<p>I think your notation makes this more confusing that is necessary since <span class="math-container">$A$</span> and <span class="math-container">$B$</span> represent sets of alternatives in one setting and alternatives in the other setting.</p>
<p>So write IIA as:</p>
<p>If <span class="math-container">$x\in C\big(\{x,y\}\big)$</span> and <span class="math-container">$y\notin C\big(\{x,y\}\big)$</span>, then <span class="math-container">$y\notin C\big(x,y,z\big)$</span>.</p>
<p>To prove this from choice coherence, simply choose <span class="math-container">$A$</span> and <span class="math-container">$B$</span> appropriately.</p>

CC BY-SA 4.0
2020-09-07T19:57:48.727Z
2020-09-07T19:57:48.727Z
Michael Greinecker
<p>Prove that Choice Coherence implies Independence of Irrelevant Alternatives (IIA).</p>
<p>From the definition of choice coherence, we have this:
A choice function c satisfies choice coherence if, for every pair x and y from X and
A and B from A, if x, y are in A and B, x in c(A), and y not in c(A), then y is not in c(B).</p>
<p>From the definition of IIA, we have this:
If A is preferred to B out of the choice set {A,B}, introducing a third option X, expanding the choice set to {A,B,X}, must not make B preferable to A.</p>
<p>The converse is not true (IIA does not imply choice coherence) since we can construct a counterexample, as per here:</p>
<p>C({x,y,z}) = {x}</p>
<p>C({x,y}) = {x}</p>
<p>C({x}) = {x}</p>
<p>C({y}) = {y}</p>
<p>C({x,y}) = {x,y}</p>
<p>Since IIA is the weaker assumption of the two, how can we show that choice coherence implies IIA? Thanks for your help.</p>

<microeconomics><preferences><choice-theory>
Prove that Choice Coherence Implies IIA
Anon
# Prove that Choice Coherence Implies IIA

(Asked by: username_0 on 2020-09-07)

Prove that Choice Coherence implies Independence of Irrelevant Alternatives (IIA).

From the definition of choice coherence, we have this:
A choice function c satisfies choice coherence if, for every pair x and y from X and
A and B from A, if x, y are in A and B, x in c(A), and y not in c(A), then y is not in c(B).

From the definition of IIA, we have this:
If A is preferred to B out of the choice set {A,B}, introducing a third option X, expanding the choice set to {A,B,X}, must not make B preferable to A.

The converse is not true (IIA does not imply choice coherence) since we can construct a counterexample, as per here:

C({x,y,z}) = {x}

C({x,y}) = {x}

C({x}) = {x}

C({y}) = {y}

C({x,y}) = {x,y}

Since IIA is the weaker assumption of the two, how can we show that choice coherence implies IIA? Thanks for your help.

**Tags:** microeconomics, preferences, choice-theory

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## 1 answers

### Answer by username_1 on 2020-09-07. Score: 2 | Accepted answer

I think your notation makes this more confusing that is necessary since $A$ and $B$ represent sets of alternatives in one setting and alternatives in the other setting.

So write IIA as:

If $x\in C\big(\{x,y\}\big)$ and $y\notin C\big(\{x,y\}\big)$, then $y\notin C\big(x,y,z\big)$.

To prove this from choice coherence, simply choose $A$ and $B$ appropriately.


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