# Independence and Reduction Axioms

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

## 1 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: $$\mathcal{L}''=\alpha\left(\beta A+(1-\beta)B\right)+(1-\alpha)\left(\gamma C+(1-\gamma)D\right)$$.