<p>Rationality requires the following:</p>

<p><strong>Completeness</strong></p>

<p>For all <span class="math-container">$x, y \in X$</span>, either <span class="math-container">$x \succsim y$</span> or <span class="math-container">$y \succsim x$</span> or both.</p>

<p><strong>Transitivity</strong></p>

<p>For all <span class="math-container">$x, y, z \in X$</span>, if <span class="math-container">$x \succsim y$</span> and <span class="math-container">$y \succsim z$</span>, then <span class="math-container">$x \succsim z$</span>.</p>

<p>Also note that if <span class="math-container">$x=y$</span> then completeness implies that <span class="math-container">$x\succsim x$</span>. So reflexive preferences follow from completeness. So I would say Completeness and Transitivity are required for rationality. Reflexive preferences just follow from complete preferences.  </p>

CC BY-SA 4.0
2019-11-28T17:24:59.203Z
2019-11-28T17:24:59.203Z
lunar_props
<p>I've been thinking about preferences alot recently and have been specifically thinking about the reflexivity requirement.</p>

<p>That is:
<span class="math-container">$$x \succsim x$$</span></p>

<p>Though this is apparent and obvious, I have been wondering why this is a required condition for rationality and if its possible to have a preference relation that is complete but non-reflexive.</p>

<p>Is it possible?</p>

CC BY-SA 4.0
2019-11-28T15:52:31.273Z
Interesting.  Would it be too limiting to express this as "I feel worse-off if I have to give up a unit widget in exchange for the exact same unit widget?"
heh
CC BY-SA 4.0
2019-11-28T15:56:24.400Z
@heh reflexivity states an object is at least as good as itself. What im asking is that what if $x \succsim x$ is not true? Does this mean that if presented with this item $x$ you would not consume it on its own?
EconJohn
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2019-11-28T16:08:40.090Z
If you took the inverse of reflexivity logically - using your words - it would be that "An object is worse than itself".  This seems to be a different concept than whether one would prefer the object "on its own", no?  Not trying to be pedantic - this is an interesting question and I just want to clarify what you are asking.
heh
CC BY-SA 4.0
2019-11-28T16:12:43.843Z
@heh no worries! Im just as confused. I think completeness may be violated but not transitivity. In anycase im not sure
EconJohn
CC BY-SA 4.0
2019-11-28T16:19:55.857Z
I'm not confident enough to post this as an answer, but I think you lose transitivity.  Consider: $y \succ x \prec x$.
heh
<microeconomics><mathematical-economics><preferences>
Can we have a Non-Reflexive Preference Relation?
EconJohn
# Can we have a Non-Reflexive Preference Relation?

(Asked by: username_0 on 2019-11-28)

I've been thinking about preferences alot recently and have been specifically thinking about the reflexivity requirement.

That is:
$$x \succsim x$$

Though this is apparent and obvious, I have been wondering why this is a required condition for rationality and if its possible to have a preference relation that is complete but non-reflexive.

Is it possible?

**Tags:** microeconomics, mathematical-economics, preferences

> **Comments:**
> **username_1**: Interesting.  Would it be too limiting to express this as "I feel worse-off if I have to give up a unit widget in exchange for the exact same unit widget?"  
> **username_0**: @username_1 reflexivity states an object is at least as good as itself. What im asking is that what if $x \succsim x$ is not true? Does this mean that if presented with this item $x$ you would not consume it on its own?  
> **username_1**: If you took the inverse of reflexivity logically - using your words - it would be that "An object is worse than itself".  This seems to be a different concept than whether one would prefer the object "on its own", no?  Not trying to be pedantic - this is an interesting question and I just want to clarify what you are asking.  
> **username_0**: @username_1 no worries! Im just as confused. I think completeness may be violated but not transitivity. In anycase im not sure  
> **username_1**: I'm not confident enough to post this as an answer, but I think you lose transitivity.  Consider: $y \succ x \prec x$.  

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## 1 answers

### Answer by username_2 on 2019-11-28. Score: 4 | Accepted answer

Rationality requires the following:

**Completeness**

For all $x, y \in X$, either $x \succsim y$ or $y \succsim x$ or both.

**Transitivity**

For all $x, y, z \in X$, if $x \succsim y$ and $y \succsim z$, then $x \succsim z$.

Also note that if $x=y$ then completeness implies that $x\succsim x$. So reflexive preferences follow from completeness. So I would say Completeness and Transitivity are required for rationality. Reflexive preferences just follow from complete preferences.


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