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

CC BY-SA 4.0
2023-12-11T15:12:57.957Z
2023-12-11T15:12:57.957Z
Michael Greinecker
<blockquote>
<p>Theorem ([Debreu 1959][1] page 9, 10) Let <span class="math-container">$X_i$</span> be space of real numbers. <em>If <span class="math-container">$\succsim$</span> is continuous, rational, independent and at least three factors are essential, then there exist functions <span class="math-container">$u_i:X_i\rightarrow \mathbb{R}$</span> such that</em>
<span class="math-container">$$
x\succsim y\iff \sum_{i=1}^n u_i (x_i) \geqslant \sum_{i=1}^n u_i (y_i).
$$</span></p>
</blockquote>
<p>Independent means: <span class="math-container">$x_Iy\succsim x_Iy'\implies x'_Iy\succsim x'_Iy'$</span></p>
<p><strong>Question</strong>: can the preference relation in independence be changed to <strong>equivalence</strong> relation?</p>
<p>Which means:  <span class="math-container">$x_Iy\sim x_Iy'\implies x'_Iy\sim x'_Iy'$</span>.</p>
<p>Perhaps we need to add some monotonicity conditions.</p>

CC BY-SA 4.0
2023-12-11T14:59:44.917Z
Could you please explain the notation $x_Iy$?
Michael Greinecker
<microeconomics><decision-theory>
Can Debreu's axiomatization of cardinal utility use equivalent relation instead of preference relation?
High GPA
# Can Debreu's axiomatization of cardinal utility use equivalent relation instead of preference relation?

(Asked by: username_0 on 2023-12-10)

> 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.

**Tags:** microeconomics, decision-theory

> **Comments:**
> **username_1**: Could you please explain the notation $x_Iy$?  

---

## 1 answers

### Answer by username_1 on 2023-12-11. Score: 3

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.


---

