Suppose that I am very skilled in a gambling game, and any day that I bet $x$, I get back $2x$ with probability $\frac 12+\delta$ (and nothing with probability $\frac 12-\delta$). My goal is to double my money as fast (pick your favorite definition) as possible, but every day I can bet only once. (And obviously I can bet at most as much I have.)



> 
> 
> > 
> > (1) What is the optimal strategy?
> > 
> > 
> > 
> 
> 
> 


We can modify the above question in several ways. One possibility (that is probably also needed to solve (1)) is to replace the goal of doubling with getting $r$ times our initial capital.



> 
> 
> > 
> > (2) For which $r$ should we bet $x=r$ in the first round?
> > 
> > 
> > 
> 
> 
> 


A nice variant is that we can make several independent bets on the same day that have a different return (similarly to [this question](https://mathoverflow.net/questions/216072/optimisation-of-betting-strategy)), where $\delta$ is replaced by $\delta\_1\ge\delta\_2\ge\ldots$ and $x$ by $x\_1,x\_2,\ldots$. In an optimal strategy obviously $x\_1\ge x\_2\ge \ldots$ but it depends on several parameters whether $x\_2>0$ or we should only use our best bet.



> 
> 
> > 
> > (3) Can anything interesting be said about when $x\_2=0$?
> > 
> > 
> > 
> 
> 
> 


And of course, the returns of the bets can have some more complicated distribution, even these distributions can be generated randomly each day etc.



What's the best betting strategy to double money if we have $\delta$ advantage?
All you want to know can be found here: <https://www.wikiwand.com/en/Kelly_criterion>


