<p>I understand that the social preference relation need not be an ordering. But is it also not necessary that it be complete? I have read conflicting viewpoints on this.</p>
<p>We haven't been taught what the exact functional form of a SCR is supposed to be, just that it is a function of the individual preference relations, so it seems to me that if we are allowed to use piecewise functions, we could construct a function from the individual preferences that could be incomplete, but I am not sure how to go about that.</p>
<p>Please help me understand this.</p>

CC BY-SA 4.0
2023-12-17T13:36:55.397Z
Normally, you should not learn about specific functional forms here but precise definitions. These tend to include the relevant domains and codomains. Usually, the codomain of a SCR is an alternative and not a relation.
Michael Greinecker
CC BY-SA 4.0
2023-12-17T13:55:28.943Z
What? Isn't the SCR defined as $f: \mathbb{R}^n \rightarrow \mathbb{R}$ such that $\forall (R_1,...,R_n) \in \mathbb{O}^n [f(R_1,...,R_n)]= {R}$? R here being the social preference relation? Can you please state what you're trying to say in simpler terms?
alyosha
CC BY-SA 4.0
2023-12-17T14:31:31.473Z
$\mathbb{R}$ does usually denote the set of real numbers, and $\mathbb{R}^n$ the set of $n$-term ordered lists of real numbers. I don't know what $\mathbb{O}$ stands for. Without you providing the relevant definitions and notation, I won't be able to help you,
Michael Greinecker
CC BY-SA 4.0
2023-12-17T22:48:41.637Z
$\mathbb{R}$ denotes the set of possible preference relations. $\mathbb{R}^n$ denotes the profile of $\mathcal{n}$ such preference relations. $\mathbb{O}^n \subset \mathbb{R}^n$ denotes the set of $\mathcal{n}$ preferences profile that are orderings (reflexive, complete, transitive).
alyosha
CC BY-SA 4.0
2023-12-17T22:50:41.703Z
These things should be part of the question, not just a comment. And the question seems to reduce to how you define a "preference relation."
Michael Greinecker
CC BY-SA 4.0
2023-12-17T22:57:29.350Z
$\mathbb{R}^n = \{(R_1,...,R_n):|R_i \in \mathbb{R}, i=1,...,n$. The SCR is defined as $f:\mathbb{R}^n \rightarrow \mathbb{R}$ such that $\forall (R_1,...,R_n) \in \mathbb{O}^n[f(R_1,...,R_n)] =\pazocal{R} where \pazocal{R}\in \mathbb{R}$ My question is when this SCR fulfills the unrestricted domain condition, ie when the domain is of the function is $\mathbb{O}^n$, does this $\pazocal{R}$ have to be complete?
alyosha
CC BY-SA 4.0
2023-12-17T23:01:58.557Z
@MichaelGreinecker I thought this to be self-explanatory since most standard textbooks I've seen refer to these things pretty much the same way. In any case, now that I've properly defined the question, do you think you could provide an answer?
alyosha
CC BY-SA 4.0
2023-12-18T00:04:48.597Z
If your preference relations are completem then $f(R_1,\ldots,R_n)$ must be complete. Otherwise, not.
Michael Greinecker
CC BY-SA 4.0
2023-12-18T01:08:53.803Z
Thank you. Could you also please direct to some material that contains a proof of that statement, for some extra reading?
alyosha
CC BY-SA 4.0
2023-12-18T01:11:44.903Z
That is not a proof, that is deciphering mathematical notation. The value of a function which has only complete relations in the codomain must be a complete relation. That is tautological.
Michael Greinecker
CC BY-SA 4.0
2023-12-19T07:00:59.200Z
Thanks but my instructor has told me that this is untrue. That an SCR satisfying condition U might not be complete itself, for example if we use the Pareto criterion as an SCR the social preference relation will not be complete.
alyosha
CC BY-SA 4.0
2023-12-19T07:04:41.777Z
Then your instructor is either wrong, or did not define a SCR with a space of conplete preferences as a codomain. The Pareto ordering is indeed not complete; it is also not a preference ordering then.
Michael Greinecker
CC BY-SA 4.0
2023-12-19T11:53:46.547Z
No, it was quite obvious that the co-domain of the function is $\mathbb{R}$, the set of preference relations not the set of complete preference relations as has been clearly stated previously.
alyosha
CC BY-SA 4.0
2023-12-19T12:14:09.560Z
I asked you what your definition of a preference relation is, you answered "I thought this to be self-explanatory since most standard textbooks I've seen refer to these things pretty much the same way." I still have no idea how preference relations are supposed to be defined.
Michael Greinecker
CC BY-SA 4.0
2023-12-19T14:50:34.237Z
That's fine. My instructor managed to resolve my original doubt. Thanks for the help you could muster. Have a great day.
alyosha
<microeconomics><social-choice><social-choice-function>
Does a social preference relation with unrestricted domain have to necessarily be complete?
alyosha
# Does a social preference relation with unrestricted domain have to necessarily be complete?

(Asked by: username_0 on 2023-12-17)

I understand that the social preference relation need not be an ordering. But is it also not necessary that it be complete? I have read conflicting viewpoints on this.

We haven't been taught what the exact functional form of a SCR is supposed to be, just that it is a function of the individual preference relations, so it seems to me that if we are allowed to use piecewise functions, we could construct a function from the individual preferences that could be incomplete, but I am not sure how to go about that.

Please help me understand this.

**Tags:** microeconomics, social-choice, social-choice-function

> **Comments:**
> **username_1**: Normally, you should not learn about specific functional forms here but precise definitions. These tend to include the relevant domains and codomains. Usually, the codomain of a SCR is an alternative and not a relation.  
> **username_0**: What? Isn't the SCR defined as $f: \mathbb{R}^n \rightarrow \mathbb{R}$ such that $\forall (R_1,...,R_n) \in \mathbb{O}^n [f(R_1,...,R_n)]= {R}$? R here being the social preference relation? Can you please state what you're trying to say in simpler terms?  
> **username_1**: $\mathbb{R}$ does usually denote the set of real numbers, and $\mathbb{R}^n$ the set of $n$-term ordered lists of real numbers. I don't know what $\mathbb{O}$ stands for. Without you providing the relevant definitions and notation, I won't be able to help you,  
> **username_0**: $\mathbb{R}$ denotes the set of possible preference relations. $\mathbb{R}^n$ denotes the profile of $\mathcal{n}$ such preference relations. $\mathbb{O}^n \subset \mathbb{R}^n$ denotes the set of $\mathcal{n}$ preferences profile that are orderings (reflexive, complete, transitive).  
> **username_1**: These things should be part of the question, not just a comment. And the question seems to reduce to how you define a "preference relation."  
> **username_0**: $\mathbb{R}^n = \{(R_1,...,R_n):|R_i \in \mathbb{R}, i=1,...,n$. The SCR is defined as $f:\mathbb{R}^n \rightarrow \mathbb{R}$ such that $\forall (R_1,...,R_n) \in \mathbb{O}^n[f(R_1,...,R_n)] =\pazocal{R} where \pazocal{R}\in \mathbb{R}$ My question is when this SCR fulfills the unrestricted domain condition, ie when the domain is of the function is $\mathbb{O}^n$, does this $\pazocal{R}$ have to be complete?  
> **username_0**: @username_1 I thought this to be self-explanatory since most standard textbooks I've seen refer to these things pretty much the same way. In any case, now that I've properly defined the question, do you think you could provide an answer?  
> **username_1**: If your preference relations are completem then $f(R_1,\ldots,R_n)$ must be complete. Otherwise, not.  
> **username_0**: Thank you. Could you also please direct to some material that contains a proof of that statement, for some extra reading?  
> **username_1**: That is not a proof, that is deciphering mathematical notation. The value of a function which has only complete relations in the codomain must be a complete relation. That is tautological.  
> **username_0**: Thanks but my instructor has told me that this is untrue. That an SCR satisfying condition U might not be complete itself, for example if we use the Pareto criterion as an SCR the social preference relation will not be complete.  
> **username_1**: Then your instructor is either wrong, or did not define a SCR with a space of conplete preferences as a codomain. The Pareto ordering is indeed not complete; it is also not a preference ordering then.  
> **username_0**: No, it was quite obvious that the co-domain of the function is $\mathbb{R}$, the set of preference relations not the set of complete preference relations as has been clearly stated previously.  
> **username_1**: I asked you what your definition of a preference relation is, you answered "I thought this to be self-explanatory since most standard textbooks I've seen refer to these things pretty much the same way." I still have no idea how preference relations are supposed to be defined.  
> **username_0**: That's fine. My instructor managed to resolve my original doubt. Thanks for the help you could muster. Have a great day.  

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## 0 answers

