By Clarke Cant
Editor note: Yes, this section is actually after Chapter 6
Chapter 4A, the rest of Chapter 4.
Lets return to this equation (Xsd/G)^2=HD.
The results are that for a given number of standard deviation units we have a number of hands where we pull ahead and stay ahead, at that level of fluctuation. By staying ahead we reach infinite growth (in the ultimate long run, pit boss tolerance willing). All of the Classic Optimal bankroll examples of even money payoff games show an 88% chance of such success, and 12% chance of failure, when a fixed betting strategy is used. By approaching the same results with fixed betting I will show the same outcomes with a blackjack bankroll, as a test of that bankroll being equally optimized for growth in the same long term.
The level of breakeven success that corresponds to 12% ruin is 1.2 sd units. By itself then (1.2sd/G)^2 only shows the number of hands where such breakthrough is achieved in the long term future history of our bankroll, or our playing.
We can explain both barrier ruin and optimal bankroll requirements for this long term growth by examining all the outcome pathways that would/do take an infinite number of hands to either reach double or nothing outcomes, by looking at those paths every HD hands. With the smallest amount of upward drift from that set of paths, which we are looking at as a flat string of HD number of hand sections, we reach, “infinity and beyond,” in our results. With the smallest amount of downward drift we wipeout. That pathway is the dividing line between ALL the infinite possible paths that wipeout and ALL the infinite possible paths that achieve success (the initial HD length of hands is finite; infinity divided by any finite number is still infinity).
Bankroll requirements are thus (1.2sd)^2/G for all the infinity of outcomes that survive the first HD number of hands without ruin. Thus I consider the optimal number of bankroll units we should divide our bankroll into to achieve optimal growth over infinity to be given by this equation, in that the outcomes of this type of fixed betting match the outcomes of fixed betting in more classic examples.
Within every HD hands win/loss order can be distributed in every way . Initial, or early losses only have the initial bankroll amount to offset your bankroll level from going toward ruin, while later fluctuations are offset both by your initial bankroll amount and the accumulated expected value of your hands.
Before HD hands are finished that total offset to possible fluctuations is less than it is for the optimal infinite growth of your bankroll. The average chance of ruin is thus about double what the long term chance of ruin is, in the first HD number of hands. The barrier ruin effect rapidly decreases thereafter.
In chapter 6 we covered techniques that either add camoflage without costing expected value, or that add tremendous expected value, but would make hash of any precisely optimized betting spread. It is far more important to optimize the application of a less than optimal betting scheme than it is to depend on a scheme that is compromised by such tactics.
Once a well known poster on bj21.com and bjmath.com, claimed that I was wrong to claim that you could have an optimal bankroll requirement calculated with a less than optimal spread, and the example he gave was a schedule that was much like a typical one an opposition bettor would use (as covered in Snyder’s, Blackbelt in Blackjack). The point I tried to make then and make here is yes. Yes the ability of such a player to determine what his optimal unit size should be should not depend on whether or not he uses a precisely optimal spread.
The best methods for finding the optimal spreads are those written by Brett Harris and archived on bjmath.com I contend that my formula from 1982 –which has other labels now – gives bankroll requirements for any positive expectation spread. It does not give an optimal spread however.
But just like Brett Harris I consider hands to offset to be a better way to rate games than advantage, DI or SCORE. HD, other than being based on offsetting 1.2 sd units, rather than just 1, is the same as his H0. Since HD here also gives the hands to double a bankroll it would take on average I think it has intuitive merits as well.