<p>Usually one defines <span class="math-container">$a \sim b$</span> if <span class="math-container">$a \succeq b $</span> and <span class="math-container">$b \succeq a$</span> and one defines <span class="math-container">$a \succ b$</span> if <span class="math-container">$a \succeq b$</span> and not <span class="math-container">$b \succeq a$</span>.</p>
<p>Now assume there are only two alternatives <span class="math-container">$a$</span> and <span class="math-container">$b$</span>.</p>
<ol>
<li>If <span class="math-container">$a \succ b$</span> then <span class="math-container">$\sim$</span> is not complete as neither <span class="math-container">$a \sim b$</span> nor <span class="math-container">$b \sim a$</span></li>
<li>If <span class="math-container">$a \sim b$</span> then <span class="math-container">$\succ$</span> is not complete as neither <span class="math-container">$a \succ b$</span> nor <span class="math-container">$b \succ a$</span>.</li>
</ol>

CC BY-SA 4.0
2022-01-15T15:40:50.693Z
2022-01-15T15:40:50.693Z
tdm
<p>I have a question about preference relations and completeness.</p>
<p>Prove that:</p>
<p>i) There exists a complete relation <span class="math-container">$\succcurlyeq$</span>, such that <span class="math-container">$\sim$</span> is not complete.</p>
<p>ii) There exists a complete relation <span class="math-container">$\succcurlyeq$</span>, such that <span class="math-container">$\succ$</span> is not complete.</p>
<p>I tried work on top of the definitions of the relations, but to no avail.</p>
<p>Could anyone give me some help?</p>

CC BY-SA 4.0
2022-01-15T15:03:41.910Z
Take your favorite utility function. Chances are, the preferences represented by it are the desired counterexample for both problems.
Michael Greinecker
CC BY-SA 4.0
2022-01-15T16:01:48.233Z
A complete relation $R$ is one where for all $a,b$ you have $a\,R \,b$ or $b \, R\, a$ (possibly both)
Henry
<preferences><self-study>
A matter of completeness and preference relations
chicoluz10
# A matter of completeness and preference relations

(Asked by: username_0 on 2022-01-15)

I have a question about preference relations and completeness.

Prove that:

i) There exists a complete relation $\succcurlyeq$, such that $\sim$ is not complete.

ii) There exists a complete relation $\succcurlyeq$, such that $\succ$ is not complete.

I tried work on top of the definitions of the relations, but to no avail.

Could anyone give me some help?

**Tags:** preferences, self-study
> **Comments:**
> **username_1**: Take your favorite utility function. Chances are, the preferences represented by it are the desired counterexample for both problems.  
> **username_2**: A complete relation $R$ is one where for all $a,b$ you have $a\,R \,b$ or $b \, R\, a$ (possibly both)  

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## 1 answers

### Answer by username_3 on 2022-01-15. Score: 2

Usually one defines $a \sim b$ if $a \succeq b $ and $b \succeq a$ and one defines $a \succ b$ if $a \succeq b$ and not $b \succeq a$.

Now assume there are only two alternatives $a$ and $b$.

1. If $a \succ b$ then $\sim$ is not complete as neither $a \sim b$ nor $b \sim a$
2. If $a \sim b$ then $\succ$ is not complete as neither $a \succ b$ nor $b \succ a$.


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