<p>I am a little confused about the statement of 1.D.5 in MWG, which I will reproduce here for convenience. I have &quot;solved&quot; the problem, I just don't understand something particular.</p>
<p><span class="math-container">$\textbf{(1.D.5)}$</span> Let <span class="math-container">$X = \{x,y,z\}$</span> and <span class="math-container">$\mathscr{B} = \{ \{x,y\},\{y,z\},\{z,x\} \}$</span>. Suppose that choice is now stochastic in the sense that, for every <span class="math-container">$B \in \mathscr{B}$</span>, <span class="math-container">$C(B)$</span> is a frequency distribution over alternatives in <span class="math-container">$B$</span>. For example, if <span class="math-container">$B = \{x,y\}$</span>, we write <span class="math-container">$C(B) = (C_x(B), C_y(B))...$</span></p>
<p>So, my issue is that it seems that <span class="math-container">$C : \mathscr{B} \to \mathbb{R}_+^2$</span> is not well-defined, and would be more appropriately defined as a function <span class="math-container">$A (\subseteq X^2) \to \mathbb{R}_+^2$</span>. What am I missing?</p>

CC BY-SA 4.0
2020-09-05T11:29:47.947Z
On second thought, I suppose any function is well-defined if you explicitly enumerate each input/output pair. But the problem doesn’t do that.
Seh-kai 
CC BY-SA 4.0
2020-09-05T17:17:46.613Z
Isn't $\mathscr B\subset X^2$? The book also specifies the range of the $C$ function as the set of non-negative 2-vectors whose elements sum to $1$.
Herr K.
CC BY-SA 4.0
2020-09-05T19:29:47.893Z
I believe $\mathscr{B}$ is a subset of the powerset of $X$; i.e. all subsets of $X$. My issue is that I don't know what $C(\{ z,x \})$ is supposed to be. Is it $= (C_z(B),C_x(B))? $ Or is it $= (C_x(B),C_z(B))$? The outputs of $C$ are ordered based on the elements of $B$, but since $B$ is a set, $B$ carries no ordering.
Seh-kai 
CC BY-SA 4.0
2020-09-05T20:55:04.337Z
You're right about $B$ having no order. I missed that part.
Herr K.
<microeconomics><mwg>
Parsing problem 1.D.5 in MWG
Seh-kai 
# Parsing problem 1.D.5 in MWG

(Asked by: username_0 on 2020-09-05)

I am a little confused about the statement of 1.D.5 in MWG, which I will reproduce here for convenience. I have "solved" the problem, I just don't understand something particular.

$\textbf{(1.D.5)}$ Let $X = \{x,y,z\}$ and $\mathscr{B} = \{ \{x,y\},\{y,z\},\{z,x\} \}$. Suppose that choice is now stochastic in the sense that, for every $B \in \mathscr{B}$, $C(B)$ is a frequency distribution over alternatives in $B$. For example, if $B = \{x,y\}$, we write $C(B) = (C\_x(B), C\_y(B))...$

So, my issue is that it seems that $C : \mathscr{B} \to \mathbb{R}\_+^2$ is not well-defined, and would be more appropriately defined as a function $A (\subseteq X^2) \to \mathbb{R}\_+^2$. What am I missing?

**Tags:** microeconomics, mwg

> **Comments:**
> **username_0**: On second thought, I suppose any function is well-defined if you explicitly enumerate each input/output pair. But the problem doesn’t do that.  
> **username_1**: Isn't $\mathscr B\subset X^2$? The book also specifies the range of the $C$ function as the set of non-negative 2-vectors whose elements sum to $1$.  
> **username_0**: I believe $\mathscr{B}$ is a subset of the powerset of $X$; i.e. all subsets of $X$. My issue is that I don't know what $C(\{ z,x \})$ is supposed to be. Is it $= (C_z(B),C_x(B))? $ Or is it $= (C_x(B),C_z(B))$? The outputs of $C$ are ordered based on the elements of $B$, but since $B$ is a set, $B$ carries no ordering.  
> **username_1**: You're right about $B$ having no order. I missed that part.  

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## 0 answers

