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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.

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  • $\begingroup$ 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. $\endgroup$ Dec 17, 2023 at 13:36
  • $\begingroup$ 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? $\endgroup$ Dec 17, 2023 at 13:55
  • $\begingroup$ $\mathbb{R}$ 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, $\endgroup$ Dec 17, 2023 at 14:31
  • $\begingroup$ $\mathbb{R}$ 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). $\endgroup$ Dec 17, 2023 at 22:48
  • $\begingroup$ 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." $\endgroup$ Dec 17, 2023 at 22:50

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