<blockquote>
<p><strong>I am having trouble understanding what <span class="math-container">$C(\{x, y, z\}) = \{x, y\}$</span> means.</strong></p>
</blockquote>
<p>MWG already explain this on p.10:</p>
<blockquote>
<p>When [<span class="math-container">$C(B)$</span> contains more than one element], the elements of <span class="math-container">$C(B)$</span> are the alternatives in <span class="math-container">$B$</span> that the decision maker <em>might</em> choose; that is, they are her <em>acceptable alternatives</em> in <span class="math-container">$B$</span>. In this case, the set <span class="math-container">$C(B)$</span>  can be thought of as containing those alternatives that we would actually see chosen if the decision maker were repeatedly to face the problem of choosing an alternative from set <span class="math-container">$B$</span>.</p>
</blockquote>
<hr />
<blockquote>
<p><strong>[H]ow can we even have a situation where if the consumer is choosing between <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, he'll always choose <span class="math-container">$x$</span>, but then when he is faced between choosing <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, and <span class="math-container">$z$</span>, he will choose either <span class="math-container">$x$</span> or <span class="math-container">$y$</span>? Can someone give me a real life example of how this would be possible?</strong></p>
</blockquote>
<p>Consider a voting situation as follows. Individuals <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, and <span class="math-container">$z$</span> are potential candidates for the position of a department chair. A five-member committee is to select the chair by voting via simple majority rule. Let's label the committee members <span class="math-container">$1$</span> to <span class="math-container">$5$</span> and suppose that they each have a private ranking of the three candidates as follows (<span class="math-container">$\succ_i$</span> is member <span class="math-container">$i$</span>'s ranking):
<span class="math-container">\begin{align}
x\succ_1 y\succ_1 z \\
y\succ_2 x\succ_2 z \\
z\succ_3 x\succ_3 y \\
y\succ_4 z\succ_4 x \\
x\succ_5 z\succ_5 y 
\end{align}</span>
Suppose that each member will vote for his/her top-ranked candidate. Then when only two candidates <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are under consideration, <span class="math-container">$x$</span> will get three votes (from members <span class="math-container">$1$</span>, <span class="math-container">$3$</span> and <span class="math-container">$5$</span>) and thus always be chosen. But when all three candidates are being considered, <span class="math-container">$x$</span> and <span class="math-container">$y$</span> will each get two votes (<span class="math-container">$x$</span> from <span class="math-container">$1$</span> and <span class="math-container">$5$</span>, <span class="math-container">$y$</span> from <span class="math-container">$2$</span> and <span class="math-container">$4$</span>) and thus both be &quot;acceptable&quot; candidates.</p>

CC BY-SA 4.0
2020-10-25T05:02:23.703Z
I suppose I could have made the example based on the current [moderator election at Econ.SE](https://economics.stackexchange.com/election?cb=1), which also happens to have three candidates. But I had a hard time deciding who's $z$ `:)`
Herr K.
CC BY-SA 4.0
2020-10-26T13:21:21.780Z
The very reasonable voting example may miss the point that people are supposed to be independent of irrelevant alternatives, that is to say, OP's confusion is fair. Individual agents should NOT suddenly become interested in y after the addition of z.  Another example could be cars in colors RB vs RBG, where the buyer would always  pick R in RB but the presence of the nearby G car in choice RBG really makes the blue "pop".  It should't matter- but it sometimes does.
RegressForward
CC BY-SA 4.0
2020-10-26T14:52:19.303Z
@RegressForward: I'm not sure I understand what you mean by the comment "*people are supposed to be independent of irrelevant alternatives*". But in my example, I would say that it's not that voting members "suddenly become interested in $y$ after the addition of $z$", but rather that some of them become *less* interested in $x$ because the newly available $z$ draws vote(s) away from $x$.
Herr K.
CC BY-SA 4.0
2020-10-26T16:16:05.403Z
Right, but the question by OP is about individual agents (so voting problems are out). OP seems to be reading MWG's discussion of the irrelevance of independent alternatives, which for *individual agents* is typically assumed to hold.
RegressForward
CC BY-SA 4.0
2020-10-27T22:51:08.633Z
Thank you for explaining this!
Alex
CC BY-SA 4.0
2020-10-25T04:56:38.097Z
2020-10-25T04:56:38.097Z
Herr K.
<p>I am reading the Microeconomics Theory book by MWG, and I am having a tough time interpreting what things mean to a real life example, so any help would be appreciated.</p>
<p>For example, it gave this.  Suppose X = {x, y, z} and B, the budget set, = {{x, y}, {x, y, z}}.  It says we can define a choice rule so that C({x, y}) = {x} and C({x, y, z}) = {x, y}.</p>
<p>I am having trouble understanding what C({x, y, z}) = {x, y} means.  Does that mean that when faced with the choice of x, y, and z, the person is indifferent between x and y?  It doesn't mean he chooses both right?  It just means he will choose either or so it's random which one of the two he chooses?</p>
<p>Furthermore, how can we even have a situation where if the consumer is choosing between x and y, he'll always choose x, but then when he is faced between choosing x, y, and z, he will choose either x or y?  Can someone give me a real life example of how this would be possible?  That usually helps me understand better.</p>
<p>Thank you so much!  I appreciate any responses.</p>

<consumer-theory><preferences><choice-theory>
Understanding the Choice Rule in MWG
Alex
# Understanding the Choice Rule in MWG

(Asked by: username_0 on 2020-10-25)

I am reading the Microeconomics Theory book by MWG, and I am having a tough time interpreting what things mean to a real life example, so any help would be appreciated.

For example, it gave this. Suppose X = {x, y, z} and B, the budget set, = {{x, y}, {x, y, z}}. It says we can define a choice rule so that C({x, y}) = {x} and C({x, y, z}) = {x, y}.

I am having trouble understanding what C({x, y, z}) = {x, y} means. Does that mean that when faced with the choice of x, y, and z, the person is indifferent between x and y? It doesn't mean he chooses both right? It just means he will choose either or so it's random which one of the two he chooses?

Furthermore, how can we even have a situation where if the consumer is choosing between x and y, he'll always choose x, but then when he is faced between choosing x, y, and z, he will choose either x or y? Can someone give me a real life example of how this would be possible? That usually helps me understand better.

Thank you so much! I appreciate any responses.

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

---

## 1 answers

### Answer by username_1 on 2020-10-25. Score: 5

> **I am having trouble understanding what $C(\{x, y, z\}) = \{x, y\}$ means.**

MWG already explain this on p.10:

> When [$C(B)$ contains more than one element], the elements of $C(B)$ are the alternatives in $B$ that the decision maker *might* choose; that is, they are her *acceptable alternatives* in $B$. In this case, the set $C(B)$ can be thought of as containing those alternatives that we would actually see chosen if the decision maker were repeatedly to face the problem of choosing an alternative from set $B$.

---

> **[H]ow can we even have a situation where if the consumer is choosing between $x$ and $y$, he'll always choose $x$, but then when he is faced between choosing $x$, $y$, and $z$, he will choose either $x$ or $y$? Can someone give me a real life example of how this would be possible?**

Consider a voting situation as follows. Individuals $x$, $y$, and $z$ are potential candidates for the position of a department chair. A five-member committee is to select the chair by voting via simple majority rule. Let's label the committee members $1$ to $5$ and suppose that they each have a private ranking of the three candidates as follows ($\succ\_i$ is member $i$'s ranking):
\begin{align}
x\succ\_1 y\succ\_1 z \\
y\succ\_2 x\succ\_2 z \\
z\succ\_3 x\succ\_3 y \\
y\succ\_4 z\succ\_4 x \\
x\succ\_5 z\succ\_5 y
\end{align}
Suppose that each member will vote for his/her top-ranked candidate. Then when only two candidates $x$ and $y$ are under consideration, $x$ will get three votes (from members $1$, $3$ and $5$) and thus always be chosen. But when all three candidates are being considered, $x$ and $y$ will each get two votes ($x$ from $1$ and $5$, $y$ from $2$ and $4$) and thus both be "acceptable" candidates.

> **Comments:**
> **username_1**: I suppose I could have made the example based on the current [moderator election at Econ.SE](https://economics.stackexchange.com/election?cb=1), which also happens to have three candidates. But I had a hard time deciding who's $z$ `:)`  
> **username_2**: The very reasonable voting example may miss the point that people are supposed to be independent of irrelevant alternatives, that is to say, OP's confusion is fair. Individual agents should NOT suddenly become interested in y after the addition of z.  Another example could be cars in colors RB vs RBG, where the buyer would always  pick R in RB but the presence of the nearby G car in choice RBG really makes the blue "pop".  It should't matter- but it sometimes does.  
> **username_1**: @RegressForward: I'm not sure I understand what you mean by the comment "*people are supposed to be independent of irrelevant alternatives*". But in my example, I would say that it's not that voting members "suddenly become interested in $y$ after the addition of $z$", but rather that some of them become *less* interested in $x$ because the newly available $z$ draws vote(s) away from $x$.  
> **username_2**: Right, but the question by OP is about individual agents (so voting problems are out). OP seems to be reading MWG's discussion of the irrelevance of independent alternatives, which for *individual agents* is typically assumed to hold.  
> **username_0**: Thank you for explaining this!  

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