No, I mean the risk per spin. For a single bet, the bet risk is easy: 100%, because that's how much of your bet you lose when you lose. So your risk (to your bankroll) is your bet size. But we now have a combination of bets where your betsize is 100% of your bankroll but the most you can lose (indeed, your only possible loss) is 2.7% of your bankroll. So does the definition of "risk" stay constant or does it change with the strategy? Trading is immeasurably less predictable than this or any other casino game (leaving aside cheating or faulty equipment). So I don't see the definition of "trading risk" being strategy-dependent, but maybe that's just me. So is the risk per spin using this thread's optimal strategy 100% or 2.7%?
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 results which allow all types of bets: R16: bet on R16 R14: bet on R14 RC: bet on red color RN: bet on all individual red numbers (other than R16 and R14) BC: bet on black color BN: bet on all individual black numbers (including Green-0) The top strategy is: 10.8% on R16 8.1% on R14 0% on Red color 43.3% on all individual red numbers, equally distributed 0% on black color 37.8% on all individual black numbers, equally distributed Code: R16 R14 RC RN BC BN F(R16,R14,RC,RN,BC,BN) 10.8 8.1 0.0 43.3 0.0 37.8 0.5192641 10.8 8.1 0.0 43.2 0.0 37.9 0.5192639 10.9 8.1 0.0 43.2 0.0 37.8 0.5192606 10.8 8.1 0.0 43.4 0.0 37.7 0.5192593 10.8 8.2 0.0 43.2 0.0 37.8 0.5192591 10.8 8.1 0.0 43.1 0.0 38.0 0.5192588 10.9 8.1 0.0 43.1 0.0 37.9 0.5192582 10.9 8.1 0.0 43.3 0.0 37.7 0.5192582 10.7 8.1 0.0 43.3 0.0 37.9 0.5192581 10.8 8.2 0.0 43.1 0.0 37.9 0.5192566
So basically what you're saying is ... Divide your bankroll into 37 equal parts and bet one part on each wheel slot. This works out to 4 parts on R-16, 3 parts on R-14, and one part on each of the other 30 numbers. Interesting. k*E == 0.446, which happens to be identical to the previous strategy: Divide your bankroll into 37 equal parts. Bet one part on each of the not-red numbers, bet 18 parts on Red, bet 3 parts on R-16 and bet 2 parts on R-14. How exactly are you measuring these strategies?
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?
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 checked again, and my k*E values are identical for the two strategies. Can you write out the formula for F(...)? Thanks.
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 - redNumber + black + 35 * (blackNumber / 14)) where pR16 = 4 / 37 pR14 = 3 / 37 pRedSingle = 16 / 37 pBlack = 14 / 37 r16 = percent of the bankroll to bet on R16 r14 = percent of the bankroll to bet on R14 red = percent of the bankroll to bet on red color redNumber = percent of the bankroll to bet on all red numbers other than R16 and R14 black = percent of the bankroll to bet on black color blackNumber = percent of the bankroll to bet on all black numbers (including Green-0) log1p(x) = ln(1 + x)
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.
After fixing the typo, the top 10 strategies are almost identical to what they were before. It's just F(.) was reduced: Code: 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 0.0 43.1 0.0 37.8 0.4784707 10.9 8.1 0.0 43.2 0.0 37.7 0.4784705 10.7 8.1 0.0 43.3 0.0 37.8 0.4784697 10.7 8.1 0.0 43.2 0.0 37.9 0.4784695 10.8 8.2 0.0 43.1 0.0 37.8 0.4784691 10.8 8.2 0.0 43.2 0.0 37.7 0.4784689 10.8 8.0 0.0 43.3 0.0 37.8 0.4784682