Independence axiom for lottery preferences

Mixing two lotteries with the same third lottery and the same positive weight preserves their strict preference order. For affine utility on probability measures, the difference of the two mixed utilities is the original difference multiplied by that weight. At weight zero the strict preference disappears, so positivity is essential.

Assume $0\le q<p\le 1$ and write $L_q=qA+(1-q)B$. The independence axiom for lottery preferences gives $A\succ L_q$ because $1-q>0$. Set $\lambda =(p-q)/(1-q)>0$; then $L_p=\lambda A+(1-\lambda )L_q$. A second use of the same axiom gives $L_p\succ L_q$. Thus a higher probability of the strictly preferred outcome strictly improves a two-outcome lottery.

Apply the independence axiom for lottery preferences in both weak directions of the indifference relation. At $p=0$ the two mixtures are identical. Repeating this substitution and using transitivity lets one replace any finite collection of indifferent component lotteries, after which reduction of compound lotteries combines their outcome probabilities.