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Theorem ([Debreu 1959][1] page 9, 10) Let $X_i$ be space of real numbers. If $\succsim$ is continuous, rational, independent and at least three factors are essential, then there exist functions $u_i:X_i\rightarrow \mathbb{R}$ such that $$ x\succsim y\iff \sum_{i=1}^n u_i (x_i) \geqslant \sum_{i=1}^n u_i (y_i). $$

Independent means: $x_Iy\succsim x_Iy'\implies x'_Iy\succsim x'_Iy'$

Question: can the preference relation in independence be changed to equivalence relation?

Which means: $x_Iy\sim x_Iy'\implies x'_Iy\sim x'_Iy'$.

Perhaps we need to add some monotonicity conditions.

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  • $\begingroup$ Could you please explain the notation $x_Iy$? $\endgroup$ Dec 11, 2023 at 14:59

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Here is a partial answer: The two conditions are equivalent if preferences are strongly monotone. To see this, let $x_Iy\succ x_Iy'$. Let $e=(1,1,\ldots,1)\in\mathbb{R}^n$. By strong monotonicity and continuity, there exists (a unique) $\alpha>0$ such that $x_Iy\sim x_I(y'+\alpha e)$. Then, $x_I'y\sim x_I (y'+\alpha e)\succ x_I'y'$ by the condition and strong monotonicity.

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