Optimal Betting and Blackjack Attack, Ch 10
by ML
Posted by ML on 30 Oct 1997, at 6:58 p.m.
This post outlines a simple method to calculate bets in the range of optimal bets for most games available using only a calculator and BJ Attack tables.
Some of the more expert posters like Don S., T-Hopper, and others are not as enamored with the idea of betting the "Kelly" bet at all ranges as I happen to be and there are definitely choices to be made. "Kelly" or "Yamashita" sizing has associated with it a reasonably high risk of ruin, at least 13%, maybe more. And, while one can approach optimal bet sizing, one cannot be terribly precise. Preciseness would require betting 1.767 of a unit bet under certain circumstances.
But despite these real problems associated with the concept, I would submit the bj player who comes closer to betting the optimal percentage of his bankroll at all times should ultimately do better than one who does not. The reason is simple. Getting closer and closer to optimal bet sizing allows the player to get more money on the table while not changing risk.
For persons who have not read theory, the calculation of the optimal bet is a twofold process. First, a unit or small bet is calculated as a percentage of bankroll and then the best bet at each level is calculated until the level at which the highest bet is made. That is, given a certain game and given the player is limited to some spread like one to ten, the correct small bet, the correct bet for a +1 count, the correct bet for higher counts, and the level at which the ten bet should be made can be precisely calculated.
And after some work by some outstanding mathematicians, the "rules" of the calculations have been shown to be fairly simple.
The small bet (expressed as a percentage of bankroll) is simply (actually just very close to) the advantage of the game divided by variance of the game. The largest bet in the spread is the top spread multiple of the small bet and it should be made at the level where the advantage divided by variance of that level is just smaller than the multiple of the small bet expressed as a percentage of bankroll. That is, if you have determined you are going to spread ten, then you multiply the small bet (expressed as a percentage of bankroll) by ten and the first true count which has an advantage divided by variance greater than the number is the place where the ten should be bet.
All bets in between the smallest and largest bets are calculated by simply dividing the unit bet into the advantage divided by variance of true counts between.
Don and John in Chapter 10 have given us enough information to make such calculations simple.
Using Table 10.9, Page 200, 6D. DAS. 5.0 Penetration. Spread 8 as an example.
Looking in the middle section numbers which are the statistics for the whole game we find.
Win/100 (advantage) is 1.22 or .0122 per hand.
SD/hand is 2.65. Since variance is SD squared, 2.65^2 is 7.0225 per hand.
The unit bet for the game described should be .0122/7.0225=.001737 of bankroll.
The required bankroll expressed in unit bets is 1/.001737 or about 575 small bets.
Now Don and John figured out optimum bets for all counts but they expressed these optimum bets in terms of a bankroll of 400. Since we are expressing terms in percentages of bankroll, we need to divide their figures for optimum bet in the left section of the chart by 400.
+1 1.2/400=.003 .003/.001737~1.7
+2 2.92/400=.0073 .0073/.001737~4.2
+3 4.36/400=.0109 .0109/.001737~6.2
+4 6.16/400=.0154 .0154/.001737~8+
So rounding.
<+1 bet 1 unit
+1 2 units
+2 4 units
+3 6 units
>+3 8 units.
If this is simmed out on SBA using a 576 bankroll, it comes close to performing optimally.
Now realize this is the same game but a different bankroll than charted. With the larger bankroll, the bets are raised at lower counts and 8 units are bet at a lower count. Notice also the ruin level for 400 units shown at the right of the charts of 25%.
For the more mathematically precise, the calculation would require another iteration because the levels calculated above would end up with with a different advantage and variance than the Chapter 10 tables. The optimum bet would be a smaller percentage of bankroll because, while expectation is raised by betting more, variance is raised even a higher percentage.
If one has SBA available, the iteration is simple. When it is made, it is found a bankroll of about 600 with the same change of bets is closer to optimal.
If one does not have SBA available, the calculation shown above will be about as close as one can get since there has to be a lot of rounding involved but a safe fudge factor would be to add 10% to the bankroll requirements originally figured.
These simple calculations should raise quickness of winning a percentage of the player's bankroll and at the same time keep risk of ruin at reasonable levels, not more than 15% if the player kept the same unit bet even after he had lost a fair percentage of his bankroll.