A pretty common situation.

This is really not complex, but I was never 100% if I am getting it right or not:

Hand 9 out of 10, BR2 2000 chips first to act and BR1 2001 chips last to act.

BR2 bet 100 chips (or more), then BR1 exactly match BR2`s bet of 100 (or more)

BR2 gets "XY", BR1 gets Blackjack (pays 3 to 2) or pair of 10s, dealer up card "Z".

BR2 double down on his "XY" and ends up with a stiff hand. - Should BR1 correlate and double/split too in order to keep his tiny lead assured on hand 10 ?

PS: Two additional questions. 1 But if BR2 bets more than 100, say 500+ ? And 2. But if BR2 ends with 17 instead of a stiff hand ? (I correlate in all cases)

 

Nice One!

Excellent teaser! I had this happen on a final hand once, and it was a no brainer. Having it occur on the second last hand where you will be acting first on the final hand makes it interesting. The choice is between taking a guaranteed micro-lead into the final hand by correlating or taking a chance at a larger lead at the possible cost of losing it.

Let me start by saying that it would depend on what the maximum bet is. If the lead we could generate by standing on the blackjack is less than a 1/2 max bet, then it carries no additional advantage and we should correlate for the guaranteed micro-lead.

If the lead we could generate is a 1/2 max bet or more, then it becomes tempting to take the chance. If the possible lead is a max bet or more, then it becomes really tempting! The largest lead we could generate would be 3.5 x max plus one chip (if both bet max) which would be a virtual lock.

I would also like to note that by standing on the blackjack, we cannot end up a 1/2 max bet or more behind (if both bet max and BR2 wins his double, then we end up behind by 1/2 max bet minus one chip).

Finally, it would probably depend on the dealer's up card.

To come up with hard numbers we would have to consider the probability that BR2 wins/pushes(in the case of 17)/loses his double combined with our probability of prevailing in each situation. We would then compare this result to our probability of prevailing with our micro-lead intact. We could then determine what the threshold ratio of bet/max makes it worthwhile to stand.

If you could provide the min/max bets, then one of the number-crunchers here could certainly come up with the answer. I'm suspecting that it's the usual 100 min/1000 max game with surrender that your teasers generally arise from.

 

- I`ll start with the end to answer this. Yes, you are very right, this originally appeared few times in some games at 21Gnet network where the betting limits are min/max 100/1000 with surrender and doubling in blackjack allowed. (surrender after split is also allowed)

- But if it can be of any further help for some other similar situations (splitting tens only once - double after split, no surrender, no double in blackjack)

- So, I am thinking it can be also used for a min/max bet of 100/500 - for the GameAccount network.

- If "tempting" means the better decision for BR1, then I can understand and work it out.

But I have one question, if we can not generate a +1/2 max bet lead, but we can achieve something very close to it, we should still correlate and split/double ?

*Lets say that the min/max bet is 100/1000 (increments of 1 - and surrender allowed). If we do not correlate BR2 action and we are hopeing for a bigger lead, in case it happens, say will generate us a lead between 401-499. Now on the last hand we still can take the straight high and low with a single bet of 600.

- We should still go and correlate our opponent action by splitting/doubling, or not really ?

PS: But by correlating with a double on blackjack or by splitting 10s, we can even further larger our lead rank, by winning while our opponent lose his double !

 

Good point. This lead does have more value than the micro-lead, but not as much as the 1/2 max+ lead. Is it good enough not to correlate? I don't know. It's another case to be examined.

Another good point to be factored into what will certainly be a complex analysis.

 

*** reposted to the intended thread ***

I've started looking into designing a series of simulations which will solve this teaser. PlayHunter originally posted starting bankrolls of 2000 and 2001 with matching bets of 100, but also asked about the case where the bets were 500. My intent was to to solve this in general so as to determine the bet threshold (if any) for which doubling/splitting to correlate becomes better or worse than standing. I had hoped to do this for both of the proposed scenarios (BR1 has BJ and BR1 has a pair of 10s).

The problem is that, with the given bankrolls and maximum, the larger the common bet is, the more crippled BR1 and BR2 become when they lose, making the scenarios different for many potential bet values. For example, with the proposed bets of 100, both still have plenty of bankroll to work with on the final hand, however if the bets were 500, then they would only have 1 max bet to work with. If both were to bet the maximum (1000), then BR2 would be instantly eliminated, should he lose his stiff double. If both bankrolls were increased, or the maximum was decreased such that both players could split and double to the full extent (I assume 4 hands doubled), then a general solution could be computed.

So, I propose the following:

1. to find the bet threshold (if any) for a BR1 blackjack with the bankrolls and maximum as posted
2. to find the bet threshold (if any) for a BR1 pair of 10s with the bankrolls and maximum as posted
3. to find the bet threshold (if any) for a BR1 blackjack with bankrolls and maximums such that both players will have full split+double capabilities on the final hand.

The first two items will answer PlayHunters specific questions from his original post. The third will solve the issue in the general case. For the third case, I hope to end up with a spreadsheet in which one can enter the probabilities of the various swing possibilities for BR1 and BR2. This could be used to more quickly compute the threshold (if any) for BR1 and BR2 hands other than those posted.

This will take a while, but the results are of interest to me, as are some intermediate results which will come from this process, so I don't mind.

 

Some Answers

I have an answer for this particular situation and question. It took a few sims to work out all the branches of the probability tree and I had to make some assumptions along the way.

• I assumed that both players would play optimally. This is not too much of a stretch since, in most cases, in my opinion, the optimal strategy was within the grasp of a reasonably skilled player.
• I also assumed that both players would make strong (if not optimal) bets on the final hand. You can be the judge of whether the bets that I chose are reasonable.

A spreadsheet summarizing the results can be found at http://gronbog.org/results/blackjac...ommon Situation/A Pretty Common Situation.xls.

The first tab summarizes the results where BR1 (Player 2) has a blackjack and the bets were both 100. The possible outcomes of the second last hand are limited by BR2 (Player 1)'s stiff double and Player 2's blackjack. There are only two possible outcomes if Player 2 stands and their probabilities are listed along with their resulting bankrolls. There are 4 possible outcomes if he doubles.

For each outcome, I then decided on what the bets for the final hand might be and ran simulations to determine the probability that Player 2 would prevail (i.e. end up first). Multiplying this by the probability of that situation occuring in the first place yields the probability of Player 2 advancing in that situation. Summing the probabilities for each situation gives us the overall probability of Player 2 advancing.

We can see that if Player 2 stands on his blackjack, on the second last hand, his probability of advancing is 0.5164 or 51.64%. If he doubles, then he has a slightly better chance at 55.11%.

The next tab is the same analysis for Player 2 having a pair of 10s. Once again, we see that he is better off correlating by splitting his 10s (and not busting).

Note that to the right on each tab are results for other final hand bets which turned out to be not as good as the ones I eventually used.

Results for the same situation with 500 bets are on the way. It won't take as long, since some of the sim results can be reused.

 

Results for 500 bet

I have updated the spreadsheet at http://gronbog.org/results/blackjac...ommon Situation/A Pretty Common Situation.xls with the results for 500 bets. Once again there are two tabs, one for the case where BR1 (Player 2) has a blackjack and the other for when he has a pair of 10s.

These new results clearly illustrate two of my original points, which are that the decision of whether to correlate or not depends on the the size of the lead one could generate in either case and how crippled BR2 (Player 1) would be should he lose.

• The case where Player 2 has blackjack is dominated by the 70.38% possibility that he could gain a lock by standing on his blackjack. That possibility alone outweighs all of the possible outcomes to be gained by doubling. We now know that the bet threshold for preferring to stand over doubling is somewhere between 100 and 500. Bets of 400 each would still allow for the lock when standing and would likely yield a similar overall result to the 500 bets. I'm not sure now that I have the time or patience to iterate to find the actual threshold.
• In the case where player 2 has a pair of 10s, there is the possibility of a lock when splitting, but the probability of it occurring is low enough that it contributes the least to the overall chance of success. A near lock is possible when standing and it has a high probability of occurring. For this hand, standing is also now the right choice.

 

Thank you for clearing this teaser, because it is common and is great to know what is optimal to do in this cases.

Though one question still remain: - There is any changes if our opponent end up with 17 instead of a stiff hand ?

 

Results for Player 1 17

I have added a "100 BJ 17" tab to the spreadsheet on my web site. As expected, Player 2's chances are slightly reduced in each case if Player 1 has 17 instead of a stiff hand. However doubling still appears to be his best option on the 2nd last hand.

 

possible sim adjustments

Gronbog’s simulations are very inspirational to me, thanks.

It was a pleasure to look at the stats and check for the right bets and dig out various scenarios. Here are some possibilities that may (or may not) change a few hundreds of percent. Gronbog it would be helpful if you on your spreadsheets include some of the rules, the two most important are doubling for full amount only and surrender.

In the table 13-16, “Go for Gold”, after Player stands and have lose/lose:
........P2.............P1
..brl 401........brl 900
bet...401...........449
I initially disregarded the bankrolls and thought only of betting less than half the stack. But the maximum P2 can get is blackjack that brings him to 1002.5, so all the P1 has to bet is 103 to no more than196 to cover bj by winning single bet and surrender (if available) anything but P2’s bj. This will be about 2.5%

In the table “A pretty common situation”, 100bj, after P2 doubled has stiff hand
...........P2.............P1
....brl 2151.......brl 2200
bet....1000...........1000
With bankrolls bigger than twice max bet most benefits comes from betting a chip more than two and half times the gap, and 5 times the gap if the opponent makes the mistake of overbetting though still keeping a chip(s) more than BR2 remaining stack. A bet that could be split and doubled a few times should improve the chances for P2. For the “Masonuc maneuver” we should bet no more than 2/7 of the brl, and with rules of double for full amount only it is betting less than 1/5 or less than 1/3 of the brl. So against good/computer players it could be bet of 124, and against real players 245, and I don’t see any downside with this bet sparring with the optimal play. Actually, at the table with the bankroll of 2151 I would make bet of 251 so the opponent can clearly see that I have left 1900 and hope he keeps 1901 – now I beat him if I double and so does he and we both lose.

In the same paragraph of that table the probability for w(dd it is a stiff)/bj is 29.62%. Is this the average of dealer’s hands busted? If so, the results will be right for the case when the players don’t see the dealer’s upcard. There will be separate and different results for every dealer’s upcard. The weaker the upcard the easier it is for the doubled stiffed hand to win and so it is for bj double to win. The stronger the upcard the less dealer will bust and at the same time double on bj will work less. I think that it may be to bj hand benefits to not double against dealer Ace, close (but still double) against dealer Ten, and definitely double on any other hand with the best results against dealer 6.

Still on the same page, for the lose/bj
...........P2.............P1
....brl 2151........brl 1800
bet.....700..............900
P2 prevails 62.29%
The most of the wins for P1 comes from winning double, which P2 can’t cover by his doubling. What will be the numbers for P2 bet range of 725 to 799? It gives away surrender versus P2 push, but it gains by locking out P2 by winning a doubled bet.
Also, though it may not be the best bet, what are P1 chances with 697 response to P2 bet of 700?

Thank you, again, for your wonderful work.
S. Yama

 

Thanks! As long as folks are interested, I don't mind doing this kind of work.

Done

I can verify this if anyone wants. S. Yama's numbers are generally very reliable.

Some excellent considerations here and more examples of how your mileage may vary depending on the skill level of your opponent and the read you have on him. I tried to set things up as two expert players, each knowing that the other is an expert.

Yes, the situation as posted did not specify the dealer's up card. I probably should have asked, since it was based on a real situation. For the simulation, the dealer was dealt random up cards.

These strategy variations can probably be verified by plugging in probabilities available from existing tables into the spreadsheet. If not, then I'll run the sims if anyone is interested.

This sounds like a request. If so, then I would be happy to run it.

Thank you all for your interest!

 

Just curious about S. Yama and gronbog

Do you two know each other? I'm one of the fortunate that knows both of you and appreciates the contributions by both of you.

Billy C

 

If so, I guess it does not only affect the probabilities on the blackjack hand, but also the probabilities of when the player gets that pair of tens, right ?

Now I can see how important it is to know the dealer up card for this situation at hand ! I did not specified the dealer up card, because I wanted to make it a general case. It seems I was unsuccessful on this. :sad: - Well, I am one guy interested, but I can see there is a lot of work on this issue so I can not ask for it.

 

Hey Billy! While I have had the pleasure of meeting you in person, I have not yet been fortunate enough to meet S. Yama in the flesh. We do correspond offline about various things. Perhaps some day our paths will cross -- hopefully not on an important hand at a tournament table!

 

This means that you advice to double on our blackjack hand on anything else except when the dealer has an Ace up card, even if when the bets are 500 ?

Also if that is true, then we should not split when we have a hard 20 ? (on anything else, except when dealer have an Ace or a 10 up card when we stand)

 

I can add something to this case too: - just two lines below when P2 doubles and win and P1 lose, on the final hand P2 (with a 401 lead) bet 800 and P1 bet 900, P2 will prevail 63.99%. - In this case, (last hand BR1 lead 401 and bet 800 when max bet 1000) Monkeysystem (in a previous thread) would advice an optimal bet for BR2 (P1) on the last hand of 797 instead of 900. So, what would be P2 winning chances when P1 will respond with a bet of 797, instead 900 ?

 

Can you post a link to Monkeysystem's post? I can't see how a bet of 797 would apply in this situation.

 

Tis is the thread link http://www.blackjacktournaments.com/bb/showthread.php?t=7428 and Monkeysystem`s post is #8 in there. (dunno how to link to post directly)

 

Wow! That thread is more than a year old. Maybe you do have the memory skills to remember all of these situations!

 

I assume that you meant to say "What will be the numbers for P2 bet range of 725 to 799? It gives away surrender versus P1 push, but it gains by locking out P1 by winning a doubled bet."

If P2 bets 725 (instead of 100), then his chances of prevailing increase to 0.6236 (from 0.6229). So it looks like it is worth giving up surrender vs push in order to cover a double.

If P1 responds with a bet of 697 (allowing him to surrender for the low, but his blackjack no longer beats a P2 win), then his chances are 1.22% worse.