Choice function and IIA

Question

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?

Answer 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). 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

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.