TITLE: Coin Tossing Game Optimal Strategy
QUESTION [17 upvotes]: I was recently asked this question in an interview, but was completely stumped as to how to even begin answering it - it's been bugging me ever since, and I thought it was quite a nice question, so hopefully someone on here can help me out. Any help would be appreciated! Here goes:
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 $x-S$. How do you choose your stake so as to maximise your expected winnings from the game, not including the initial balance?
Cheers,
Boris

REPLY [4 votes]: In my opinion this calls for the Kelly Criterion (http://en.wikipedia.org/wiki/Kelly_criterion). In this case, the fraction of your wealth that should be bet is $\frac{0.5 \times 3-1}{3-1}$.