Let $\succsim$ complete and $x, y, z \in X$. Suppose $x \succsim y$ and $y \succsim z$. Is it OK to write $x \succsim y \succsim z$ just knowing this?

Or, on the contrary, such expression automatically implies $x \succsim z$?

  • 5
    $\begingroup$ This seems more like a math question. (Math notation specifically, so very little microeconomics is involved.) Also, it boils down to whether writing $$ 5 > 2 < 3 $$ is okay. These are two different inequalities, but I was lazy and did not write the symbol 2 again. But this clearly does not imply anything about the relationship of 5 and 3. $\endgroup$ – Giskard Oct 1 '19 at 17:37
  • 4
    $\begingroup$ Strictly speaking, $\succsim$ symbolizes a binary relation, and so should be used as one. Writing "$x\succsim y\succsim z$" as a shorthand for "$x\succsim y$ and $y\succsim z$" is an abuse of notation and should only be used within an appropriate context where confusion/misinterpretation is unlikely. $\endgroup$ – Herr K. Oct 1 '19 at 21:11
  • 1
    $\begingroup$ To be clear, you can't conclude that $x \succsim z$ without assuming transitivity. $\endgroup$ – Art Oct 2 '19 at 16:45

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