# Can Debreu's axiomatization of cardinal utility use equivalent relation instead of preference relation?

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.

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

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.