# Properties of orders and preference relations

Suppose I have alternatives $A$, $B$, and $C$. If I have strict preferences, that means that for any $x,y \in \{A,B,C\}$ such that $x \ne y$, either $x \succ y$ or $y \succ x$. Assume transitivity, non-reflexivity, completeness.

Things get a lot less clear to me when I'm trying to talk about it as an order. How would I say that an agent has an order over the alternatives that would allow some ties? And how would I say an agent has an order over the alternatives that would not allow ties? Is it just the intuitive weak total order and strict total order, over alternatives where the weak and strict carry the same meaning as for preference relations and total is a synonym for complete?

You are right, total is a synonym for complete, it simply means that two elements can always be compared.

Regarding your other point, I don't think that your definitions of strict order and weak order are common but they make perfect sense. Notice, however, that they differ from some definitions used by mathematicians and therefore you have to specify what you are talking about to avoid possible confusions (you can check for instance "strict weak ordering").

• Thanks. Indeed, reading about strict weak orderings has given me a real headache. Let me restate my question: If I want to talk about orders in a way that makes sense to economists and is formally correct to mathematicians, how best would I refer to the two alternatives I'm considering? Nov 24 '15 at 21:20
• @Shane I don't see a best option than simply "total order" and "total order without indifference". I am not aware of a better convention, and I think each author uses his/her own definition and explanation for it.
– Oliv
Nov 24 '15 at 21:37
• Ahh, I see... That's a good bad answer. Good in that it answers my question. Bad in that it is rather unsatisfying! I'll leave it open for a little while to see if anybody else wants to chime in, but thank you for your advice. Nov 24 '15 at 21:39
• My pleasure! Please leave it open, if someone comes with a better answer I would be happy to know it as well.
– Oliv
Nov 24 '15 at 21:41

I don't think there's really a name for the type of ordering on preferences, and preferences are between vectors, which aren't necessarily ordered. It looks like because you have a finite number of alternatives though, you can easily have utility function representation, where

$U$ represents $\succcurlyeq$ on the set of alternatives $X$ if $\ \forall x,y \in X, x \succcurlyeq y \implies U(x) \geq U(y)$

Even though $U$ would be discontinuous in this case, you could probably call the domain of $U$ totally ordered (since $U \in \mathbb{R}$), and that should be sufficient in characterizing the preferences as well. We can't necessarily describe it as monotone, since we don't know the size of $\{A, B, C\}$, or what each element in the vectors are.