Mixture monotonicity
= Mixture monotonicity
{title2=$A\succ B,\ p>q\ \Longrightarrow\ pA+(1-p)B\succ qA+(1-q)B$}
Assume $0\le q<p\le1$ 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>.