<p>You're right. The proof is provided for the proposition
<span class="math-container">$\alpha &gt; \beta \implies \alpha \cdot a \ \oplus (1-\alpha) \cdot b \succ \beta \cdot a \ \oplus (1-\beta) \cdot b$</span></p>
<p>But note that proving the converse</p>
<p><span class="math-container">$\alpha \cdot a \ \oplus (1-\alpha) \cdot b \succ \beta \cdot a \ \oplus (1-\beta) \cdot b\implies \alpha &gt;\beta$</span></p>
<p>is same as proving its contrapositive</p>
<p><span class="math-container">$\beta \geq \alpha \implies  \beta \cdot a \ \oplus (1-\beta) \cdot b \succsim \alpha \cdot a \ \oplus (1-\alpha) \cdot b$</span></p>
<p>and here <span class="math-container">$\beta =\alpha$</span> case is trivial, and the proof for the case when <span class="math-container">$\beta &gt; \alpha$</span> holds is exactly the same as for <span class="math-container">$\alpha &gt; \beta$</span> (provided in the question). We just need to reverse the roles of <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>.</p>

CC BY-SA 4.0
2024-06-22T23:33:56.823Z
Oh, I see. Many thanks!
Riccardo Iorio
CC BY-SA 4.0
2024-06-22T23:28:34.590Z
2024-06-22T23:28:34.590Z
Amit
<p>I'm struggling to understand a proof in Osborne and Rubinstein's <em>Models in Microeconomic Theory</em> (p. 35). The relevant lemma is</p>
<blockquote>
<p>Let <span class="math-container">$Z$</span> be a set of prizes. Assume that <span class="math-container">$\succeq$</span> is a preference relation over <span class="math-container">$L(Z)$</span> and satisfies the independence property. Let <span class="math-container">$a$</span> and <span class="math-container">$b$</span> be two prizes with <span class="math-container">$[a] \succ [b]$</span>, and let <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> be two probabilities. Then <span class="math-container">$$\alpha &gt; \beta \iff \alpha \cdot a \oplus (1 - \alpha) \cdot b \succ \beta \cdot a \oplus (1 - \beta) \cdot b.$$</span></p>
</blockquote>
<p>The proof they provide is the following:</p>
<blockquote>
<p>Let <span class="math-container">$p_{\alpha} = \alpha \cdot a \oplus (1 - \alpha) \cdot b$</span>. Because <span class="math-container">$\succeq$</span> satisfies the independence property, <span class="math-container">$p_{\alpha} \succ \alpha \cdot b \oplus (1 - \alpha) \cdot b = [b]$</span>. Using again the independence property, we have <span class="math-container">$$p_{\alpha} = (\beta/\alpha) \cdot p_{\alpha} \oplus (1 - \beta/\alpha) \cdot p_{\alpha} \succ (\beta/\alpha) \cdot p_{\alpha} \oplus (1 - \beta/\alpha) \cdot b = \beta \cdot a \oplus (1 - \beta) \cdot b.$$</span> <span class="math-container">$\square$</span></p>
</blockquote>
<p>I don't understand how this proves both directions of the if-and-only-if statement in the lemma. Doesn't it prove only one direction?</p>

<microeconomics><mathematical-economics><uncertainty>
Proof that independence implies monotonicity in Osborne and Rubinstein
Riccardo Iorio
# Proof that independence implies monotonicity in Osborne and Rubinstein

(Asked by: username_0 on 2024-06-22)

I'm struggling to understand a proof in Osborne and Rubinstein's *Models in Microeconomic Theory* (p. 35). The relevant lemma is

> Let $Z$ be a set of prizes. Assume that $\succeq$ is a preference relation over $L(Z)$ and satisfies the independence property. Let $a$ and $b$ be two prizes with $[a] \succ [b]$, and let $\alpha$ and $\beta$ be two probabilities. Then $$\alpha > \beta \iff \alpha \cdot a \oplus (1 - \alpha) \cdot b \succ \beta \cdot a \oplus (1 - \beta) \cdot b.$$

The proof they provide is the following:

> Let $p\_{\alpha} = \alpha \cdot a \oplus (1 - \alpha) \cdot b$. Because $\succeq$ satisfies the independence property, $p\_{\alpha} \succ \alpha \cdot b \oplus (1 - \alpha) \cdot b = [b]$. Using again the independence property, we have $$p\_{\alpha} = (\beta/\alpha) \cdot p\_{\alpha} \oplus (1 - \beta/\alpha) \cdot p\_{\alpha} \succ (\beta/\alpha) \cdot p\_{\alpha} \oplus (1 - \beta/\alpha) \cdot b = \beta \cdot a \oplus (1 - \beta) \cdot b.$$ $\square$

I don't understand how this proves both directions of the if-and-only-if statement in the lemma. Doesn't it prove only one direction?

**Tags:** microeconomics, mathematical-economics, uncertainty


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## 1 answers

### Answer by username_1 on 2024-06-22. Score: 5 | Accepted answer

You're right. The proof is provided for the proposition
$\alpha > \beta \implies \alpha \cdot a \ \oplus (1-\alpha) \cdot b \succ \beta \cdot a \ \oplus (1-\beta) \cdot b$

But note that proving the converse

$\alpha \cdot a \ \oplus (1-\alpha) \cdot b \succ \beta \cdot a \ \oplus (1-\beta) \cdot b\implies \alpha >\beta$

is same as proving its contrapositive

$\beta \geq \alpha \implies \beta \cdot a \ \oplus (1-\beta) \cdot b \succsim \alpha \cdot a \ \oplus (1-\alpha) \cdot b$

and here $\beta =\alpha$ case is trivial, and the proof for the case when $\beta > \alpha$ holds is exactly the same as for $\alpha > \beta$ (provided in the question). We just need to reverse the roles of $\alpha$ and $\beta$.

> **Comments:**
> **username_0**: Oh, I see. Many thanks!  

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