I'm looking at a simple coin game where I have \$100, variable betting allowed, and 100 flips of a fair coin where H=2x stake+original stake, T=lose stake.
- If I'm asked to maximise the expected final net worth $N$, am I meant to simply bet a fraction of $\frac{1}{4}$ (according to the Wikipedia article on the Kelly criterion)?
- What if I'm asked to maximise the expectation of $\ln(100+N)$? Does this change my answer?
Thanks for any help.
Best Answer
On (1) the optimal strategy to maximise the expectation of your final net worth is to be everything you have all the time. With probability $\frac{1}{2^{100}} \approx 8 \times 10^{-31}$ you will end up with $100 \times 3^{100}\approx 5 \times 10^{49}$; otherwise you end up with nothing. So your expected final net worth with this all-or-nothing strategy is $100 \times \left(\frac32\right)^{100} \approx 4 \times 10^{19}$ despite the overwhelming likelihood that you will end up with nothing; whatever happens, the final outcome will not be close to the expected final outcome.
If instead you bet $\frac14$ of your net worth at each stage as a Kelly Strategy, your expected final net worth is $100 \times \left(\frac98\right)^{100} \approx 1.3 \times 10^{7}$, much less than the all-or-nothing strategy, even though with the Kelly Strategy you are very likely to end up ahead and quite likely to end up with something large.
On (2) the position is reversed. The all-or-nothing strategy gives an expected outcome for $\ln(100+N)$ of $\frac{2^{100}-1}{2^{100}} \times \ln(100)+\frac1{2^{100}}\times \ln(100+100\times 3^{100}) \approx 4.6$, rather less than less than the $\ln(100+100)\approx 5.3$ expectation if you were not to bet anything ever.
But betting $\frac14$ of your net worth at each stage would give an expected outcome for $\ln(N)$ of $\ln(100) + 100 \times \frac12\left(\ln\left(\frac32\right)+\ln\left(\frac34\right)\right) \approx 10.5$ and then expected outcome for $\ln(100+N)$ would be very close to this. This Kelly Strategy is much better for this expectation than either an all-or-nothing strategy or a never-bet strategy: note that $e^{4.6}-100 \approx 0$ while $e^{5.3}-100 \approx 100$ while $e^{10.5}-100 \approx 36000$.