<p>You can find in many textbooks (for example Mas-Collel, Winston and Green) that the Independence of irrelevant alternatives axiom implies that preferences over lotteries are linear with respect to the probabilities of each event. </p>

<p>Then, a compound lottery is of the form: <span class="math-container">$\mathcal{L}''=\alpha\mathcal{L}+(1-\alpha)\mathcal{L}'$</span>, If <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are the outcomes of the simple lottery <span class="math-container">$\mathcal{L}$</span> and <span class="math-container">$C$</span> and <span class="math-container">$D$</span> are the outcomes of the other simple lottery, <span class="math-container">$\mathcal{L}'$</span>. Then, the compound lottery can be seen as a simple lottery with four outcomes, given the linearity in probabilities: <span class="math-container">$\mathcal{L}''=\alpha\left(\beta A+(1-\beta)B\right)+(1-\alpha)\left(\gamma C+(1-\gamma)D\right)$</span>. </p>

CC BY-SA 4.0
2019-05-27T16:44:20.353Z
2019-05-27T16:44:20.353Z
Regio
<p>I have read that the Independence of Irrelevant Alternatives axiom in expected utility theory implies the fact that compound lotteries are equally preferred to their reduced form simple lotteries. However, I am unable to prove this. I'm sure that the proof should be easy, but I just can't see it.
I read it here: <a href="http://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html" rel="nofollow noreferrer">http://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html</a></p>

<microeconomics><expected-utility><proof-reference>
Independence and Reduction Axioms
Student
# Independence and Reduction Axioms

(Asked by: username_0 on 2019-05-24)

I have read that the Independence of Irrelevant Alternatives axiom in expected utility theory implies the fact that compound lotteries are equally preferred to their reduced form simple lotteries. However, I am unable to prove this. I'm sure that the proof should be easy, but I just can't see it.
I read it here: <http://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html>

**Tags:** microeconomics, expected-utility, proof-reference

---

## 1 answers

### Answer by username_1 on 2019-05-27. Score: 1 | Accepted answer

You can find in many textbooks (for example Mas-Collel, Winston and Green) that the Independence of irrelevant alternatives axiom implies that preferences over lotteries are linear with respect to the probabilities of each event.

Then, a compound lottery is of the form: $\mathcal{L}''=\alpha\mathcal{L}+(1-\alpha)\mathcal{L}'$, If $A$ and $B$ are the outcomes of the simple lottery $\mathcal{L}$ and $C$ and $D$ are the outcomes of the other simple lottery, $\mathcal{L}'$. Then, the compound lottery can be seen as a simple lottery with four outcomes, given the linearity in probabilities: $\mathcal{L}''=\alpha\left(\beta A+(1-\beta)B\right)+(1-\alpha)\left(\gamma C+(1-\gamma)D\right)$.


---

