<p>You are correct that the motivation regards issues surrounding completeness. As you point out, when <span class="math-container">$\succcurlyeq$</span> is complete <span class="math-container">$y \not\succcurlyeq x$</span> implies <span class="math-container">$x \succcurlyeq x$</span> (hence, <span class="math-container">$x \succ y$</span>).</p>
<p>If, however, we do not want to assume complete preferences then (2) does not imply (1), but strict preference can still constructed from weak using the more complex definition. (One could also take the set difference of weak preference and indifference).</p>

CC BY-SA 4.0
2020-09-29T19:31:59.307Z
2020-09-29T19:31:59.307Z
201p
<p>Strict Preference usually states that</p>
<p>x is strictly preferred to y if  :  &lt; x is weakly preferred to y and not y is weakly preferred to x &gt;.</p>
<p>Let me split the &lt; &gt; part into two segments:</p>
<ol>
<li>x is weakly preferred to y</li>
</ol>
<p>and</p>
<ol start="2">
<li>not y is weakly preferred to x</li>
</ol>
<p>Why do you need the first condition to satisfy the strict preference relationship? The second condition seems to be enough to satisfy the strict preference relation. I mean, &quot;not y is weakly preferred to x&quot; means the same thing as &quot;x is strictly preferred to y&quot; right? So 1. seems to be unnecessary to me when trying to explain Strict Preference Relation.</p>
<p>Is it because we assume that &quot;completeness&quot; of preference isn't satisfied when we define strict preference relation? Therefore there is a third option of &quot;I don't know&quot; if we just write the 2nd condition which requires us to add the first condition as well to cover for &quot;non-completeness&quot;?</p>

<consumer-theory><preferences>
Question about Strict Preference Relation
Robin311
# Question about Strict Preference Relation

(Asked by: username_0 on 2020-09-29)

Strict Preference usually states that

x is strictly preferred to y if : < x is weakly preferred to y and not y is weakly preferred to x >.

Let me split the < > part into two segments:

1. x is weakly preferred to y

and

2. not y is weakly preferred to x

Why do you need the first condition to satisfy the strict preference relationship? The second condition seems to be enough to satisfy the strict preference relation. I mean, "not y is weakly preferred to x" means the same thing as "x is strictly preferred to y" right? So 1. seems to be unnecessary to me when trying to explain Strict Preference Relation.

Is it because we assume that "completeness" of preference isn't satisfied when we define strict preference relation? Therefore there is a third option of "I don't know" if we just write the 2nd condition which requires us to add the first condition as well to cover for "non-completeness"?

**Tags:** consumer-theory, preferences

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## 1 answers

### Answer by username_1 on 2020-09-29. Score: 3

You are correct that the motivation regards issues surrounding completeness. As you point out, when $\succcurlyeq$ is complete $y \not\succcurlyeq x$ implies $x \succcurlyeq x$ (hence, $x \succ y$).

If, however, we do not want to assume complete preferences then (2) does not imply (1), but strict preference can still constructed from weak using the more complex definition. (One could also take the set difference of weak preference and indifference).


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