By Clarke Cant
Appendix C – Estimating Bankroll Requirements
The first treatment of the “ what is an optimal bankroll?” came from the famous J.L. Kelly paper on information rate, published by Bell Labs, in 1956. The conclusion was that the highest growth of your money, in a betting game with some edge involved, was to bet in proportion to that edge. In the most general form, his conclusions can be stated:
For the optimal growth bet, b=bankroll size in units, sd= standard deviation per hand, ev=expected value per hand, 1/b=ev/(sd^2), is the proper fraction of your bankroll to be each hand.
Myself, in the 1982 manuscript of the same title as this I never published, and later, in posts on bjmath.com, by Brett Harris and Winston Yamashita, the following formula was developed that is the equivalent to a bankroll requirement b, where the goal is to find a bankroll that will allow play long enough for the expected value to exceed one standard deviation unit of fluctuation. It is the same result that the Kelly formula gives, but it is derived as what bankroll is required for a FIXED betting system to survive long enough to pull one standard deviation unit measure of fluctuation. This formula is based on endpoint only.
For the optimal bankroll, result, b>=sd*sqr(h), h*ev>=sd*sqr(h). Here h is the number of hands.
The result is that the required minimum bankroll, that meets these criteria is: b in required bankroll units, b=(sd^2)/ev.
But this ignores the RWI type effect that raises the total negative impact of fluctuations. For a 50/50 game the RWI=2, and the results are general enough to confirm those effects. It is easiest to follow by starting with the number of hands involved, or the long run index, as Yamashita and Harris termed it. I used hands to double as my term originally and in the main text here.
Long run index, h*ev=sd*sqr(h); sd*sqr(h)=h*ev; sd=sqr(h)*ev; sd/ev=sqr(h); h=[sd/ev]^2.
If we start with the simple results formula, for a result of zero, and model how, for a 50/50 game, ev comes from winning hands, and sd works against us only coming from losing hands, the negative impact per losing hand is: -sd*sqr(RWI*h*.5), while the gain from each winning hand is: 2*.5*ev; here .5 adjusts for the probability of each outcome. To get ev from every hand, in a 50/50 game, the ev from every winning hand must be 2*ev. The RWI and the probability cancel (as the sqr(1)).
The simple result formula is then the same as the original and so is the estimate of an optimal bankroll, even with RWI effects included. Similar happens if we model a game where all ev is compacted well.
The optimal growth of your bankroll is the optimization of the endpoint of your bankroll anyway. So intuitively as well, nothing changes, with the effects of how a normal distribution retraces itself, with how you calculate an optimal bankroll.
But there is a rise in your element of ruin from this retracing/RWI effect, even if it does not change what the EKB is, or is not. That rise in ruin is covered in appendix A. It is still a fair question as to what effects the well known drop in your advantage, when you apply proportional betting, on your EKB estimates has. That is described in Wong’s paper, What Proportional Betting Does to Your Win Rate.
A win causes your betting size to rise by the rato: [1+(1/EKB)] , while a loss causes your betting size to decrease by [1-(1/EKB)]. For any number of n trials, with the simple assumption that we are dealing with a 50/50 game, your return has to be adjusted by the size of your bet changing every hand. The total change in your betting size becomes, continuing for h number of hands or trials—.[(1-(1/EKB))*(1+(1/EKB))]^(h/2).
Including this term in the presumed formula for how our bankroll should grow after n trials—which is:
— has to be done by a Taylor series approximation, to avoid making this a totally re-interative formula. The result, messy and hard to type, makes clear the: total bankroll instead grows as an exponent of h/2 and not h.
The first differential of the formula for bet unit size changes, delta/delta EKB, reduces down to (1+(1/EKB))^(h/2) when you drop the term (1-(1/EKB))^(h/2), as you presume that a high power of a number just under 1 tends to reach zero. The same conclusion is reached, about proportional betting cutting advantage, as you substitute ev, for (1/EKB), where, for a near 50/50 game, betting one unit each time, the ev is the inverse of EKB, and then you estimate the exponential growth of the bankroll as being the same as the exponential growth of the unit size (which is another way of saying, in a long-winded way, that you have proportional betting etc.). What is misleading, in the example of the betting size changes, is that the initial betting size change formula has a limit of 1.0
That same result can be modeled from the tendency of the mirror effect, or the RWI effect (same). For ruin and the probability of any bankroll level, the negative impact of unfavorable variance is increased by the RWI, so is the positive impact of favorable variance. The true measure of the Wong effect is how the measure ev/variance, or ev/ (sd^2) changes with proportional betting.
For proportional betting the betting unit size is proportional to the bankroll level. Coupling the betting unit size to the bankroll level, increases the variance per hand by the degree of coupling and the RWI. For full coupling, AKA proportional betting with adjustment every hand, the level of variance increase that predicts bankroll level, is the level of variance increase per hand.
But this is only in terms of the growth of the bankroll and not it turns out for the required optimal bankroll. The actual degree of coupling is low as well. Over a limited range of hands, or for a given goal ruin probability rises given no bet size adjustment, and hands to double grows, making a long run index 4 times as long, but not the optimal bank to bet ration.
The rounding that is done, between ideal unit size and the unit size that is practical for common chip denominations, to avoid having to use huge piles of chips, causes minimal impact (there is some from the rounding itself) in the ideal bankroll growth, but cuts the degree of coupling between the betting unit size and the bankroll size. The probability that the next decision will require us to change our betting size is in the range of 1 chance in 500, for typical blackjack spreads, conditions, and chip denominations. The result is that the Wong effect can be ignored, for bankroll optimization, as it has little impact on actual casino play.
One definition of the term Certainty Equivalent is to mislabel adjustments for the Wong effect with this label, abbreviated CE. Another, the accurate definition, is to label the long term fraction of our betting unit, compared to the fraction indicated by the EKB, that we would use. While the bankroll size that will grow our capital the quickest is EKB units, or using a betting unit of 1/EKB, betting 1/EKB fraction of our capital, as a betting unit size, is not the fraction that will minimize the boundaries of our fluctuations. In fact it is the fraction that will MAXIMIZE the boundaries of our fluctuations, when future fluctuations are compared to the initial bankroll size. For betting a fraction 2/EKB, our capital will never grow and the boundaries of our fluctuations will be a large orbit around the original betting unit size. Only a smaller unit size than 1/EKB, or over-betting, betting more than 2/EKB, will reduce the average future fluctuation to betting unit ratio. The over-betting option is not recommended in that our capital, as is well proven, in numerous references, will shrink toward zero, even with an “advantage.”
For betting with a unit size of 1/EKB, AKA with a CE of 1, future results will be a continuous probability spread where the probability of having our bankroll be 10% of the highest prior level will be that same 10%. We will be at 1% of our peak 1% of the time.
If you have read the main text you realize that I advocate combining earnings from couponomy in an aggressive fashion, with regular play, to subsidize our regular playing. I also advocate that you top off your betting level at some point. Growth has a safety factor all of its own. Attempting to keep raising our bankroll unit size without any top-off, is what would make our bankroll continue to be subject to this CE relationship. It is for this reason, that I don’t think that the CE concept fully properly applies to the period of our playing history where we are most trying to “grow our bankroll.” Much of what a CE adjustment would gain for us is already reachable, without delaying initial growth, by having that top-off goal.
The bankroll formula that I gave is that which is used to find the long-run index survival bankroll, but is based on an offset of 1.2 sd units instead of the normal 1 sd unit offset. The tradeoff, in CE based reductions in fluctuations, and maximum growth, is a good compromise between growth overall, and a complex of considerations, such as initial increases in ruin probability, at the first instance of raising your bet size, and the availability of couponomy earnings, and the likely amount that a person, who has “kept their day job,” would be willing to allocate toward bankroll replentishment. I once thought it was the proper offset to approach other criteria for optimal growth, but I was wrong.
What is correct however is that the adjustment produces a CE (here a CE=.6944) spectrum of results that is closer to the more intuitive pattern you might expect from an optimal betting system, where the probability is not 10% of being 90% down from your peak bankroll level, as it is with CE=1, but more the type or pattern we might expect, by estimating, as once thought, that fixed betting use of an EKB has a ruin probability of ~13.5% Fixed betting of an EKB sized bankroll, has a higher element of ruin, than this, much higher.. An optimal growth bankroll was once thought to be one that when a fixed betting scheme was applied had a ruin probability of 13.5% Ruin tolerance varies, and ruin probability varies with an EKB in various games . That is an individual decision. Regardless of your risk preferences you should never:
(1) Round up your betting size more than 1/(EKB*.6)
(2) Attempt to use trip ruin plan to divide your bankroll into short “shots” at a high element of ruin, without any plan for bankroll replentishment. The couponomy subsidy plan in the main text is a replentishment plan.
(3) Use any trip ruin plan to bet more than your current bankroll, in terms of original bankroll + current winnings. Once any initial bankroll grows past .6*EKB, that should be what you consider your total bankroll.
(4) Fail to recalculate your betting size with any significant change in your playing bankroll—results always change things.
(5) Take money from your bankroll until you have met your top-off goals. You can modify your top-off goal downwards, if you find too-much heat for your act and playing style as you move up, and enjoy your winnings that way, but you never really have winning available for any other purpose until you reach some form of top-off point. It might be a good example to temporarily top-off at a $25 unit size, set aside some money for expenses and a few nice things for yourself, while leaving enough for an EKB based on $25 units, but don’t withdraw ANY money while attempting to keep moving your bankroll and betting level upward.
The example I gave of not buying that DeLorean should be enough to show how CE effects hit me in the early 80s, and then an outside problem cut my playing such that I was not able to reach that level of bankroll, from playing, for sometime. I rebuilt my current stake by starting at zero, unable to work or get disability, as mentioned in the main text, but taking free rolls of nickels and other coupons, using the formulas in the main text to add riskier coupons as my stake grew, and began adding regular 21 play, in other words without coupons, at .6*EKB, for the games in the area I was at. Today I would have known that I could also treat money above $150, which involves a very low element of ruin for couponomy plays, as a small trip bankroll. Luckily then there were some very good $1 single deck games available. I did top-off at $25 units for about 2 months. I bought a moped. I bought a VW van when I topped off at $100 units.
Last Update: 04/19/05