soft 12?

soft 12?

A thought just occured to me. If you are playing in a game that allows you to hit on and resplit aces (once), what do you do in the following situation:

Dealer has 6 as up card.
Player is dealt A,A
Player splits, gets another ace.
Player splits, gets another ace.
Now the player cannot split anymore, and is left with (A,A) or (A,1) against 6

This play is not accounted for in basic strategy. Should the player double or hit (as basic strategy)?

Hit

"Should the player double or hit (as basic strategy)?"

Hit.

Don

Can't do it...

Almost every BJ game (excluding the carnival types) have the same rule regarding splitting aces. That is: only one card to each ace, and if they allow RSA, then ok. However, if you have the usual 4 hand allowed rule, then in your situation, you are not allowed to take another card, much less double down.

Even in a DAS game, it does not usually apply to Aces..

Let me know if you find a game where you can double after splitting aces and still get 3/2 on BJ.

PB

Many MI Indian joints (nt)

Cool! A Question..

I wonder what this rule adds to the player advantage in terms of percentage.

Any math types care to jump in (please?).
Thanks
PB

I would Double

I don't think you indicated S17 or H17.

I did a little quick Combinatorial Analysis. Based on 6 decks. For S17, the correct BS is to Hit, For H17 it is Double. Here is the output

S17 

      Split   Mean =   67.71189 %   SD =  1.542

      Hit     Mean =   18.59536 %   SD =  0.956

      Dble    Mean =   18.43762 %   SD =  1.946





H17

      Split   Mean =   67.43800 %   SD =  1.556

      Dble    Mean =   20.04665 %   SD =  1.946

      Hit     Mean =   18.71503 %   SD =  0.955

Notice how close the S17 numbers are. I believe that in your Hypo, the removal of all those Aces would be enough to shift the result to double.

More importantly, you would douible this in plus counts. I am fairly confident that the Counters Basic Strategy for this play would be Double. If you are not going to learn an index number, it is better to learn CBS than regular BS.

Also, I will be VERY surprised if the BS for fewer decks was double, even with S17.

My initial reaction was "Of course double". But I was actually thinking of a different situation. There are Indian casinos in my area which all you to double, but no hit, after splitting Aces. You can double for less. The decision would be whether or not to double for less, or double for the full amount. Of course, here, we would double for all we can.

Where do I find CBS? (nt)

Questions

1) How many times can you split? (Maximum hands allowed)
2) # decks?
3) S17 or H17?
4) You said a pair of aces can be treated like a pair of eights. Is that correct?

Thanks.

Cacarulo

thanks...

...for the info all. For those who are wondering where these rules cans be found (resplitting and hitting on split aces), I have no idea. My question was purely hypothetical. It was just a thought that popped into my head. Also, I didn't have any particular rule set in mind, but obviously the answer may vary based on #decks, H17/S17, etc...

--Thanks, Homer J.

Not so Hypothetical

I know of places where this rule can be found but don't ask me where ;-). If you don't have a particular set then I will answer with the following: 6D,S17,DOA,DAS,SPL3 (Aces are treated as any other card)

1) 6D,S17,DOA,DAS,SPL3 (Aces are treated as any other card)

A,A vs 6

Standing  = -14.56985770378374%

Hitting   =  18.79735207889857%

Doubling  =  19.07085975358308%

Splitting =  81.65824732277233%

If four aces are on the table and you receive another ace the answer for this particular hand will be:

Standing  =  -13.24972842459168%

Hitting   =   19.33305307218343%

Doubling  =   20.93816914825459%

So doubling would be the correct answer as MathProf have already said.

2) 6D,H17,DOA,DAS,SPL3 (Aces are treated as any other card)

A,A vs 6

Standing  = -11.51592444501268%

Hitting   =  18.95361146626500%

Doubling  =  20.63117654095774%

Splitting =  80.86942908370871%

Again, If four aces are on the table and you receive another ace the answer will be:

Standing  = -10.55370217004014%

Hitting   =  19.48500178714592%

Doubling  =  22.29815471605918%

doubling again!

Sincerely,
Cacarulo

Check Out....

Kewadin Casino Sault St Marie MI at Trackjack.com. Double split aces is worth +.1% according to Trackjack. Click on the question mark at Trackjack to get a breakdown of player expectation.

It Varies

There was a fascinating paper by Hal Marcus at the 11th International Gambling Conference sponsored by UNR in June 2000. It discussed CBS for various games. I believe that Hal's software will generate it for you.

The problem with CBS is that there is no single CBS. It varies from game to game. Most importantly, it depends on your bet spread.

However, there are some plays that would be CBS in any game. Standing 16v10 is one such play. In the post above, I was expressing my belief that doubling AA vs. 6 would fall into a similar category. I say "belief", because I haven't made a detailed analysis of it. But I would be VERY much surprised if there were a counter-example.

Discrepancies

There were some differences between our numbers. I went back and saw that I had several parameters incorrectly set in my calculation. I actually made two, somewhat offsetting errors. (Basically, I computed the expectations over a full 52 card pack, rather than a 309 card remainder.)

Then I recomputed them. These are my new figures for BS, (from the 309 card remainder that we have after removing the Ace, Ace, and 6):

S17

      Dble    Mean =   19.15784 %   SD =  1.946

      Hit     Mean =   18.88763 %   SD =  0.956



H17

      Dble    Mean =   20.69651 %   SD =  1.946

      Hit     Mean =   19.02624 %   SD =  0.954

Now I believe that you are computing the exact Hitting expectation, whereas I am doing an approximation. So the discrepancy here is not surprising, and your figure is the more accurate. But I think I am using an Exact Doubling Algorithm, and so our figures should agree? And they are off in the third decimal place.

Index number

With the difference between the hitting and doubling expectations as close as they are, it would seem that an EV-maximizing index for such a play would probably be very close to zero. Therefore, I'd expect the R-A index to be somewhat higher.

Don

Cacarulo match

My numbers for this case exactly match Cacarulo's except for splitting, where I believe his numbers are more accurate. It never hurts to verify CA numbers from multiple sources, but as a general rule if your numbers don't agree with Cacarulo's, there is a very high probability that Cacarulo's numbers will ultimately prove to be correct.

Surprise!

> Also, I will be VERY surprised if the BS for fewer decks was double, even with S17.

With a single deck, doubling is worth 32.93% compared to 22.6% for hitting, and the numbers are the same for S17 and H17 since all the aces are gone.

For S17 games, doubling remains the better play until 18 decks are used. For H17 games, doubling is better with any number of decks.

Why the surprise? Is your CA program unable to compute numbers for single deck games?

Cards removed

For the first part of the problem I computed the play with 3 cards removed (6,A,A) and for the second part 6 cards were removed (6,A,A,A,A,A).

Let me know what you get.

Sincerely,
Cacarulo

Found It

Yes I found what I did wrong. I thought I was using my Best (and slowest) algorithms to compute the numbers. This should have given the exact answer, and a very good approximation for hitting. However, I wasn't doing that.

The way my software works is it does a quick approximation for all possible plays. It then does the top choice, in this case splitting, using the best algorithm. If it is a close play, it also does the runners-up that way. However, here it decided that Splitting was so much better than either doubling or hitting, then no-one would be interested in the hitting/doubling numbers.

I simply outlawed splitting, and ran the numbers again. I will post below the numbers from the standard 309 card remainder. The doubling numbers match Cac's to the amount of precision that I have printed. The hitting numbers are, as expected, off from his exact numbers in the fourth or fifth decimal place.

BTW, I did not realize until just now that even with S17, the exact numbers show that doubling is the BS. To respond to Don's issue of Risk-Aversion, the critical fractions are also listed.

//New  S17



Results  for:      AC AC  (Soft 12) vs. 6C 



We Computed 2 Choices 

      Dble    Mean =   19.07086 %   SD =  1.946

      Hit     Mean =   18.80238 %   SD =  0.956





Risk Averse Data - Critical Banks 



Dble:   535.08 (  0.187% )  with Var Method





//Following H17



Results  for:      AC AC  (Soft 12) vs. 6C 



We Computed 2 Choices 

      Dble    Mean =   20.63118 %   SD =  1.946

      Hit     Mean =   18.95811 %   SD =  0.955



Risk Averse Data - Critical Banks 



Dble:    85.90 (  1.164% )  with Var Method

If a player was betting only 1 unit from a 400 unit Kelly Bank, he/she would double. If they had 2 units out, they would hit in the S17-BS strategy situation. For the H17-situation, they would need to have 1.12% of bank out. This is actually a fairly large bet if you are playing two spots; your total action would be 2.4%.

Since this rule would only be used after splitting, we should probably use a 308-card remainder: Remove 2 aces and the dealer 6. Split the Aces and draw another Ace. I thought the Homer's rule did not Re-splitting fo Aces. If Re-splitting were allowed, then we should remove 4 Aces, the dealers 6, and another Ace, as done by Cac. The critical fractions would be much higher then.

Kelly critical fractions

> The way my software works is it does a quick approximation for all possible plays. It then does the top choice, in this case splitting, using the best algorithm. If it is a close play, it also does the runners-up that way. However, here it decided that Splitting was so much better than either doubling or hitting, then no-one would be interested in the hitting/doubling numbers.

Interesting. That is essentially the same method Griffin used to create his basic strategy charts -- quick/dirty approximations combined with slow/accurate computations. This led to the two known (but very minor) errors in his basic strategy, due to misjudgements that the approximate results were "far enough apart" to preclude a strategy error.

It is nice to not have to worry about approximations most of the time. One of these days I've got to upgrade my pair splitting so that I never have to worry about approximations. Ah, the luxury of a blazing fast algorithm... ;-)

> BTW, I did not realize until just now that even with S17, the exact numbers show that doubling is the BS. To respond to Don's issue of Risk-Aversion, the critical fractions are also listed.

Do you support log-optimal strategy at all decision points, or are you approximating the critical fractions? Also, do you support any other utility functions, or only the linear and log() cases? I have support for power utility functions in addition to Kelly.

> Dble Mean = 19.07086 % SD = 1.946
> Hit Mean = 18.80238 % SD = 0.956

> Risk Averse Data - Critical Banks
> Dble: 535.08 ( 0.187% ) with Var Method

OK, I guess that answered one of my questions. I compute the crossover at 0.19036% for this decision.

> //Following H17
> Dble Mean = 20.63118 % SD = 1.946
> Hit Mean = 18.95811 % SD = 0.955

> Risk Averse Data - Critical Banks
> Dble: 85.90 ( 1.164% ) with Var Method

This is closer -- I get 1.1661% as the crossover here.

> Since this rule would only be used after splitting, we should probably use a 308-card remainder: Remove 2 aces and the dealer 6. Split the Aces and draw another Ace. I thought the Homer's rule did not Re-splitting fo Aces. If Re-splitting were allowed, then we should remove 4 Aces, the dealers 6, and another Ace, as done by Cac. The critical fractions would be much higher then.

The critical fractions go up to 0.809% for S17 and 1.693% for H17.

Full Output

Here is the full output, for the H17-BS case. "List" refers to the list of outcomes. That is, it is the entire probability distribution. Keep in mind that the doubling numbers are supposed to be exact, while the Hitting Number are an approximation. I can also produce an approximation of the distribution for Splitting as well.

Risk Averse Data - Critical Banks 



Dble:    85.90 (  1.164% )  with Var Method

Dble:    86.10 (  1.161% )  with SS  Method



Results  for:      AC AC  (Soft 12) vs. 6C 



We Computed 2 Choices 

      Dble    Mean =   20.63118 %   SD =  1.946

      Hit     Mean =   18.95811 %   SD =  0.955



List for Dble

          -2.00   42.694

           0.00    4.297

           2.00   53.009

List Mean =  0.206312   SD =   1.945648



List for Hit

          -1.00   37.875

           0.00    5.292

           1.00   56.833

List Mean =  0.189581   SD =   0.954536







Risk Averse Data - Critical Banks 



Dble:    85.90 (  1.164% )  with Var Method

Dble:    86.10 (  1.161% )  with SS  Method

Dble:    85.98 (  1.163% )  with Log Method

This code is several years old. It needs a major re-engineering, which I really haven't had time to do. I did not have power utilities in there, but I suppose I could. My recollection is that I found the log-optimal fraction with Newton's method, but it has been a long time since I looked at it.