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Strict Preference usually states that

x is strictly preferred to y if : < x is weakly preferred to y and not y is weakly preferred to x >.

Let me split the < > part into two segments:

  1. x is weakly preferred to y

and

  1. not y is weakly preferred to x

Why do you need the first condition to satisfy the strict preference relationship? The second condition seems to be enough to satisfy the strict preference relation. I mean, "not y is weakly preferred to x" means the same thing as "x is strictly preferred to y" right? So 1. seems to be unnecessary to me when trying to explain Strict Preference Relation.

Is it because we assume that "completeness" of preference isn't satisfied when we define strict preference relation? Therefore there is a third option of "I don't know" if we just write the 2nd condition which requires us to add the first condition as well to cover for "non-completeness"?

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You are correct that the motivation regards issues surrounding completeness. As you point out, when $\succcurlyeq$ is complete $y \not\succcurlyeq x$ implies $x \succcurlyeq x$ (hence, $x \succ y$).

If, however, we do not want to assume complete preferences then (2) does not imply (1), but strict preference can still constructed from weak using the more complex definition. (One could also take the set difference of weak preference and indifference).

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