Coins in two piles game — where have I gone wrong?

Question

In a certain game, a player can put his money in two piles labelled A and B. A fair coin is tossed. If it lands heads, then all the money in pile A is multiplied by a factor $\alpha$ and returned to you. Likewise, if it lands tails, then all the money in pile B is multiplied by a factor $\beta$ and returned to you. Determine an optimal allocation of money into the two piles if i) $\alpha=2$ and $\beta=0.5$; ii) or $\alpha=2$ and $\beta=1.5$.

My thought process is: suppose we place a fraction $f\in[0,1]$ into pile A and $(1-f)$ into pile B. Then, double our expected returns (so that we can ignore the factor of half) are $f\alpha-(1-f)\beta=(\alpha-\beta)f+\beta$. This is a linear relation between the payoff and $f$. My gut tells me I should not put all my money in the same pile, but the monotonicity implies otherwise. Where have I gone wrong?

Answer 1

As long as you have some money, you can play the game again, which changes the expectation from linear to exponential.

Answer 2

What is your definition of optimal? Unless otherwise specified, we take it as expected value. In that case, put all your money in the pile where it is multiplied the most. See this Your intuition that going broke is infinitely bad is wrong.

Note that as defined betting on pile A is break-even and betting on pile B is losing in both cases. If you are not going to put everything in pile A you should not play at all.