Hi folks--The casino I play at has decided to move to a pure accumulation format (to my displeasure, but I need to adapt). Rules are: BJ pays 2:1; insurance but no surrender; min bet 5, max bet 500, 500 units to start; dealer hits soft 17; 15 hands per round; all entry fees ($20 per round) paid 50% to highest score in all of the rounds, 30% to second highest score, and 20% to third highest score. Based on the previous hybrid format, I anticipate that there will be approximately 10 rounds, and that the highest score will be approximately 6-8 times the original bankroll (i.e., 3000-4000 units). I anticipate playing in every round, unless and until I score either 3500 units, or the top three spaces on the leader board each exceed 3500 units. (Those numbers might increase over time as players adjust to the higher scores necessary to win and start betting even more aggressively). Question 1: Based solely on intuition, I suspect that I have about a 5-10% chance of achieving a score of at least 3000 in any round playing basic strategy. Is there an exact percentage that can be mathematically derived, which would then tell me whether it even makes sense for me to pay up to $200 to try to win a portion of the assumed $1400 pot? Question 2: I know that Gronberg's general advice for accumulation formats is to make maximum bets until you either bust or hit your desired number. Does the fact that the BJ payoff creates a small positive variance, and the presumed participation in 10 rounds of 15 hands each, create any reason to prefer betting 50% of your bankroll until you have enough to make 500 unit max bets? Question 3: With such a high target number, I assume my strategy should be to increase variance (since I predict the other players will continue to play basic strategy or occasionally sub-optimal BS). I also assume that the best way to increase variance is to double and split more often than BS would otherwise specify. But is there any reference work you can direct me to that would mathematically rank those deviations in doubling and splitting BS strategy from the plausible (eg., splitting 10s when the dealer shows 2-6) to the debatable (eg., doubling on 12 or 13) to the ugly risky (eg., doubling on 17)? And assuming such a ranking of deviations can be accessed somewhere, where would you draw the line between good risky and bad stupid? Thanks in advance for your always thoughtful replies. Best--Acercher

Good thoughtful questions! Question 1: There exist double barrier formulas for calculating the probability of reaching a goal from a given starting bankroll with a given unit before busting out. The one you want also specifies a limit on the number of hands. The formulas are extremely complex but can be found in Blackjack Attack, 3rd Edition by Don Schlesinger, in the chapter on Risk of Ruin. I have always used my own simulator to get these kinds of answers. Question 2: In the long run, 2:1 for blackjack does create a small positive player edge for most games. The problem is that you do not have until the long run. You mentioned 15 hands per round. On average, you get a blackjack once in approximately every 21 hands. There is a good chance that you will never see one during a given round. In general, when an improvement in your edge is due to a relatively rare event, you will be less likely to change your strategy. I do not change my strategy in this situation. If I get a blackjack, then great. It is worth noting however that this rule does slightly increase your overall chance of reaching your goal even without changing the strategy. Question 3: I have often seen general advice about doubling or splitting more often in accumulation rounds, but no specific research. One way to approach it would be to look at doubles which have a relatively low (card counting) index. For example, 9 vs 2; 8 vs 5 and 6; A,7 vs 2; A8 vs 5 and 6; and even A,9 vs 5 and 6. These are normally hits, but the EV of doubling these hands is very close to the EV of hitting. You get the benefit of the increased variance without giving up much of the EV. I'm not saying to count the cards and make these doubles at the index. I'm saying you might want to consider making these doubles all the time. For splitting, you could consider a similar methodology, but I would consider splitting tens to be the most useful deviation because you get pairs of tens more frequently than any other pair and because you generally have an advantage against most dealer up-cards when starting with a ten. You can generate opportunities to place another bet when you know you have the advantage, as opposed to your initial bets which are all placed at a disadvantage, unless you are counting, which is waste of mental resources, in my opinion, for a 15 hand round. You can bet with an advantage by splitting tens vs a dealer 9 or less. Other splits to consider might be 2,2 vs 8; 3,3, vs 8; 6,6 vs 7; 9,9 vs 7. Of course, near the end of the round, if you need to make several max bets, you should be doubling and splitting everything. Ken Smith wrote an interesting chapter in one of his e-books showing how you are more likely to win the vast majority of doubles and splits than you are to win 2 hands in a row. Good luck!

I am trying to learn also. That is interesting with the double having better win rate than 2 singles in row. Plan on doing some trials using WONGS program.

Gronbog--Thank you for your reply after I misspelled your name.... Your answer was precisely what I was looking for: if I understood you correctly, I can increase variance without noticeably degrading my results by doubling in situations where the expected value of doubling or splitting is very close to the expected value of hitting. I am unclear, though, if the examples you gave were all of the situations that fell into that category, or just some of them. I tried to find a chart online that would show such a comparison of expected values, and believe I did at http://web.archive.org/web/20060214092652/http://www.bjmath.com:80/bjmath/ev/6dh17.htm. Am I correct in believing that I can populate the set of hands where the expected value of doubling or splitting is very close to the expected value of hitting by looking at the columns in the link for hitting and doubling and selecting those hands where the numbers specified (no matter what they are) are virtually the same, such as A vs. 8,2 or 7,3 or 6,4? Or have I completely misunderstood either you or the linked table? Thanks--acercher

Acercher, you have the right idea. I may have led you astray in that I said that all those hands were basic strategy hits. In fact, some of them are stands. So you should be looking to double or split in situations where the EV of doing so is close to the EV of the correct basic strategy action.