the Kelly criterion and then some
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The Kelly criterion is a way to optimise an unlimited sequence of bets under the following circumstances: a probability p of winning each bet, a loss of a fraction a of the sum bet, a gain of a fraction b of the sum bet, and a fraction f of the current fortune as the sum bet. Then
is the fraction optimising the growth
![\mathbb E[ \log\{X_n/X_0\}^{1/n}]](https://s0.wp.com/latex.php?latex=%5Cmathbb+E%5B+%5Clog%5C%7BX_n%2FX_0%5C%7D%5E%7B1%2Fn%7D%5D&bg=000000&%23038;fg=B0B0B0&%23038;s=0&%23038;c=20201002)
Here is a rendering of the empirical probability of reaching 250 before ruin, when starting with a fortune of 100, when a=1, p=0.3 and f and b vary (on a small grid). With on top Kelly’s solutions, meaning that they achieve a high probability of avoiding ruin. Until they cannot.
The Ridder is asking for a variant of this betting scheme, when the probability p to win the bet is proportional to 1/(1+b), namely .9/(1+b). In that case, the probabilities of reaching 250 (using the same R code as above) before ruin are approximated as follows
with a maximal probability that does not exceed 0.36, the probability to win in one go, betting 100 with a goal of 250. It thus may be that the optimal choice, probabilitiwise is that one. Note that in that case, whatever the value of b, the Kelly criterion returns a negative fraction. Actually, the solution posted by the Riddler the week after is slightly above, 0.3686 or 1−(3/5)9/10. Which I never reached by the sequential bet of a fixed fraction of the current fortune, eps. when not accounting for the fact that aiming at 250 rather than a smaller target was removing a .9 factor.
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