We both put 20 USD into a box. Then, we each generate a number in the interval (0,1) with uniform distribution. The person with the higher number wins and takes 40 USD, whilst the loser is left with 0 USD. I offer to sell you an option that allows you to regenerate your number after you see both of our numbers. What is the price of the option?
This is my approach. After we get our number, there is a 50% chance that yours is higher than mine. If this is the case, I choose to regenerate my number to try and win, which gives me another 50% chance of winning. However, if my number is initially higher than yours (50% chance), then I do not use the option. This gives the expected probability P of winning of:
P = 0.5(1) + 0.5(0.5) = 0.75
Hence price of option would be (0.75*40) - 0.5(40) = $10
However, the answer is 20/3. May someone please explain where I am going wrong?