I happen to know that an apparently fair coin actually has probability $p$ of heads with $1>p>1 / 2$. I play a very long sequence of games of heads and tails in which my opponent pays me back twice my stake if the coin comes down heads and takes my stake if the coin comes down tails. I decide to bet a proportion $\alpha$ of my fortune at the end of the $n$th game in the $(n+1)$ st game. Determine, giving justification, the value $\alpha_{0}$ maximizing the expected logarithm of my fortune in the long term, assuming I use the same $\alpha_{0}$ at each game. Can it be actually disadvantageous for me to choose an $\alpha<\alpha_{0}$ (in the sense that I would be better off not playing)? Can it be actually disadvantageous for me to choose an $\alpha>\alpha_{0}$ ?

[Moral issues should be ignored.]