Monotonicity

Question: Show that if a relation is complete, transitive and satisfies the independence axiom, then it satisfies the monotonicity property.

Answer: Without loss of generality, assume $$ p \succ q. $$ We need to show that for any probability a and b, $$ ap + (1-a) q \succ bp + (1-b) q $$ iff $$ a >b $$

Rewrite the left hand side as $$ (a-b)p + bp + (1-a) q.$$ Rewrite the right hand side as $$ (a-b)p + bp + (1-a)q.$$

Then the indepence axiom implies the inequality.