I'm looking at a betting game where I have \$100 and want to double my money by repeatedly betting on a biased coin; it shows heads with probability $p<\frac{1}{2}$ in which case I win even money.
I imagine my best strategy is to go all-in, but I also wanted to investigate what happens if I bet a constant fraction $0<f<1$ of my wealth each toss, so I implemented this on a computer. As I'd never go broke doing this, I actually implemented a bet size of $\max(fw,0.0001)$, where $w$ is current wealth instead. My findings are here (I realise as a new user I'm not allowed to post images):
My question is why does a dip in success probability occur? Shouldn't such a graph be monotone increasing? (I'm hoping this isn't a simple implementation error!)
Any input greatly appreciated, John
No, the function shouldn't be monotonically increasing. It jumps whenever the net result of a certain combination of heads and tails crosses the target line, and these jumps mostly go up but sometimes go down.
For example, consider the case where you exactly reach the target line with $4$ heads and one tails. Then $(1+f)^4(1-f)=2$, which has a real solution at $\hat f\approx0.81618$. The derivative with respect to $f$ is $(1+f)^3(3-5f)$, and this is negative at $\hat f$, so by increasing $f$ beyond $\hat f$ you move the result for this case below the target line; so there's a downward jump in the graph at $\hat f$.