I recently asked a question, that was answered excellently:
Here I'd like to complicate the question slightly. This part stays the same:
You start off with £100 and you toss a coin 100 times. Before each toss you choose a stake S which cannot be more than your current balance x (so your maximum stake for the first toss is £100). If the coin comes up heads, you win 2S and your new balance is x+2S. If it comes up tails, you lose your stake and have x−S.
This time, though. Instead of maximizing the expected profit $E[P]$, we want to maxmise $E[\log(100+P)]$. In both cases the initial balance of £100 is not included in any profit. How should we choose our bets in this case? Now, my first thought and indeed my answer was that whichever strategy maximizes the first case must also maximize the second.. but the very fact that it was asked makes me doubt myself? Any thoughts greatly appreciated!