What an interesting trick. A starting point to intuit a resolution: yes you have a 50% chance of doubling and a 50% of halving. But the doubling only happens if you have x dollars, and the halving only happens if you have 2x. So you can see you either gain x or lose x, so your expected return is 0.
When the amount you multiply your money by depends on the amount of money you currently have, you have to factor in your initial money to each case to compute expected return.
When the amount you multiply your money by depends on the amount of money you currently have, you have to factor in your initial money to each case to compute expected return.