From the bjmath Workshop

Case Study: Optimal Strategies when BJs Pay 2 to 1

Posted by MathProf on Saturday, 31 October 1998, at 8:59 a.m., in response toUsing RA-Fractions and to Optimize DI or RoR., posted by MathProf on Thursday, 29 October 1998, at 6:07 p.m.

To illustrate the concept discussed above, I have analyzed the Basic Strategy for a game in which Naturals pay 2 to 1. Here I am looking at a strategy for a non-counter playing from the complete deck. However we are interested in the "optimal strategy", taking Risk Aversion into account. I used 6 decks, and LS, Rs2 (but no DAS). I wanted to include Surrender among the possible Risk Averse decisions; but did not want to DAS or resplitting, because I did not wish worry about differences between pre-split and post-split strategy (an interesting question in its own right, but beyond are current scope.)

The 2-1 bonus gives the players an advantage off the top. However if the players do not get a natural, they are playing standard full-deck Black Jack. This makes analyzing the strategy very easy; we do not have to worry about methods of index generation, but can simply do combinatorial analyses of the full-pack.

Here are the most important Risk Averse plays. Sorted by Critical Fraction. To conserve bandwidth, I only took fractions up to 5% (which is probably wasting too much bandwidth as it is.) I can make available the full list if anyone wants it (not as neatly formatted).

The entries with * indicate those where Surrender is the alternative play.



	Play		Fraction

	A2 v 5		0.12%

	A4 v 4		0.14%

	88 v 10*	0.57%

	22 v 4		0.71%

	33 v 4 		0.72%

	A5 v 4		1.30%

	A3 v 5		1.54%

	A2 v 6		1.74%

	9 v 3		1.89%

	A7 v 3		1.99%

	A6 v 3		1.99%

	10 v 9		2.07%

	55 v 9		2.21%

	99 v 2		2.55%

	66 v 3		2.66%

	A4 v 5		2.67%

	A3 v 6		3.21%

	A5 v 5		3.70%

	33 v 5		3.82%

	22 v 5		4.03%

	11 v 10		4.05%

	9 v 4		4.44%

	A6 v 4		4.46%

	A5 v 6 		4.50%

	A7 v 4		4.89%

	11 v 9		4.91%

	99 v 3		4.94%

Now the Conventional Basic Strategy for this game, gives the following results:


	Mean    1.762%   

	SD      1.152

	DI is   15.297

	Kelly Bank:  75.315 units

	Kelly Fraction is   1.33%

The Kelly player would optimally bet 1.33% of Bank. She could improve the game somewhat by adopting those RA plays whose critical fraction is below this. These are the top 6 plays in the table above, which have fairly low frequency of occurrence. With this Optimal Strategy, we obtain a smaller EV, but smaller SD and a marginally better DI: 


	Mean  1.758%   

	SD    1.146

	DI is 15.343

	Kelly Bank: 74.704

	DI has changed From     15.297  TO     15.343

	No Changed From   4273.43260  rounds to   4248.13233 rounds 

	An Improvement of  0.59 %

 

Note that here we do our Kelly fraction has increased, but it is still less than 1.34%. This is well below our next play, (A3 v 5, critical fraction of 1.54%) so our process is complete.

Now let us suppose that the table is very high for our player, and he wishes to reduce his overall Risk of Ruin. According to the methods discussed earlier, we double our optimal fraction. We get the value 2(1.33%) = 2.66% . The player adopts those RA plays who fraction is less than 2.66%. This includes the next series of plays, through AA4 v 5. Note that A4 v 5 (and 66 v 3) are right on the borderline. If we carry out the calculations with great precision, then we actually need two passes. On the first pass (starting from baseline Conventional strategy) they came off just above the cutoff. However, when we try our improved strategy, our optimal Fraction goes up. We double that and try again, and then we pick them up on the first round.

The result of our new strategy is a lower DI. However our Bankroll requirements (ratio of Var/EV), has been reduced. By about 1.4%. That is, for any desired RoR, the required bank is lower by 1.4%. 


Game  Summary:  Bj Pays 2 for 1 (RoR Strategy)

	Mean   1.731%   

	SD     1.134

	DI    15.269

	Kelly Bank: 74.266

	

	Banks Changed from  75.31 to 74.27 (1.4%)

What is perhaps is interesting is how little Risk Aversion is proper. If the player adopted even more conservative strategies, by adopting plays with higher Critical Fractions, the bankroll requiremetns would go up, or the Risk of Ruin would increase! Note that most of the popular Risk Averse plays (like 11 v 10, 10 v 9 ) are way up there on our list. Players who made these changes would actually be increasing their RoR.

Of course, if we were facing a "hot shoe", we would be betting at higher levels, and higher levels of Risk version would be appropriate. There would be greater savings to the player from RA strategies. Of course, where our baseline strategy would not be basic strategy, but be the strategy based on Index Numbers for the Hot Shoe.


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