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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?

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  • $\begingroup$ 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?" $\endgroup$
    – heh
    Nov 28, 2019 at 15:52
  • $\begingroup$ @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? $\endgroup$
    – EconJohn
    Nov 28, 2019 at 15:56
  • $\begingroup$ 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. $\endgroup$
    – heh
    Nov 28, 2019 at 16:08
  • $\begingroup$ @heh no worries! Im just as confused. I think completeness may be violated but not transitivity. In anycase im not sure $\endgroup$
    – EconJohn
    Nov 28, 2019 at 16:12
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    $\begingroup$ I'm not confident enough to post this as an answer, but I think you lose transitivity. Consider: $y \succ x \prec x$. $\endgroup$
    – heh
    Nov 28, 2019 at 16:19

1 Answer 1

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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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    $\begingroup$ You are correct that Complete + Transitive (together sometimes dubbed Rationality) imply Reflexive. However, it is possible to have a preference that is Complete but not Reflexive. Example. Let the set of alternatives be $A=\{x,y\}$ and consider the preference $\succsim$ over $A$ given by $x\succ x$ and $x\succsim y$. Then $\succsim$ is Complete but not Reflexive. $\endgroup$
    – user18
    Feb 18, 2020 at 3:43

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