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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?

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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 $\beta$.

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  • $\begingroup$ Oh, I see. Many thanks! $\endgroup$ Commented Jun 22 at 23:33

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