## the Kelly criterion and then some

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

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 followswith 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.

August 26, 2022 at 8:14 am

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