# 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 '19 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 '19 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 '19 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 '19 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 '19 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. – Kenny LJ Feb 18 at 3:43

Upon discussing this idea with my peers it seems that the reflexivity axiom just means that the group of goods which we are comparing are well defined. The traditional reading of the definition being "a good is at least as good as itself" is an extremely exact way of summing up this idea.

An example of non-reflexive preferences could be explained via example:

Say you walk into a pizza store and have decided to choose a pepperoni slice of pizza out of all the variety of items on the menu , the slices that you see for sale over the counter are cut in a extremely uneven way.

Thus we can see that though we can have complete and transitive preferences, we do not find reflexivity! (since some slices are bigger than the others).

This is a result of misspecified commodity groups or inability to separate them into their own groups (organizing slices not only by type but by size as well).

Is your preference across all these slices the same?

• For which slice of pizza $x$ do you strictly prefer $x$ to $x$? – Kenny LJ Feb 18 at 3:44
• @KennyLJ its with reference to having well defined goods. Meaning that though we label all the pepperoni slices as an $x$ we have dispersion within the group. This is what i think though as of now. – EconJohn Feb 18 at 3:49
• So, you first declare that all pizza slices are exactly identical. But to arrive at your desired conclusion, you then argue that some slices are bigger (and therefore more desirable) than others. Unfortunately, this latter argument contradicts your initial declaration. – Kenny LJ Feb 18 at 3:55
• @KennyLJ no, i said pepperoni slices would be prefered to all menu items, however there is preference in which particular slice of the pepperoni pizza to choose. – EconJohn Feb 18 at 3:57