<p>Let <span class="math-container">$f$</span> be a function with range of <span class="math-container">$\{-1,1\}$</span> and <span class="math-container">$f(x,y)=-f(y,x)$</span>.</p>
<p>Let the preference <span class="math-container">$\succ\subset X\times X$</span> where <span class="math-container">$x\succ y \iff f(x,y)=1$</span>.</p>
<p>In math we can define <span class="math-container">$f$</span> to be a constant function on a subset <span class="math-container">$Y\times Z\subset X\times X$</span>:</p>
<p><span class="math-container">$f(Y\times Z)=-1 \ OR \ 1$</span>.</p>
<p>In economics, without defining a function, is it standard to say that the preference is a &quot;constant&quot; at the subset?</p>
<p>For example: is it standard and clear to say &quot;<span class="math-container">$\succ$</span> over <span class="math-container">$Y\times Z$</span> is the same&quot;, or, &quot;for each <span class="math-container">$(y,z)\in Y\times Z$</span> the preference between <span class="math-container">$y$</span> and <span class="math-container">$z$</span> is the same&quot;?</p>
<hr />
<p>It is true that:</p>
<p>For each <span class="math-container">$y,z$</span> (the preference between <span class="math-container">$y,z$</span> is constant)</p>
<p><span class="math-container">$\iff$</span> Either for each <span class="math-container">$y,z$</span>: <span class="math-container">$y\succ z$</span>, or for each <span class="math-container">$y,z$</span>: <span class="math-container">$y\prec z$</span>.</p>
<p>However, we still don't know what is equivalent to the proposition in bracket (the preference between <span class="math-container">$y,z$</span> is constant).</p>
<p>It is possible that the proposition itself is ill-defined. But since we often say that a function is constant, I don't think saying a preference is constant is wrong.</p>
<p>I think the proposition in bracket is equivalent to:</p>
<p><span class="math-container">$\forall y',z': y\succ z \iff y'\succ z'$</span></p>
<p>(Of course, this assumes that <span class="math-container">$\succ=\not\prec$</span>. And there is one further problem in this expression.)</p>

CC BY-SA 4.0
2022-05-30T21:11:25.063Z
I have never seen anything like that. You could simply write "Either every element of $X$ is preferred to ever element of $Y$ or every element of $Y$ is preferred to $X$."
Michael Greinecker
CC BY-SA 4.0
2022-05-30T21:21:51.193Z
Hi @MichaelGreinecker I agree that your supplied statement might be equivalent to "constant preference". However, note that the supplied statement is not a [Prenex normal form](https://en.wikipedia.org/wiki/Prenex_normal_form). I will include more details in the question.
High GPA
CC BY-SA 4.0
2022-05-30T21:24:45.747Z
Is there a special reason you want to write everything in prenex normal form?
Michael Greinecker
CC BY-SA 4.0
2022-05-30T21:41:04.933Z
Hi @MichaelGreinecker I updated the question and while writing I think a find a possible (albeit a bit complicated) answer.
High GPA
CC BY-SA 4.0
2022-05-30T21:44:51.607Z
If you really want everything in prenex normal form: $\forall x\forall y:(x\in X\wedge y\in Y~\implies~x\succ y)\vee(x\in X\wedge y\in Y~\implies~y\succ x)$.
Michael Greinecker
CC BY-SA 4.0
2022-05-30T21:50:55.267Z
@MichaelGreinecker Thanks for your help, let me digest it a bit.
High GPA
<microeconomics><preferences><decision-theory>
Say a preference is "constant“, by analogy with a constant function
High GPA
# Say a preference is "constant“, by analogy with a constant function

(Asked by: username_0 on 2022-05-30)

Let $f$ be a function with range of $\{-1,1\}$ and $f(x,y)=-f(y,x)$.

Let the preference $\succ\subset X\times X$ where $x\succ y \iff f(x,y)=1$.

In math we can define $f$ to be a constant function on a subset $Y\times Z\subset X\times X$:

$f(Y\times Z)=-1 \ OR \ 1$.

In economics, without defining a function, is it standard to say that the preference is a "constant" at the subset?

For example: is it standard and clear to say "$\succ$ over $Y\times Z$ is the same", or, "for each $(y,z)\in Y\times Z$ the preference between $y$ and $z$ is the same"?

---

It is true that:

For each $y,z$ (the preference between $y,z$ is constant)

$\iff$ Either for each $y,z$: $y\succ z$, or for each $y,z$: $y\prec z$.

However, we still don't know what is equivalent to the proposition in bracket (the preference between $y,z$ is constant).

It is possible that the proposition itself is ill-defined. But since we often say that a function is constant, I don't think saying a preference is constant is wrong.

I think the proposition in bracket is equivalent to:

$\forall y',z': y\succ z \iff y'\succ z'$

(Of course, this assumes that $\succ=\not\prec$. And there is one further problem in this expression.)

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

> **Comments:**
> **username_1**: I have never seen anything like that. You could simply write "Either every element of $X$ is preferred to ever element of $Y$ or every element of $Y$ is preferred to $X$."  
> **username_0**: Hi @username_1 I agree that your supplied statement might be equivalent to "constant preference". However, note that the supplied statement is not a [Prenex normal form](https://en.wikipedia.org/wiki/Prenex_normal_form). I will include more details in the question.  
> **username_1**: Is there a special reason you want to write everything in prenex normal form?  
> **username_0**: Hi @username_1 I updated the question and while writing I think a find a possible (albeit a bit complicated) answer.  
> **username_1**: If you really want everything in prenex normal form: $\forall x\forall y:(x\in X\wedge y\in Y~\implies~x\succ y)\vee(x\in X\wedge y\in Y~\implies~y\succ x)$.  
> **username_0**: @username_1 Thanks for your help, let me digest it a bit.  

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## 0 answers

