<p>In an economic context, the most natural interpretation of a choice function is that it indicates the alternative(s) that most satisfies a consumer. Thus, every consumer could have a potentially different choice function.</p>
<p>IIA imposes a consistency requirement that filters out a <em>type</em>, rather than any specific number, of choice functions. Imagine the following (partial) choice function:
<span class="math-container">\begin{align}
c(\{x,y\}) &amp; = \{x\} \\
c(\{x,y,z\}) &amp; = \{y\} \\
c(\{x,z\}) &amp; = \{x\} \\
c(\{y,z\}) &amp; = \{y\} \\
\end{align}</span>
So the consumer chooses <span class="math-container">$x$</span> when only <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are feasible, but chooses <span class="math-container">$y$</span> when <span class="math-container">$x,y,z$</span> are all feasible, and <span class="math-container">$z$</span> is deemed inferior to both <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. This behavior is not uncommon in reality (e.g. <a href="https://en.wikipedia.org/wiki/Decoy_effect" rel="nofollow noreferrer">decoy effect</a>). However, this would be the type of choice function that IIA seeks to rule out, because the introduction of a clearly inferior and thus irrelevant alternative <span class="math-container">$z$</span> should not reverse the ranking <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
<hr />
<p><strong>Reponse to <a href="https://economics.stackexchange.com/questions/59131/choice-function-and-iia/59136?noredirect=1#comment98254_59136">comment</a></strong></p>
<p>To determine the <em>unique</em> number of possible choice functions for a given set of alternatives, we need some notations. Recall that a choice function is a set-valued function (or correspondence). Given a set <span class="math-container">$X$</span> of alternatives, let <span class="math-container">$\mathcal A$</span> be the set of all possible nonempty subsets of <span class="math-container">$X$</span>, where <span class="math-container">$A\in\mathcal A$</span> is interpreted as a <em>feasible</em> set of alternatives (e.g. those within a consumer's budget), and <span class="math-container">$\mathcal A$</span> is a collection of all possible such feasible sets. A choice function <span class="math-container">$c(\cdot)$</span> maps each feasible set <span class="math-container">$A$</span> to one of its own subsets, i.e. <span class="math-container">$c(A)\subseteq A$</span>. Since a choice function is typically assumed to be nonempty, then for each <span class="math-container">$A$</span>, the number of possible value <span class="math-container">$c(A)$</span> is <span class="math-container">$2^{|A|}-1$</span>.</p>
<p>Let's define two choice functions <span class="math-container">$c_1$</span> and <span class="math-container">$c_2$</span> to be distinct if there exists at least one <span class="math-container">$A\in\mathcal A$</span> such that <span class="math-container">$c_1(A)\ne c_2(A)$</span>. Then the problem becomes finding the number of ways to combine elements from all <span class="math-container">$A\in\mathcal A$</span>. Therefore, the number of unique possible choice functions on <span class="math-container">$(X,\mathcal A)$</span> is
<span class="math-container">\begin{equation}
\prod_{A\in\mathcal A}(2^{|A|}-1) \label{num} \tag{1}
\end{equation}</span></p>
<p>Suppose <span class="math-container">$X=\{x,y,z\}$</span> and let <span class="math-container">$\mathcal A$</span> be the power set of <span class="math-container">$X$</span> (excluding <span class="math-container">$\varnothing$</span>). More explicitly,
<span class="math-container">\begin{equation}
\mathcal A = \Bigl\{
\{x\},\{y\},\{z\},
\{x,y\},\{x,z\},\{y,z\},
\{x,y,z\} \Bigr\}
\end{equation}</span>
According to <span class="math-container">$\eqref{num}$</span>, the unique number of possible choice functions is
<span class="math-container">\begin{equation}
1\times1\times1\times7\times7\times7\times127=43,561
\end{equation}</span></p>
<p>To find out how IIA cuts down on this number is tedious though not impossible. It also depends on the version of IIA being used. I'll leave that exercise to you.</p>

CC BY-SA 4.0
2024-09-12T07:15:23.183Z
Thanks this clarifies a bit. I was thinking that isn't it possible to compute all possible different choice functions when the options are limited. If we consider infinite amount of people for example surely people must have identical choice functions when the set is limtited?
Diego
CC BY-SA 4.0
2024-09-12T15:27:03.823Z
@Diego: I've updated my answer.
Herr K.
CC BY-SA 4.0
2024-09-11T03:09:50.120Z
2024-09-12T15:26:32.283Z
2024-09-12T15:26:32.283Z
Herr K.
<p>I'm new to microeconomics and struggle to understand choice function in practice. E.g. lets say I have a set X=(a,b,c). How many possible choice functions I could in theory have? Then other concept that relates to this is the IIA (Independence of irrelevant alternatives). Lets say that the choice function should fulfill this property, how it narrows down the number of choice functions?</p>

<microeconomics><choice-theory>
Choice function and IIA
Diego
# Choice function and IIA

(Asked by: username_0 on 2024-09-10)

I'm new to microeconomics and struggle to understand choice function in practice. E.g. lets say I have a set X=(a,b,c). How many possible choice functions I could in theory have? Then other concept that relates to this is the IIA (Independence of irrelevant alternatives). Lets say that the choice function should fulfill this property, how it narrows down the number of choice functions?

**Tags:** microeconomics, choice-theory


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## 1 answers

### Answer by username_1 on 2024-09-11. Score: 1

In an economic context, the most natural interpretation of a choice function is that it indicates the alternative(s) that most satisfies a consumer. Thus, every consumer could have a potentially different choice function.

IIA imposes a consistency requirement that filters out a *type*, rather than any specific number, of choice functions. Imagine the following (partial) choice function:
\begin{align}
c(\{x,y\}) & = \{x\} \\
c(\{x,y,z\}) & = \{y\} \\
c(\{x,z\}) & = \{x\} \\
c(\{y,z\}) & = \{y\} \\
\end{align}
So the consumer chooses $x$ when only $x$ and $y$ are feasible, but chooses $y$ when $x,y,z$ are all feasible, and $z$ is deemed inferior to both $x$ and $y$. This behavior is not uncommon in reality (e.g. [decoy effect](https://en.wikipedia.org/wiki/Decoy_effect)). However, this would be the type of choice function that IIA seeks to rule out, because the introduction of a clearly inferior and thus irrelevant alternative $z$ should not reverse the ranking $x$ and $y$.

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**Reponse to [comment](https://economics.stackexchange.com/questions/59131/choice-function-and-iia/59136?noredirect=1#comment98254_59136)**

To determine the *unique* number of possible choice functions for a given set of alternatives, we need some notations. Recall that a choice function is a set-valued function (or correspondence). Given a set $X$ of alternatives, let $\mathcal A$ be the set of all possible nonempty subsets of $X$, where $A\in\mathcal A$ is interpreted as a *feasible* set of alternatives (e.g. those within a consumer's budget), and $\mathcal A$ is a collection of all possible such feasible sets. A choice function $c(\cdot)$ maps each feasible set $A$ to one of its own subsets, i.e. $c(A)\subseteq A$. Since a choice function is typically assumed to be nonempty, then for each $A$, the number of possible value $c(A)$ is $2^{|A|}-1$.

Let's define two choice functions $c\_1$ and $c\_2$ to be distinct if there exists at least one $A\in\mathcal A$ such that $c\_1(A)\ne c\_2(A)$. Then the problem becomes finding the number of ways to combine elements from all $A\in\mathcal A$. Therefore, the number of unique possible choice functions on $(X,\mathcal A)$ is
\begin{equation}
\prod\_{A\in\mathcal A}(2^{|A|}-1) \label{num} \tag{1}
\end{equation}

Suppose $X=\{x,y,z\}$ and let $\mathcal A$ be the power set of $X$ (excluding $\varnothing$). More explicitly,
\begin{equation}
\mathcal A = \Bigl\{
\{x\},\{y\},\{z\},
\{x,y\},\{x,z\},\{y,z\},
\{x,y,z\} \Bigr\}
\end{equation}
According to $\eqref{num}$, the unique number of possible choice functions is
\begin{equation}
1\times1\times1\times7\times7\times7\times127=43,561
\end{equation}

To find out how IIA cuts down on this number is tedious though not impossible. It also depends on the version of IIA being used. I'll leave that exercise to you.

> **Comments:**
> **username_0**: Thanks this clarifies a bit. I was thinking that isn't it possible to compute all possible different choice functions when the options are limited. If we consider infinite amount of people for example surely people must have identical choice functions when the set is limtited?  
> **username_1**: @username_0: I've updated my answer.  

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