<p>On <span class="math-container">$\mathbb R^2$</span>, define <span class="math-container">$x=(x_1,x_2)&gt;(y_1,y_2)=y$</span> if <span class="math-container">$x_i\geq y_i$</span> for all <span class="math-container">$i$</span> and <span class="math-container">$x_j&gt;y_j$</span> for some <span class="math-container">$j$</span>.</p>
<p>Let <span class="math-container">$\sim $</span> be an equivalence relation that <span class="math-container">$x\sim y$</span> implies <span class="math-container">$x\not&gt; y$</span>.</p>
<p>Define preorder <span class="math-container">$\succ$</span> this way:</p>
<ol>
<li><span class="math-container">$x&gt;y\implies x\succ y$</span>, and</li>
</ol>
<ol start="2">
<li><span class="math-container">$x\sim y&gt;w\sim z$</span> implies <span class="math-container">$x\succ z$</span> and similar for <span class="math-container">$\prec$</span>.</li>
</ol>
<p>That is, <span class="math-container">$\succsim$</span> is extended from <span class="math-container">$\geq$</span> and <span class="math-container">$\sim$</span> relation.</p>
<p><strong>Question:</strong> Do we need additional assumption to conclude that <span class="math-container">$\succsim$</span> is represented by a strictly increasing utility function?</p>
<p>Perhaps we need some continuity for <span class="math-container">$\sim$</span>?</p>
<hr />
<p><strong>Solution attempt:</strong></p>
<p>One assumption on <span class="math-container">$\sim$</span> seems both sufficient and necessary:</p>
<p>A1. The set <span class="math-container">$\{x:x\not\sim y\}$</span> is the union of two disjoint open sets.</p>
<p>Proof idea: Suppose <span class="math-container">$x_1&gt;x_2$</span> and <span class="math-container">$x=(x_1,x_2).$</span></p>
<p><span class="math-container">$(x_1,x_1)\succ x\succ (x_2,x_2)$</span>.</p>
<p>A1 implies that, for each <span class="math-container">$x$</span>, there exists a real number <span class="math-container">$a$</span> such that <span class="math-container">$x\sim (a,a)$</span>. Then define <span class="math-container">$u(x)=a$</span>.</p>

CC BY-SA 4.0
2022-06-11T11:33:02.820Z
Is the first line meant to say that $\succ$ extends the usual greater than relation on $\mathbb{R}$? I don't get the question.
Michael Greinecker
CC BY-SA 4.0
2022-06-11T12:03:36.160Z
@MichaelGreinecker Yes you are right. I'll add more explanation.
dodo
CC BY-SA 4.0
2022-06-11T18:12:10.463Z
If I understand you correctly, there is no proper extension like that.
Michael Greinecker
CC BY-SA 4.0
2022-06-12T06:41:55.930Z
@MichaelGreinecker Is it possible that an equivalence relation is represented by a function: $x\sim y\iff u(x)=u(y)$?
dodo
CC BY-SA 4.0
2022-06-12T07:07:19.017Z
Yes. All you need is that there are no more equivalence classes than real numbers.
Michael Greinecker
CC BY-SA 4.0
2022-06-14T05:28:20.720Z
@MichaelGreinecker I think I might have a solution updated. But I think the assumption can be further relaxed.
dodo
<microeconomics><reference-request><preferences><decision-theory>
Representation theorem for $\succsim\supset>\cup\sim$
dodo
# Representation theorem for $\succsim\supset>\cup\sim$

(Asked by: username_0 on 2022-06-11)

On $\mathbb R^2$, define $x=(x\_1,x\_2)>(y\_1,y\_2)=y$ if $x\_i\geq y\_i$ for all $i$ and $x\_j>y\_j$ for some $j$.

Let $\sim $ be an equivalence relation that $x\sim y$ implies $x\not> y$.

Define preorder $\succ$ this way:

1. $x>y\implies x\succ y$, and

2. $x\sim y>w\sim z$ implies $x\succ z$ and similar for $\prec$.

That is, $\succsim$ is extended from $\geq$ and $\sim$ relation.

**Question:** Do we need additional assumption to conclude that $\succsim$ is represented by a strictly increasing utility function?

Perhaps we need some continuity for $\sim$?

---

**Solution attempt:**

One assumption on $\sim$ seems both sufficient and necessary:

A1. The set $\{x:x\not\sim y\}$ is the union of two disjoint open sets.

Proof idea: Suppose $x\_1>x\_2$ and $x=(x\_1,x\_2).$

$(x\_1,x\_1)\succ x\succ (x\_2,x\_2)$.

A1 implies that, for each $x$, there exists a real number $a$ such that $x\sim (a,a)$. Then define $u(x)=a$.

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

> **Comments:**
> **username_1**: Is the first line meant to say that $\succ$ extends the usual greater than relation on $\mathbb{R}$? I don't get the question.  
> **username_0**: @username_1 Yes you are right. I'll add more explanation.  
> **username_1**: If I understand you correctly, there is no proper extension like that.  
> **username_0**: @username_1 Is it possible that an equivalence relation is represented by a function: $x\sim y\iff u(x)=u(y)$?  
> **username_1**: Yes. All you need is that there are no more equivalence classes than real numbers.  
> **username_0**: @username_1 I think I might have a solution updated. But I think the assumption can be further relaxed.  

---

## 0 answers

