# Optimal Stake for coin flip game

I'm doing some prep work for a job interview and one of their difficult sample questions is as follows:

You flip a coin 100 times, but instead of having a fixed stake, you can freely choose the stake for each flip. You start with £100. After each flip, if it comes up heads you win twice your stake (and your stake is returned), if it comes up tails you lose your stake. So if you start with $x$ and select a stake of $S$, then after the flip you will either have $x+2S$ or $x-S$. You can never make your stake larger than your current balance.

If your profit is $P$ how should you select your stake, $S$, to maximise:

a) $E[P]$
b) $E[\log (P+100)]$

Edit: to clarify $0<S<100$ is required so you are betting an amount but you are also protecting yourself from losing all of your money (of course successive losses would whittle your total money to zero eventually)

• For a, see my answer here The bet is in your favor. Take it. All of it. – Ross Millikan Jan 23 '17 at 21:18
• That's not correct because a single tail causes you to go bust. I'm after a mathematical based argument – user403033 Jan 24 '17 at 9:29
• If you want the expectation, it is correct. Yes, you are very likely to go bust, but if you don't you have lots of money. The expectation is very high. If you aren't allowed to bet it all, you should bet as much as you can. What happens if you get down to $1$. Can you bet then? – Ross Millikan Jan 24 '17 at 14:47
• you can still bet when you get down to 1, but you would have to bet everything by that point. You want S to be a percentage of your total so that you can maximise your gain whilst protecting against bankruptcy. i.e. S should be a reasonable value so that if you ran a simulation you would end in the money. – user403033 Jan 24 '17 at 16:54
• That depends on your utility function. You are blindly assuming that betting everything is the wrong choice because you can go broke, but if you want the highest expectation value you should. In my other answer, I show the computation for the expectation value betting everything on every flip. Show me a better strategy. – Ross Millikan Jan 24 '17 at 17:03

For the second, you are trying to maximize the expectation of the log of how much you have. For one flip, you start with $1$ and bet $x$. Your expectation is then $\frac 12(\log (1-x)+\log (1+2x))$ We take the derivative, set to zero, solve for $x$. That gives $\frac {-1}{1-x}+ \frac 2{1+2x}=0$ or $x=\frac 14$ so you should bet one quarter of your bankroll at each step. A win and a loss multiply your bankroll by $\frac 98$, so fifty wins and fifty losses will have you at about $36110$ at the end.
If you are required to bet at least $1$ every time there is no strategy in b that has positive expectation. You could lose every time, winding up with $0$ and the log is negative infinity. When you do questions like this it is essential to get the rules specified precisely.
• Should it not be $\log (100-S) + \log (100+2S)$? – user403033 Jan 24 '17 at 18:14
• I was scaling to $1$ as your current bankroll. This ignores the granularity of betting, which was not specified, as well as an upper limit to what you can bet. – Ross Millikan Jan 24 '17 at 19:27