By Clarke Cant
Appendix A – Ruin and Distribution
List the values of Pascal’s triangle by centering each new row under the previous row. Each entry is the total of the just above entries, one space left and one space right. You add a new 1 entry, on each side, for every new row. Find a result in the last row you wrote down. Look up at the column above. Pretty soon, about 20 rows, the total above each entry is just about the last result entry, in the last row. If a result has more than 20 entries, in the column above it, this is nearly exactly so.
Hogben’s, Mathematics for the Million, has several series summation exercises that prove this. If there is some edge involved, new entries do tend to pull this distribution to one side, but for edges of under 2%, it takes more than 50 rows to pull the arrangement over by one unit. By this time neighboring entries are also tending to equalize. The trend of past results equaling end results is much stronger.
The bell curve is the approximation of this distribution if it were carried onwards to an infinite number of rows. It is handy, but you lose the information from prior columns, as there is simply no meaning in the idea of going back from infinity any finite number.
The tendency of results to repeat is not meaningless, and does preserve the consequences of this column effect. Weingar took this repetition tendency to follow this formula; RWI=random walk index, sd=standard deviation per hand; RWI=[1+(1/(sd^2))]^[sd^2]. But it is more complicated than just that formula. Suppose while flipping coins, the game Pascal’s triangle is used to explain, we took a standard die and tossed it first, and bet the number of the face of the die units, each coin toss? Our standard deviation would go up, but the tendency for repetition of results would not.
Weingar explained this tendency as having another limit which was the greater of the inverse of the probabilities of each result, at each decision. In complex games a weighted average is used for final calculation of this lower limit. Blackjack, excluding ties, has a win probability of about 48%, and a loss probability of about 52%, that does not change much with different deck compositons (ie different counts) and is not changed much by considering double downs and pair splits. This tighter limit can be estimated as somewhere above 2.084. The RWI would then be for blackjack somewhere just above 2.084
Eventually this analysis branched off and became known as the study of the Weiner measure in statistics. It would be totally obscure today, except that the RWI concept won the Nobel prize, in physics, early in the 20th century, in application to the Brownian movement and the general boundaries and reversion to prior conditions of any random walk—some guy with bad hair named Einstein.
Packel, in his, The Mathematics of Games and Gambling, was the first to apply this concept to blackjack results, in a hint, in one exercise, concerning the boundaries of prior blackjack results.
So rather than involving any special insight by Don Schlesinger, noted colleagues, and his fans posting on various websites, the total probability of prior results has a long and distinguished history behind it, as does the number I call the RWI. It is old and basic stuff, and does not involve different ruin criteria or special considerations of tendencies of early hitting a barrier pulling against tendencies to end at some endpoint, to paraphrase Schlesinger.
The most basic equation for analyzing results is, H=number of hands, ev=expected value per hand, sqr()=square root of (what’s enclosed), Z= number of standard deviation units and the number to lookup in a chart of the areas of the tail of a normal curve; result=H*ev-Z*sd*sqr(H). There are other conventions, but since we are usually only concerned with fluctuations that work against us this equation will be used.
The tendency for a 50/50 game, is for every outcome to have been arrived at an equal number of times previously. The general tendency is for every result to have been arrived at RWI total times as often as the endpoint result that the bell curve is able to estimate, at the final number of hands, or other goal, examined.
The total path, in reaching that result, according to Einstein’s paper, and its mention of otherwise obscure Weingar, is RWI times as long as the start to finish straight line, with its continuously distributed expected value and continuously distributed fluctuations, that standard normal analysis models. Einstein used this to establish his intermediate conclusion that the tendency, of a particle in a random walk to return to a previous position, was equal to the tendency of that random walk to exceed the straight-line path the bell curve models. Then he preceded to his final conclusion, that a particle has a tendency to arrive at its origin that is equal to the tendency of the boundaries of its path to expand. Einstein’s paper was the missing key to several parts of the Kinetic Theory of Gasses, and totally depended on the model used above, for the relationship between the mirror tendencies, in a normal distribution, and the total, endpoint and prior, deflection from a mean path in statistics.
The total deviation from that straight line path, that the bell curve models, is the same as a straight line, standard normal path, RWI times as long as the H number of hands (trials). The results formula is modified to become:
results = H*ev-Z*sd*sqr(H*RWI), if we are looking at bankroll to survive a given Z fluctuation in a given number of hands, where the probability, given by the area of the tail in a bell curve chart etc. is found from that Z, then we are looking for results of, b=bankroll, results=-b; substitute -b for results and solve. All other formulas commonly used in standard normal analysis have similar modifications, when derived with the RWI lengthening the path that adverse fluctuations occur along.
For an infinite goal:
EKB=equivalent to Kelly bankroll, alpha =b/EKB; ruin=RWI*e^(-2*alpha), which is the Samuelson/Sharpe infinite formula modified.
If you specify a bankroll, ruin=RWI*e^[b*ev/(sd^2)], which is the Sileo infinite goal formula modified.
For the limited goal, of winning a given amount, tw=target win:
ruin=[e^(2*ev*tw/(RWI*(sd^2)))-1]/[e^(2*ev*(tw+b)/(RWI*(sd^2)))-1] which is Sileo’s goal ruin formula modified.
My own trip ruin, for ruin before a target number of hands is derived similarly to the modified results formula, and results in the modification of Chris Cummings excellent endpoint formula:
f(x)=cumulative normal distribution of x; x=-[(H*ev)+b]/[sd*sqr(H*RWI)]
That f(x) can be taken to mean that: this x, when used as your z statistic, with a bell curve table, finds the probability of ruin as the area of the tail listed in that table. If you wish to solve working in the other direction, you find the probability of survival that you wish to have, lookup the z statistic, and use that value for x, in the x= formula, and solve.
The estimate of the proper RWI is not easy for blackjack. For hands survival of between 500 and 5000 hands, Schlesinger’s trip ruin formula, approximates the results of a RWI between 2.05 and 2.1, in the ratio of the probability for the first term, as f(x), explained above, and the total probability that results. Its derivation is highly flawed however. The RWI is demonstrable to not change appreciably, for numbers of hands from a few hundred to infinity. The ratio of total probability to endpoint measured probability is also the RWI. The Schlesinger formula certainly changes in that ratio, for different numbers of hands.
A bridge between the two limits Weingar used is to divide the overall standard deviation by the average bet size, and then use his overall standard deviation formula. Then some useful numbers can be found, for the RWI, this way, simply, as it is easy to prove that overall standard deviation is easier to find than this complex weighted average. Usually this will suffice, as this method converts the overall standard deviation to a one unit equivalent. For 50/50 games the answer given by both methods is the same.
The Schlesinger trip ruin also treats endpoint measures of ruin and ruin in prior to endpoint number of hands as separate ruin criteria. In fact the second term is simply modified for a larger number of hands, and is presumed to vanish at infinity. This is an improper combination of ruin criteria, even assuming it is permissible to treat barrier and endpoint ruin as separate criteria. Such combinations have to have a large degree of independence to be validly combined. The more limited goal of the first term precludes the largest part of the second term, in probability of ruin. The combination only achieves its high accuracy by the total approximating the effects of the RWI. This is one example however where the “wrong formula” still gives useful results. There is nothing wrong with such formulas, the Blackjack Formula is like this as well, but further development from such can be erroneous. It is appears that the error, in combining a longer run term with a hands limit term, cancels the error from the overall formula not preserving the ratio of total ruin probability to the endpoint estimate. This should not be taken as any proof however, that there is any different form of hitting ruin, at an endpoint, and by dropping down to having zero funds.
In actual use, between the two trip ruin formulas you chose between an accurate formula that is simple but has definite limits for hands survival, and one that is slightly more complex, also accurate, but only has fuzzy limits.
The two trip ruins share the fact that where the hands to survive is >b/ev, the growth of the bankroll, as expected value accumulates, is added to the starting bankroll, and the results become more of an approximation of infinite goal ruin than any limit. The Schlesinger formula results are still reasonable past this point, in that he assumed that limited goal ruin would become the conventional endpoint type of ruin as the number of hands grows to infinity, and approached hands survival ruin as the total of that infinite goal and a conventional limited endpoint. Actually the infinite goal is still subject to RWI effects. That formula has ruin approaching one half of actual ruin. The limit is fuzzy in that reasonable figures are still produced as the results only drift down slowly. The Cant formula drops off dramatically. For both the most reasonable limit appears to be H<[b/(2*ev)]. Any more hands is really a fairly long run for your trip bankroll, and what you may actually be doing is over-betting by rationalizing over-betting and taking many high risk short term shots with your overall bankroll. Trip ruin should only be taken as a recommendation as to what amount of money to have instantly available.
Finally the proper combination formula is, r1=ruin criteria 1 probability, r2=ruin criteria probability 2, rn=ruin criteria probability n, rt= combined ruin probability; rt=(r1+r2…+rn)*(1-r1)*(1-r2)….*(1-rn). It is not as difficult as claimed to combine ruin criteria. It is difficult to type-in directly, but a combination of hands survival and win goal ruin will easily plug-in to this combination formula.
The most accurate ruin formulas of all appear to be those proposed by Evgeny Sorokin. His approach is to use a recursive formula to find the ruin probability for a one unit bankroll, and to base estimates of actual ruin, for a more reasonable bankroll, as a power of the number of units in that final bankroll. The RWI adjustment of infinite goal ruin near exactly matches Sorokin type estimates. Within the limits given above, the same is true for RWI trip and win goal ruin estimates, and for the Schlesinger trip ruin formula.
APPENDIX B – Showing the maximum profit and BJF for continuous shuffle machine games
APPENDIX C – Estimating Bankroll Requirements
Last Update: 04/19/05