - how to show they are equivalent
subrat
# Weak axiom of revealed preference and choice coherence - how to show they are equivalent

(Asked by: username_0 on 2015-06-01)

$B$ and $B'$ are elements of the family of subsets of $X$

WARP  
For every pair $x,y \in B \cap B' $ and if $x \in c(B)$ , then if $y \in c(B'), x$ must $\in c(B').$

Choice Coherence  
For very pair $x,y \in B \cap B'$ and if $x \in c(B)$ and $y \notin c(B)$, then $y$ must $\notin c(B').$

Are these two equivalent. If yes, how can we prove it?

**Tags:** microeconomics, choice-theory
> **Comments:**
> **username_1**: you should attempt to prove it by your own means or at least show effort, this seems to be a homework question.  

---

## 2 answers

### Answer by username_2 on 2015-06-01. Score: 1 | Accepted answer

Yes, they are equivalent. Here is a formal proof by contradiction.

**WARP $\Rightarrow$ Choice coherence**

Suppose that WARP holds but that choice coherence is not true. There exists $B,B'$, $x,y \in B \cap B'$ such that $x \in c(B)$, $y \notin c(B)$ and $y \in c(B')$.

But WARP applied to the conditions $y \in c(B')$ and $x \in c(B)$ implies $y \in c(B)$. This is a contradiction.

**Choice coherence $\Rightarrow$ WARP**

Suppose now that choice coherence is true but that WARP is falsified. There exists $B,B'$, $x,y \in B \cap B'$ such that $x \in c(B), y \in c(B')$ and $x \notin c(B')$.

But choice coherence applied to the conditions $y \in c(B')$, $x \notin c(B')$ yields $x \notin c(B)$. This is a contradiction.

> **Comments:**
> **username_3**: This seems to have been a homework question.  
> **username_0**: No, This was due to the different versions of the definitions in MWG and Kreps. Former mentions it as weak axiom of revealed preference while Kreps calls it choice coherence. I just wanted to be sure of my understanding that economically they both mean the same. Doing self study for now, would be nice if people can point to some good sources to actually practice and build up on these concepts.  

---

### Answer by username_4 on 2015-06-17. Score: 0

The new Kreps is clearer about it! Choice coherence and WARP are contrapositives.

Starting from choice coherence, write the contrapositive as:

For all x,y in B and B',

y in C(B') => not { [x in C(B)] and [y not in C(B)]}

which is equivalent to

y in C(B') => [x not in C(B)] or [y in C(B)]

So if x is in C(B), then y must be in C(B).


---

