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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

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How many simulations did you carry out to produce the graph? If the number of simulations is too small the dip may be due to the sample of bets being unrepresentative. –  Till Hoffmann Nov 27 '12 at 11:45
    
I did this for 10000 simulations, and I re-did the large f with a smaller minimum bet size than 0.0001; the same results appear to hold. –  John Shih Nov 27 '12 at 13:54
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1 Answer 1

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 $\hat f$.

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Actually, I suspect that all jumps near $f=1$ are downward, and the upward slope on your graph at the end is just an artefact of discretization; I suspect that the function goes downward all the way up to $f=1$ and only then jumps upward. –  joriki Nov 27 '12 at 12:58
    
Thanks for your insight; I agree with the jump as I had derived the above graph for f = [0.05, 0.1, ... 1.0] only. Could you explain how one would find the maximum f then? I'm not sure if this is related to the Kelly criterion because I thought the optimum fraction is only derived for favourable games (well, favourable when you take the payouts into account). –  John Shih Nov 27 '12 at 13:53
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