# What is the difference between preferences lacking “completeness” and being indifferent?

If two preferences are complete, at least one must have a relationship to the other. If neither has a relationship, doesn't that mean that the consumer doesn't care which one he/she purchases?

• I don't understand what you are asking. Preferences describe how you evaluate bundles, But you seem to be talking about the bundles themselves. Perhaps you could provide an two good example to clarify what you. – BKay Feb 4 '16 at 17:02
• The consumer wouldn't know if she cares or not if completeness doesn't hold. E.g. I don't know if or how much I like caviar compared to bread. – BB King Feb 4 '16 at 18:32
• I'm referring to the completeness theorem for rational consumers. – bloopton Feb 4 '16 at 20:33
• @BBKing so a lack of completeness is a lack of enough information to decide one's preference? – bloopton Feb 4 '16 at 20:33
• If you never had a pizza or a hot dog before, and I asked you which one you liked better, saying "I don't know" doesn't mean you are indifferent. It means your preferences are incomplete. Colloquially in English, "not having a preference" can either mean you know you don't prefer either option more strongly OR it means you don't know at all how much you like one or both of the options, so they are incomparable to you. – Kitsune Cavalry Feb 4 '16 at 21:46

As alluded to in the comments the distinction is roughly:

• Indifference: The decision maker knows she will receive the same utility from the consumption of $x$ or $y$.
• Incompleteness: The DM does not know her preference between $x$ and $y$. (Note, this could stem from a lack of information, or because no preference exists)

So, from a conceptual vantage, the difference is that indifference is the existence of knowledge, whereas incompleteness is a lack of knowledge. While interesting in a foundational/philosophical sense, the distinction is subtle and likely not the reason economists have spent so much thought on the subject. However, the difference has behavioral implications! and this is important for economics.

For example, imagine various choices between "pizza", "hot dog", or "hot dog and $\$0.05$". Perhaps, the decision maker has never had pizza or hot dogs and therefore finds them incomparable. As such, from the choice set$\{p,h\}$she would choose either (since neither option is known to be inferior[1]). So$C(\{p,h\}) = \{p,h\}$. Now, the decision maker also likes money, so although she does not know her preference for a hot dog, she will always choose more money to less, all else equal:$C(\{h, h+5\}) = \{h+5\}$. But her indecisiveness regarding the two foods might not be resolvable for 5 cents, so, it is reasonable that$C(\{p, h, h+5\}) = \{p, h+5\}$. But this violates the weak axiom of revealed preference. Such a choice function cannot be rationalized by a complete and transitive relation. Notice if the decision maker was actually indifference between$h$and$p$then,$C(\{p, h, h+5\}) = \{h+5\}\$. In summary: incompleteness can cause "thick" indifference curves, and this can violate the basic tenets of rational choice theory.

[1] There are other interpretations of choice under incompleteness, for example, allowing the choice function to return an empty set.

• "There are other interpretations of choice under incompleteness, for example, allowing the choice function to return an empty set." Do you have a reference for how to construct a choice function for incomplete preferences? – user7935 Jul 6 '16 at 10:16
• Eliaz and Ok (2006) find necessary and sufficent conditions for a choice function to return all undominated alternatives. I am not aware of an explicit reference regarding a choice function that allows for not choosing anything, but, my intuition suggests: non-emptyness of singleton sets if any only if reflexivity; WARP if and only if transitivity. Therefore the movement from generic non-emptiness to non-emptyness of singletons corresponds to the movement from rationalizing weak order to rationalizing pre-order. – 201p Jul 6 '16 at 19:19