<p>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</p>

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

<p>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.</p>

CC BY-SA 3.0
2015-11-25T05:18:33.573Z
2015-11-25T05:18:33.573Z
Kitsune Cavalry
<p>You are right, total is a synonym for complete, it simply means that two elements can always be compared.</p>

<p>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").</p>

CC BY-SA 3.0
2015-11-24T21:20:53.857Z
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?__
Shane
CC BY-SA 3.0
2015-11-24T21:37:40.650Z
@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
CC BY-SA 3.0
2015-11-24T21:39:29.753Z
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.
Shane
CC BY-SA 3.0
2015-11-24T21:41:12.893Z
My pleasure! Please leave it open, if someone comes with a better answer I would be happy to know it as well.
Oliv
CC BY-SA 3.0
2015-11-24T21:17:31.240Z
2015-11-24T21:17:31.240Z
Oliv
<p>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.</p>

<p>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 <strong>would</strong> allow some ties? And how would I say an agent has an order over the alternatives that <strong>would not</strong> allow ties? Is it just the intuitive <strong>weak total order</strong> and <strong>strict total order</strong>, over alternatives where the weak and strict carry the same meaning as for preference relations and total is a synonym for complete?</p>

<mathematical-economics><decision-theory>
Properties of orders and preference relations
Shane
# Properties of orders and preference relations

(Asked by: username_0 on 2015-11-24)

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?

**Tags:** mathematical-economics, decision-theory

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## 2 answers

### Answer by username_2 on 2015-11-24. Score: 3 | Accepted answer

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").

> **Comments:**
> **username_0**: 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?__  
> **username_2**: @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.  
> **username_0**: 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.  
> **username_2**: My pleasure! Please leave it open, if someone comes with a better answer I would be happy to know it as well.  

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### Answer by username_1 on 2015-11-25. Score: 1

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.


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