Calculating Kelly Criterion of a tiered payout

Question

There is a bet, which can payout different amounts based on the result. possible results, in units, would be:

+30 +10 +6 +4 +2 +1 0 -1

How do you calculate the Kelly Criterion for this? I know the Expected Value of the bet, which is 1.08%. so what I have done, for each, is taken 1.08% / the value (from those listed above), and summed them. So 1.08% / 30 = 0.00036 etc... Then sum them together.

Is this the right way to get the Kelly Criterion for the bet?

(Answer I get is 1.13%)

Answer 1

Just to stop this question being unanswered, the Kelly criterion bet depends on the full distribution, not just the expected value. Take four examples, all with expectation $0.0108$:

• $P(X=1)=0.5054$ and $P(X=-1)=0.4946$. This is also a usual Kelly criterion type of question with two opposite but equal payoffs of $\pm1$, so the optimal proportion of bankroll to bet is $0.0108$.
• $P(X=30)=\frac{2527}{77500}\approx 0.0326$ and $P(X=-1)=\frac{74973}{77500}\approx 0.9674$. This is also a usual Kelly criterion type of question with two unequal payoffs, and the optimal proportion of bankroll to bet is $0.00036$, which is $\frac1{30}$ times the previous solution. It is likely to take many more bets to achieve any specific target. As Ross Millikan commented, the variance of the second distribution is much higher, and it is higher risk for the same reward, so it is not a surprise that the bet should be smaller.
• A more complicated $P(X=30)$ $=P(X=10)$ $=P(X=6)$ $=P(X=4)$ $=P(X=2)$ $=P(X=1)=0.01$ and $P(X=0)=0.4208$ and $P(X=-1)=0.5192$. Trying to optimise this analytically might involve finding the root of a degree-$7$ polynomial, so instead numerical methods are probably needed and it seems that here the optimal proportion of bankroll to bet is close to $0.001$.
• $P(X=1)=0.0108$ and $P(X=0)=0.9892$. You can never lose a bet here so you may as well bet everything, making the optimal proportion of bankroll to bet $1$ (or more if you can borrow at zero interest).

So indeed the Kelly criterion bet for multiple outcomes varies with the details, not just varying the expected gain.