By Clarke Cant
Chapter 4, the truly optimal bankroll, to inifinity and beyond. (or my 1982 views on bankroll)
I couldn’t resist saying that.
Actually advantage is not a very good way to rate games. Advantage implies some simple game like the biased coin flip that analysis of optimal betting was first based on. When I first started playing, back in 1977, the only way to approach bankroll estimates was to follow a proportional betting scheme and hope for the best. The Blackjack Formula is just one approach to estimate your gains per hand too. But you should know my feelings from the first part of chapter 3 and my poor forestry service friend. There is even a quirk to the BJF that makes for higher accuracy in estimating gains than in estimating advantage.
The BJF based formula for G, or gains per hand is:
G=AB*BJF/100 (the 100 is there to convert from a percentage to a decimal figure)
AB is the average bet per hand; pr is the probability of making your average high bet. The probability of your high bet is based upon your betting optimization (the best work on this is by Brett Harris as given in his count booklets and his numerous archived postings on bjmath.com), mainly at what true count you raise your bets at. It also has to involve the probability of that true count, which itself is derived from the number of decks in the pack and the depth of the shuffle point, in conventional games, and with continuous shufflers it is either dependant on the usual delay in the reapearence of cards or the relative depth you choose to use (be patient). Sources for such information would have to start with Snyder’s book series on Beating the _decks, where a separate book was written for 1,2,4,6 and 8 deck games, or a simulator that can give you a probability of a given true count, given the number or decks in the pack and conventional shuffle point.
If you are jumping your bets between a high bet and low bet, your average bet is:
If you are using a proportional betting schedule that calls for different bets at each true count you expand out and:
A similar set of formulas is derived from Wong’s Proffesional Blackjack for calculating standard deviation per hand. Once G and sd are found for one hand approximations can be used from the same book, or similar mentions in Theory of Blackjack by Peter Griffin, to adjust for math measures such as variance and covariance between hands, if you typically play more than 2 hands. Blackjack Attack by Don Schlesinger is useful here too, but the famous Chapter 10 will comein for harse words over its overly timid safety dancing recommendations for your bankroll.
The simple formula for sd is…
In the formula for proportional betting schedules the H and pr terms expand as:
H1^2*pr1+H2^2*pr2…..,+H^2*prn; we take the sum, multiply by the FBV and take the SQR.
Fortunately FBV, or flat bet variance is mostly tied to the probability of your double downs and for virtually any game is:
1.2 for d10 games where you can double on 10 and 11 only,
1.28 for DOA games, and
1.32 for DA$ games,
with very little if an variation for other sets of rules, the only exception to which is in the rare DA$d10 games (I knew I had that entry for a reason) go ahead and use FBV=1.32
I first developed my own estimates of required bankroll back in 1982, in the first version of this book that Arnold Snyder talked me out of publishing. I was very impressed with the Blackjack Forum Articles on the Gwynn and Seri simulations that were written up asking, How True is Your True Count? I quickly discovered a version of what is now known as the Yamashita equation or the number of hands needed to break even against a known level of standard deviation units of downward fluctuations. I came to this by trying every possible way to do away with advantage as a measure of game earnings and only using standard deviation and expected value, here labeled G. VERY important was that I strictly followed all of the rules for such ingredients in that I strictly treated every outcome as a combination of expected value and deviation, and that the number of trials involved had to be closed and set to a definite number. By closed I mean that each trial had to be analysed from the view that you had to have no knowledge of the paths of outcomes in each trial, just the results.
First of all the bankroll had to be: B >=X*sd*SQR(T) where X is the number of standard deviation units you were interested in and T is the number of trials. The only real revision, from 1982, of this is that today, with knowledge of barrier ruin instead of bankroll you might change the term to total funds potentially used in surviving T trials. As will be seen later however barrier ruin is a special and not general case. This relates to another rule in such analysis in that statistics is not a prediction of outcomes, but is a prediction of “future histories” or analysis of looking backward at the way you arrived at different results.
Every level of fluctuation had a breakeven point where:
The simple reserve calculations for how much reserves would have been used if I survived a number of hands to a given degree of probability is related to the number of standard deviation units of survival I wanted to examine. Classic Calculations of ruin that followed the recommendations of J. L. Kelly indicated that optimized betting would involve a long term ruin of 12% probability which is approximated by fluctuations of 1.2 standard deviation units (acctually 1 sd unit would be more correct except I am choosing to still use 1.2 Recently there have been some studies on chaotic approximations to the bell curve that call for curve broadening. We will briefly mention this in chapter 10 after a review in chapter 4A).
The maximum reserves used could be found by solving the combination of equations for the number of trials involved:
(1.2*sd/G)^2=T and B=T*G or B=(1.2*sd)^2/G;
Personally I decided to use the label HD or hands to double instead of T.
It would approximate the characteristics of long term optimized betting, even though here I was studying a double or nothing model based on a fixed betting strategy in that it could be demonstrated that long term, after this survival, 88% of all paths could be shown to result in infinite growth and 12% in ruin.
Put in simple terms anytime you breakeven, after playing a given number of hands there is a standard deviation measure for the fluctuations you have survived and that sd measure will allow you to extract a measure of the maximum use within those bounds of your reserves. The more you play, the less likely it is that you will fall to any breakeven point. Your playing safety tends to increase as the SQR of the number of hands that you play.
A simple thought experiment can easily show how boundary ruin vanishes, long term and how the above expression long term is a practical optimization of the number of units required for your playing bankroll.
Everytime you at least breakeven, you have both decreasing probabilities of being behind and an increase in the maximum reserve usage, over the number of hands you have played (and keep in mind that strange future history statement above). If you consider the entire history of your play and bankroll, the maximum reserve usage grows without any new moneys being put into your bankroll, unless you meet ruin.
Clearly the more you play, the more your bankroll safety increases not from the original money put in play, but from the total expected value being more and more being able to offset your fluctuations.
This is a simple fact that people who commonly advocate drastic underbetting fail to realize. Bankroll safety arises not from initial safety of your betting but from having the optimal combination of expected value PULLING your fluctuations.
My worst nemesis on this point has had to have been Don Schlesinger. Speaking against his Blackjack Attack is difficult due to his reputation, but usually when I have posted on the internet on these topics he has posted to claim, Clarke Cant is selling snake oil and directed everyone to ignore Cant and read Chapter 10 of his Blackjack Attack, Playing the Pro’s Way.
Well the Charts in Chapter 10 are indeed acurate and correct, in inclusion of this barrier element of ruin factor, which, as I say above, does tend to approximately double the element of ruin when you follow a fixed betting strategy. It happens because the closed box I mentioned above doesn’t let us know about the initial distribution of our losses until the hands have been actually played. Concerns about barrier ruin are overstated to begin, in his estimates, (which he later had quoted in several papers on bjmathcom) in that the box is opened, and the ways in which ruin is increased by early fluctuations is then recalculated and added to the initial calculations of long term ruin. So at best his famous charts are a bit exagerated.
What makes them totally useless for estimating your bankroll requirements for the best long term growth of your bankroll however is that barrier ruin is not a permanent problem. Several times in chapters 9 and 10 Schlesinger admits that barrier ruin exists only where you cannot or will not lower your betting unit size.
The fact is that barrier ruin vanishes once you have enough head room in your bankroll to lower your betting sizes by a factor of 3 to 1 and still meet minimum appropriate table game requirements. The fact is that any one of the advanced playing techniques in this book (so far only optimized couponomy has been given and it itself is enough to aleaviate the risk of barrier ruin for most $5 chip players). To keep on adjusting bankroll requirements, once barrier ruin is gone, as everyone (unless possibly personally tutored by Don) would be lead to by being directed just to Blackjack Attack, is about like insisting your child keep on using training wheels on his bicycle—forever?
I first mentioned that I was using HD or hands to double, to rate games and conditions, instead of advantage in a published letter to Blackjack Forum, that I believe “hit the stands,” in 1983. I incorporated this math in my 1985 Atari program package. Copies of the stack of letters from Snyder had to be sent along to the Library of Congress because of the derivative nature of that package. This work also derives its expected value estimates from Snyder’s work and would be covered in the same permissions for derivative use that were then given. Nothing more needs to be said.
The earlier tables for couponomy were calculated with the following formulas for expected value and standard deviation:
For roulette even money bets wins were calculated as 18/38 probable, and losses as 20/38 probable, with an sd of 1. For craps I used wins .493 probable and losses .507 probable, with an sd of 1. For Blackjack I used a different approach from Theory of Blackjack, which has a section on couponomy, and an sd of 1.1 Some rounding was used for coupons where the HD was less than 30 hands/trials, to match more accurate T-test values for standard deviation. For initial playing barrier ruin is negligable with the suggestions to continue playing lesser coupons for at least a time as your mini bankroll grows. The subsidy factor involved of providing $140 per week to a couponomist is based on using the major coupons listed at the date of writing in Charles W. Lund’s mini board on lasvegasadvisor.com Only the coupons with easy availability to myself were included, and considered to be played within the restrictions printed on them. After approximately 12 days even the homeless person in chapter 1 would be able to approach this level of earnings (given enough bathing etc. too). After 1 week playing at the level of mini bankroll needed to play the least favorable 5-7 roulette coupons, the couponomist would have an element of ruin that is less than .003%. Applying the level of earnings above $140, each week to commonly available $5 single deck games, in the Las Vegas area, is more than enough, even limiting play to less than 4 hours per day, to have that same player betting black chips optimally in any of several single deck games within two months, even if couponomy is ceased after 3 weeks. More about this will be added in the Chapter on how to safely move on up.
Generally games of less than 10,000 hands to double will be the investment games that will build your bankroll. Lesser games are best saved in that they often will be the games, that are tough to grow your bankroll, but may have high limits that can give you a high, though fluctuating income, from playing, and profitable recreation as you move to larger investments.