Min. Bankroll Guideline: Max. Bet x ? (nt)

$25,000 in bankroll for every 1x$150 or 2x$100 of max bet for a straight counting game.

I usually play in a double deck game with double after split and S17, penetration to about .8 decks. I spread my bets 25 to $250. Haven't gotten much heat. Schlesinger's simulations say I have a long term risk of ruin of 2.3% spreading 1:8 with an 800 unit (units, not max bets) bankroll, so maybe 2.9%(?) 1:10? For a long term risk of ruin, this doesn't seem so bad to me. And since I like to spread my big bets out between two hands, I can lower my sd and risk of ruin from Schlesinger's numbers. 800 units would only be 80 max bets. Talking to Andy Bloch, from MIT, he thinks maybe 100 max bets. 125 sounds very conservative. I understand blackjack entails a high standard deviation, but I think people sometimes go overboard on these bankroll estimations.

you have a 1 in 35 or so chance of busting out. For an irreplacable bankroll this is very high. If you've still got another job and can generate or re-generate a bankroll then it's probably OK.

1 chance in 35 of going bust doesn't seem high to me. I could always get a job and build up again if I got screwed. On the other hand, cutting your bet the way you suggest costs you an awful lot.

I don't suggest making optimal Kelly bets, but there has to be a limit to conservatism. If 1 in 35 sounds high, does 1 in 100 sound high? Probably. If you are thinking it's the end of the world, then 1 in 100 probably scares the bejesus out of you too. But you'll never get the percentage of going bust down to zero, and to try and approach 0 will require such ridiculous bankrolls that you'll never make anything.

Now, your utility curve is what it is and I'm not going say mine is better than yours. But I think you do a disservice to novices (not that I am much more than a novice myself) by giving them such high bankroll numbers. The vast majority of them can go in with MUCH less than you say and make plenty of money if they play well.

Personally, depending on the games you play, I think you could get away with 800 units for a bankroll. I certainly think 1,000 is reasonable. 1,250, as has been suggested, seems unnecessarily conservative. $25,000 for every $150 of max bet, which has also been suggested, would be, what, around 1,666 units if you are going 10:1 (which you probably are not, of course). I've never read anyone suggest that you need this much. It is a bedtime story meant to scare children.

I want to a couple of points to this discussion. First the term �Max Bet� is somewhat imprecise. It depends on what you consider your Max Bet to be.

If you play 2 spots shoe games with good rules (S17-DAS-LS) and your Max Bet is not put out until a TC of 4 or 5, then your Kelly Equivalent Bank is about 30 Max Bets. If you have bank of 60 Max Bets, then you are playing at the Half-Kelly level. This has a static Risk of Ruin of 1.8%. 100 Max Bets means you playing only about one-third Kelly.

Now your Max Bet seems to be 1-spot at a double deck game, placed at about a TC of 4. This is about 1.5% Kelly Bet. Then you need about 120 Max Bets in your bank if you are playing Half-Kelly.

Now you quoted a 2.9% RoR. That is your RoR if you keep your bets sizes fixed and never resize them. But your rule �80 Max Bets� sounds like a resizing rule. That is periodically you reset your Max Bet to 1.25% of your Bank. If so, your RoR is a lot less.

If you take a simple rule of resizing once, when you lose half the Bank, then your RoR is the 3/2 power of your static RoR. In you case, it is (2.9%)^(3/2) or about 0.5%. Note this assumes all winnings are re-invested, and that never resize on the upside.

Mathprof,

In this case, my numbers come straight from Schlesinger's simulations in Blackjack Attack. Given double deck, penetration at 62 of 104 cards, s17, DAS, and a 1 to 8 spread, Schlesinger gives long term risk of ruin for an 800 unit bankroll of 2.3% (2.9% was a quick extrapolation on my part to a 1 to 10 spread). This is playing one spot.

I believe you know Schlesinger and his work, so it should be no problem to verify these numbers.

By 80 max bets, I did not mean to imply resizing. Only that, if my max bet is ten units, then 800 units is 80 max bets.

the reason you bet to a smaller kelly faction (say in in 500 versus 1 in 100 risk levels) is that you never have to worry about resizing in almost every losing streak.

At your level of risk, you will be faced with resizing quite often. When you lose bankroll betting 2x$200 and have to earn it back betting 2x$100 versus a player betting 2x$150 who never has to resize then comparing time and utility becomes more complicated. The change in time it takes to double bankrolls for different kelly fractions is not linear. Betting 1/2, 1/3, or even 1/4 kelly achieves most of the earnings power of full kelly with lower risk.

It sounds like you are using the charts that were in BJA I, and reprinted in BJA II. You should note that those charts were done before the modern theory of Constrained Optimal Betting was developed. The ramps used there are sub-optimal. The charts were re-done for BJA 3. It would be a good idea for you to read BJA 3, especially if you are interested in following the charts. It would also be a good idea to get BJRM, so that you could do these calculations for yourself.

To be more specific, Don�s chart in BJA 1 delays putting out the 8-unit bet until a TC of 2.3%. As I stated, that is sub-optimal. If you used an optimal spread, the RoR would be 4.25%.

Regarding the extrapolation to 1-10 spread: it depends on your betting schedule. If you follow the same pattern but put out the 10-unit bet at a TC of 6, then you get about a 2.3% RoR. If you put it out at 5, it goes up to 3%.

While you have the BJA charts, please look at table 10.18 which is the Vegas shoe game; S17-DAS-LS. Look at the RoR of 1-8 Wonging: it is .0.3%. Note that that is 10% of yours. Such players would only bust out 1/10 as often as you would. I mention this to point out that �Max Bet� means different things to different players.

I mention this because you quoted Andy Bloch of MIT as saying 100 Max Bet is enough. I suspect that his betting plan is closer to the second situation that the first. I also suspect that his rule is a dynamic one: means that he used it for periodic re-sizing.

On the substantive issue as to whether 2.9% RoR is �too big�, I am not sure you and BP are in that much disagreement. For a recreational player, for which BJ is a financial side-line in his/her life, 2.9% is not so bad. For a pro for whom it is their bankroll is a significant part of their net worth, it is too high

One last minor point: Rather than say �You need 100 Max Bets of Bank� I would prefer to say �You Max Bet should not be more than 1% of your Bankroll�. This usage highlights a difference between recreational and pro players. A pro says �I have a bank. How do I invest it�. A recreational player says �I want to play a particular game: what kind of Bank do I need.�

To imply that 1/4 Kelly achieves most of the earnings power of full Kelly is not mathematically correct. It may be a true statement if other factors are considered e.g. you can't find a good game that will take your full Kelly bets. I play full Kelly up to 2% of my bankroll. Resizing is no big deal. I often do it during a playing session. Sure the downturns hurt, but the upswings are incredible.

Dr 21

to say that 1/4 kelly achieves most of the earnings power of full kelly is a bit simplistic...but factoring into the equation that many fulltime players bankrolls are relatively irreplacable, frequent resizing session to session is unrealistic for many, and that all profits are not re-invested into the bankroll (we have to pay bills from our EV) betting full kelly would be hard pressed to outperform almost any kelly fraction when real world elements are factored in.

are no longer playing blackjack...they worked their bankroll up and swore that full kelly was the way to go...then got plastered and got very discouraged over their losses and quit the game.

There is a middle ground between espousing a full Kelly bet and saying you need $25,000 to bet a $10 to $150 spread. Personally, I'd call my bankroll around $25,000 and I go $25 to $250. Yeah, I have a chance of going broke, but it's pretty unlikely.

The formula that describes the relative bankroll growth for Kelly fraction k is

K*(2-K).

K=1 represents Full Kelly, for which formula gives 100%
K=2 is double Kelly, for which the formula give 0.

For half-Kelly, k=0.5 and we get 75% of the power of full Kelly betting
For quarter-Kelly, k=0.24, and we get 44%.

I did not completely understand your comment of �I play full Kelly up to 2% of my bankroll.� Do you mean your Max Bet is 2%. That would seem to be an under-bet in some games, like Surrender Shoe games.

My "bankroll" is kept totally separate from my family budget. Like you, I am a full time college professor. I do not "need" by BJ income. However, I have used it to buy numerous luxories. As I started to invest in TIAA-CREF when I was 21, my bankroll is extremely small compared to my retirement account. At age 59 1/2, if my bankroll is less than $200K, I may treat some of my retirement account as my bankroll, but not until then.

I make Kelly bets up to a 2% edge. If my edge is greater than 2%, then I generally limit my bet to 2% of my "bankroll". Because of splits and doubles, I have won or lost 10% of my bankroll on a single hand several times.

I will go over my self imposed 2% limit if the count is sky high or a good oppurtunity arises. Several years ago, a player walked by and placed a $1000 chip in the betting circle. He received an 11 vs the dealer 5. As he was out of money and he saw that I was the only player with more than $1000 in front of me, he offered the double down to me. Even though $1000 was more than 2% of my bankroll at the time, I took the double down without a second thought as I knew it was well within "Kelly."

Dr 21

playing 2D game h17 das with 1-8 spread 1/2 Kelly is $15-$120 spread on two spots for $25K bankroll and 62 cards seen.

Looking at an earlier MathProf post to me, I see he says about 120 max bets are necessary for 1/2 Kelly. And that was WITHOUT going to two hands. Also, I believe he knew I was spreading 1 to 10 when I said that, and not 1 to 8. Obviously the bank for 1 to 8 would be less. You just quoted me a whopping 208 max bets for 1/2 Kelly. This does not sound consistent at all.

Also, looking at Schlesinger's charts, the game you talk about, 62 out of 104, DAS, H17 (by the way, I usually play S17) is covered in table 10.60. It says 800 units for a 4.1% risk of ruin. You are asking for 1,666 units, more than twice the number Schlesinger shows. And he was also not spreading to two hands. Again, there is an inconsistency here. I don't need twice a bankroll that already yields 4.1% risk of ruin, especially not when I am playing under MORE favorable conditions (spreading to two hands).

I don't have BJRM, but I can't believe your number. Mathprof, perhaps you could resolve his for me?