# Why is the preference relation based on the pareto criterion used as the social choice rule not complete?

Let $$\mathbb{R}$$ be the set of all possible preference relations. Let $$\mathbb{R}^n=\{(R_1,\dots,R_n)|R_i \in \mathbb{R} \}$$ be the set of all possible profiles. A social choice rule (SCR) is a functional relations specified by the function: $$f:\mathbb{R}^n \rightarrow \mathbb{R}$$ such that $$(\forall(R_1,\dots,R_n)\in \mathbb{R}^n)[f(R_1,\dots,R_n)=\mathcal{R}]$$ where $$\mathcal{R} \in \mathbb{R}.$$

The preference relation based on the pareto criterion is defined in the following way. For any given $$x,y \in \mathbb{X}$$, where $$\mathbb{X}$$ is the set of alternatives, we define the preference relation $$\mathcal{\bar{R}}$$ as $$x\mathcal{\bar{R}}y \iff [(\forall i \in \mathbb{N})(xR_iy)]$$

Our instructor has taught us that the parteo criterion when used as an SCR is not a complete preference relation and therefore cannot be an ordering. I don't understand why that is true, or what that even really means since we have only been dealing with all this in abstract definitional terms without any proper examples or cases. Please guide me.

• I guess $\mathbb{R}$ is the set of preference relations on $\mathbb{X}$. So how are preference relations defined? Dec 29, 2023 at 20:46
• Can you explain what you mean by "how are preference relations defined". You kept posing the same question on a previous post of mine without clarifying what you mean by it, only confusing me further. If it isn't obvious, you should know I'm learning social choice theory for the first time and all the information that I have written in my question is all the information that has been given to me to make sense of. I really appreciate you replying to my question when no one else did, I'd also really appreciate it if you could be a little patient with me as I try to learn. Dec 30, 2023 at 6:16
• Usually, a preference relation on a set is defined as a complete and transitive relation. Sometimes they are conplete, transitive, and anti-symmetric relations. Or they might be "strict" relations. Usually, one works with very well-behaved individual preference relations and is more permissive about the social relation. The way you define a SCR, the social relation $f(R_1,\ldots, R_n)$ has to be just like an individual preference relation. This means that it is by definition impossible that a SCR requires the agent's preferences to be complete but allows for an incomplete social relation. Dec 30, 2023 at 9:04
• I think it is therefore likely that you either use a wrong definition or that your instructor made a mistake in setting up the formalism. Dec 30, 2023 at 9:06
• Okay, thank you, that explains things a lot better. Dec 30, 2023 at 13:47