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I think I know why you think there is equivalency here. There are 16 red singles, not 18. So, placing 1/16th of the unit on the 16 red singles wins 1.25 units (35*(1/16)-15/16 = 1.25), not 1 unit. Thus, there is no sense in betting on red color. Agreed?

This is not what I get in my simulation, which does not want to place anything on red color, but instead wants to place the bets on red numbers. I'd be very interested to see what your Excel solver gives you when you factor in the ability to bet on red numbers. In my results, the geometric...

I am not sure I understand this. I see that (4/37), (3/37), and (30/37) are probabilities. But what are (107/37), (71/37), and (-1/37)?

Something is not right. The expectancy of the second one is about 1.5 times greater than the expectancy of the first one, so the (k*E) equality means that the Kelly of the first one is 1.5 times greater than the Kelly of the second one, right? But they both bet 100% of the bankroll. So, perhaps...

You know, there is certain universality in that 1/37 strategy. It's as if it doesn't matter how many R16 or R14 spots are on the wheel. To verify, I changed the rules so that there are 8 R16 spots and 6 R14 spots. Then I calculated the top strategy. The best allocation per slot came out to be...

I added more precision in the calcs, and the top strategy converged to exactly 1/37 of the bankroll on every single number, i.e.: R16: 10.8108% R14: 8.1081% RC: 0% RN: 43.2432% BC: 0% BN: 37.8378%

After fixing the typo, the top 10 strategies are almost identical to what they were before. It's just F(.) was reduced: R16 R14 RC RN BC BN F(R16,R14,RC,RN,BC,BN) 10.8 8.1 0.0 43.2 0.0 37.8 0.4784760 10.8 8.1 0.0 43.3 0.0 37.7 0.4784735 10.8 8.1 0.0 43.1 0.0 37.9 0.4784735 10.9 8.1...

I've spotted a typo in my code. This: pRedSingle * log1p(-r16 - r14 + red + 35 * (redNumber / 14) - black - blackNumber) Should be this: pRedSingle * log1p(-r16 - r14 + red + 35 * (redNumber / 16) - black - blackNumber) I am re-running the simulation.

F(R16,R14,RC,RN,BC,BN) = = pR16 * log1p(35 * r16 - r14 + red - redNumber - black - blackNumber) + pR14 * log1p(-r16 + 35 * r14 + red - redNumber - black - blackNumber) + pRedSingle * log1p(-r16 - r14 + red + 35 * (redNumber / 14) - black - blackNumber) + pBlack * log1p(-r16 - r14 - red -...

Can you check again? The current strategy is much superior to the previous one. The performance metric, F(R16,R14,RC,RN,BC,BN), is the geometric rate of the bankroll growth per spin. I am maximizing this function, so the resulting bets are Kelly-optimal.

I didn't think of it this way, but yes, as it turns out, it is almost exactly 1/37 of the bankroll on every single number. There is a certain beauty in this, isn't there? So, what's the E*k for this?

Well, I have another surprise for us all. You were right about betting on black color -- it's worthless. But as it turned out, betting on individual red numbers is much better than betting on red color, to the point that no money should be wasted in betting on red color. Here are my latest...

The risk is that you lose 10 times in a row, each time 2.7% of the bankroll.

You know, as I was coding this, it occurred to me that we are still not doing it quite right. Specifically, we are not considering the bets on Black (the bets on black color, that is), and the bets on any red numbers other than R16 and R14.

Okay, yeah, that makes sense.

What's the k*E of your holy grail?

Yeah, I got that wrong, thanks. Should be: EXP(0.3279876)^10 = 26.57 The difference with your 26.63 is probably due to rounding errors.

The strategy that you proposed multiplies the original bankroll by a factor of 3, after the 10 spins. The strategy that I calculated multiplies the original bankroll by a factor of 91, after the 10 spins. That's the payoff for the risk, which makes the rate of growth reach the max at the...

Just for clarity, the above solution does not allow borrowing, i.e. it follows the original rules where the total bet can't exceed the current bankroll.

Ok, here are the top 10 strategies now, with the inclusion of the bet on all black numbers: R16 R14 R BN F(R16,R14,R,BN) 8.1 5.4 48.6 37.8 0.3279876 8.1 5.4 48.5 37.9 0.3279855 8.1 5.4 48.7 37.7 0.3279851 8.2 5.4 48.5 37.8 0.3279832 8.2 5.4 48.6 37.7 0.3279822 8.0 5.4 48.7 37.8...