I'm studying Decision Theory from the book 'An introduction to decision theory' by Martin Peterson, and there is a problem that I don't understand how to solve. The problem is:
You prefer a fifty-fifty chance of winning either 100 dollars or 10 dollars to a lottery in which you win 200 dollars with a probability of 1/4, 50 dollars with probability of 1/4, and 10 dollars with probability of 1/2. You also prefer a fifty-fifty chance of winning either 200 dollars or 50 dollars to receiving 100 dollars for sure. Are your preferences consistent with von Neumann and Morgenstern’s axioms?
The book proposes as solution 'No. Your preferences violate the independence axiom' without proposing the steps to reach that solution and I don't know why it is correct.
The first lottery is
$$\frac{u(100)}{2}+\frac{u(10)}{2}>\frac{u(200)}{4}+\frac{u(50)}{4}+\frac{u(10)}{2}\implies u(100)>\frac{u(200)}{2}+\frac{u(50)}{2},$$
because of the independence axiom. But the second lottery says
$$\frac{u(200)}{2}+\frac{u(50)}{2}>u(100).$$
Of course, the two are incompatible.
Another way to look at it: let $p$ be the lottery that pays $200$ or $50$ with probability $1/2$ and $q$ the one that pays $100$ for certain. The second statement of preferences implies $$p\succ q.$$ Now consider a third lottery that pays $10$ for certain and call it $r$. The independence axiom implies that $$a p+(1-a)r\succ a q+(1-a)r\quad\forall a\in [0,1).$$
However, the first statement implies $$\frac{1}{2}q+\frac{1}{2}r\succ\frac{1}{2}p+\frac{1}{2}r,$$ which is incompatible with the independence axiom.