# Can we have a Non-Reflexive Preference Relation?

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?

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

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.

• 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.
– user18
Feb 18, 2020 at 3:43