Category: Gambling and Casinos

Thorp’s Era.
If you don’t know, now we are living in 43th year of Thorp’s era. This person had such a great influence on the blackjack and all the gambling industry, that I seriously consider that he is worth of setting up a monument while alive, by the way at the casino’s expense.

Being a young scientist, Edward O. Thorp regularly read mathematic journals. When he was a student, he was pressed for money. The idea of outplaying the casino on the basis of calculations appealed to Thorp and he decided to check the results of the “excellent four” and see what will come out of it. The result of his activity was a phenomenal book Beat The Dealer, now its sales are approaching million of samples.

Thorp chose the other way – he did not make analytical calculations, but with powerful at that time computer IBM (they were also called mainframe) he wrote several programs on Fortran, creating quite original methods for 60s. By the way, Thorp together with his research instructor Claude Elwood Shannon, a great scientist, were also involved in solving the problem “how to outplay the roulette?”

From his calculations Thorp understood that dead cards had a considerable affect on the gambler’s chances this or that way. His main idea was about memorizing the dead cards in somewhat simplified way and when the situation is beneficial for the player, make high stakes. By the way, this system still remains the basis of any methods and counting systems of blackjack.

The counting system introduced by Thorp, was rather complicated for usage in real casinos, required great concentration and large amount of mentally arithmetic actions. However with good training there was nothing impossible in its application.

Thorp’s book immediately became a success and bestseller. Everyone understood that with quite simple actions you can get an advantage. Every reader dreamt of enormous prizes. However, the casinos also knew the score.

Panic seized them. And in 1962 after Thorp’s book, all Las Vegas Casinos without exception changed their rules being afraid of mass influx of “system players”. The quality of the rules worsened terribly and no counting system could ever help you to win. Though the effect for casino turned out to be quite unpredictable and reverse – people stopped playing blackjack. And during several months, all the casinos had to return to the former rules for their own survival.

Another interesting effect caused by Thorp’s book- incredible popularity of blackjack outrunning the favorite American craps. A lot of people, after reading the book considered themselves potential winners and rushed to the blackjack tables. However, most of them had a bad understanding of the mathematical principles of the game did not become winners but quite vice versa. Therefore, casino got more clients and moreover losing clients. Since then their number is only increasing.

The system presented in the first edition of Thorp’s book was quite difficult even for professionals and only the few could apply it under the conditions of a real game. Something should be simplified.

The power behind the throne.
Julian Braun is quite a unique personality. He had hardly ever played a deal in real blackjack. But he played millions and maybe even billions of deals on the computer. Braun was a good mathematician and programmer and he got interested in Thorp’s idea and offered him cooperation in the sphere of calculations and programming.

Braun became the person who first invented the counting system Hi-Lo. He was behind the development of all modern systems the authorship of which belongs to Revere, Humble, Wong and Uston. He wrote the only book How to Play Winning Blackjack, but what a book!

Braun upgraded Thorp’s system on FORTRAN and made considerable changes and the second edition of Thorp’s book also contained Hi-Lo system in its modern form. It was quite a revolution in gambling world. Braun worked in IBM corporation and had access to probably most powerful computers of that time. This fact helped to build simple and efficient tool in struggle with casino.

Using Braun’s calculations the gambler mostly known as Lawrence Revere developed his own counting system and presented his results in the form of convenient tables which are applied in most counters of the world. Lance Humble based his HiOpt systems on Braun’s experience.

Regardless the fact that Julian most probably had never been to the casino, he became a power behind the throne of the blackjack and all the mathematical modeling of the game fell on him.

The dream of capoting a casino is as old as the notion “casino” itself. Every gambler wants to win. Every other invents his own “system”. Every hundredth attempts to carefully analyze the game. And only the few of millions succeed.

The most famous achievement in the sphere of “system” game against the casino was the so called “card counting” at blackjack – mathematically based methods of game allowing to get some advantage over the casino. The casinos, surely, knows about the existence of “counters” and are trying to oppose them – from changing the rules to banning the game. Sometimes even exceeding the limits of reason and legality. In fact, all the history of the blackjack is the struggle between the gamblers and the casino which is more clever.

More than half a century passed since the first attempts of beating the casinos in blackjack, based not on the marked cards and intuition. There has been a peck of salt eaten since then, but the struggle is still going on…

Mammoth.
Probably the first person in history who applied mathematic analysis to the game blackjack was Jess Marcum (originally Marcovitch) was born on 30th December 1919.

It happened at the turn of 1949 and 1950, when Jess, being an excellent mathematician and physicist-theorist got to Las Vegas. Then his mathematic flair prompted him that most possibly not everything depends on intuition in blackjack.

Marcum started analyzing blackjack. First, the uniqueness of his attempt is that he performed absolutely all calculations with his pen on the paper, on principle without using any technologies whatever weak they might be at that time. Second he had found the solution!

Jess manually developed what is now called basic strategy and counting systems ten years before someone else has done that. He counted that theoretically he had an advantage over the casino of about 3% – under the conditions of that time it was quite real. Moreover, Markum had been playing blackjack in all known to him casinos- both in the USA and abroad. His name appeared on the pages of the newspapers as an example of “lucky man” who managed to hit the jackpot.

Jess Markum also went down to history as probably the first man thrown out from the casino because of cards. Though at that time casinos did not understand that such system existed. Jess lived in Las Vegas for about a year. The casino owners exchanged the information when they gathered at the general meeting. And they were terrified. After that Markum was not admitted at all Vegas casinos. He went to Reno. The same story- in half a year after unbelievable “luck” the casino owners started making inquiries about the incredible gambler. Then other states and cities. Then Cuba, the Bahamas. Nobody knows how much money he has won during that time. One thing is known for sure- in fact, Jess Markum started the war between the gamblers and casinos in blackjack, that is still going on and getting more and more strained.

He did not share his calculations with anyone, and completely gave up playing against casino after a wide public publication of blackjack methods by other authors.
Jess Markum died in 1992, at the age of 72.

The four of Neanderthals.
The next effort of great influence on the blackjack math was made in 1956 by the group of four mathematicians- Roger Baldwin, Wilbert E. Cantey, Herbert Maisel and James McDermott.

These people had never played in casino before, having spent a great number of m/hr, they created basic strategy of playing blackjack according to the most popular at that time rules and published it together with the calculation methods in the specialized math journal for statisticians – Journal of the American Statistical Association entitled “The Optimum Strategy In Blackjack”.

They made several slight mistakes in calculations which is nothing serious taking into account their enormous contribution. A year later they issued a small book Playing Blackjack to Win, which is now a bibliography rarity.

In fact their book and the article did not spark much furor, and remained almost without attention regardless of their innovative approach. However, the word almost turned out to be the key for all the gambling industry, as someone did paid close attention to the research of the excellent four.

Thorp managed to find out that owners of gambling houses gave their officials rather strict directions with regard to the strategies which they should stick to in the game with visitors. Control over fulfillment of these directions had its initial aim to prevent from a frame-up of a croupier with the rest of the gamblers, a chance of which could not be excluded. Assigned for a croupier strict rules determining his game strategy really substantially reduced a probability of such a frame-up, but on the other hand, allowed an “advanced” gambler to rather adequately reveal the essence of this strategy and effectively oppose it. For unlike a croupier a gambler needn’t show the first of the received cards, as well as isn’t enchained by any strict rules as regards his strategy, that is why flexibly changing his behavior he can confuse a croupier. For example, Thorp found out that practically in all gambling houses of Nevada State croupiers were strictly ordered to keep away from a widow in case the amount of points in his cards exceeded or was equal to 17, and a player, from our mathematician’s point of view did not have to miss an opportunity to make use of the knowledge of even some aspects of a croupier’s strategy for achievement of his aims. Thus, those advantages which had an official of a gambling house from the start (as we already know, he is not obliged to open his cards at the end of the game), can be compensated to a certain degree for the knowledge of a player about the strategic “tunnel vision” of a croupier.

Besides, as has been mentioned, Thorp, while building his strategy presumed that cards were not often shuffled, in particular, if after finishing of a regular game there were still cards left in a pack, a croupier did not collect the thrown-away by the gamblers cards but dealt them anew (and the next game was played), and only after complete exhaustion of a pack, an official of a gambling house collected all the cards, thoroughly shuffled them and a new “cycle” began. Naturally, if a gambler had a good memory he could change his strategy depending on the knowledge of the cards which had gone out of the game, and what cards could still be counted upon. It is important to remember that a croupier himself who was to strictly follow the directions of the casino’s owners practically without changing his strategy!

Thorp set himself a task to formulate the rules which would allow him to calculate probabilities of taking out one or another card out of an incomplete pack. Knowing these probabilities a gambler could already with reasonable assurance draw cards from the widow without being too much afraid of “a pip out”, and besides, on the basis of the knowledge of some aspects of a croupier’s strategy to make suppositions about those cards which he had, and other gamblers as well. Naturally, as a gambler was to make a decision with regard to a widow very quickly, the sought rules for calculation of probabilities were to be rather simple for a gambler to be able to use them “in mind” with the help of neither a calculator, nor a pen and paper (even if we suppose that a gambler will be given a chance to do calculation on paper, it will certainly arise suspicion). Edward Thorp managed to solve this mathematical problem having created rather simple algorithms for calculation of probabilities of taking out of one or another card from a pack, and using them to build a strategy of the game of twenty-one which would not be very complicated, allowing a gambler to considerably increase his chances of winning!

As the Hungarian mathematician A.Reni states after a few days of presenting his report on the obtained results at the meeting of the American Maths Society in 1960 in Washington “Thorp received from a businessman a letter with a check for 1 thousand dollars intended for checking of a winning strategy in practice. Thorp accepted the check and having learnt the formulated by him rules left for Nevada to try his discovery. The trial went well: less than after two hours Thorp won 17 thousand dollars.

Needless to say, the owner of a gambling house didn’t share Thorp and his companion’s delight with regard to a successful comeout of the trial and the next day did his best to prevent Thorp from joining in the game. Later on Thorp tried to penetrate into other gambling houses, but the news of him had already spread far and wide, so that the doors of all the gambling houses appeared to be closed for him. Several times having adjusted a fake beard or having got a make up of a Chinese, Thorp managed to get to the gaming-table, but in any disguise his constant gain invariably gave him away. Thorp had to refuse from further checking of the strategy developed by him”. Though “additional checks” were “necessary” only to enrich the pockets of the talented mathematician. One could hardly doubt that E.Thorp managed to create a real winning strategy!

However, since he could no longer benefit from his discovery himself, he decided to render “welfare assistance” to his colleagues having published in 1961 a small article in an American academic journal (Thorp E.O. “A favourable strategy for twenty-one”, Proc.Nat.Acad.Sci., 47, 110-112, (1961)). And despite the small size of the article and, consequently, an extremely condensed form of persentment, made it comprehensible for rather a narrow group of professionals, one can be sure that a number of American scientists and their friends certainly “improved” their material situation (owners of gambling houses were unlikely to read scientific magazines at that time).

After one more year Thorp published a book (I mentioned it at the beginning of the article) in which he rather in details, in the form comprehensible to any even a slightly literate and sensible person, set the rules of formation of a winning strategy. But the publication of the book did not only cause a quick growth of those willing to enrich themselves at the cost of gambling houses’ owners, as well as allowed the latter ones to understand the main reason of effectiveness of the developed by Thorp strategy.

First of all, casinos’ owners understood at last that it was necessary to introduce the following obligatory point into the rules of the game: cards are to be thoroughly shuffled after each game! If this rule is rigorously observed, then a winning strategy of Thorp cannot be applied, since the calculation of probabilities of extracting one or another card from a pack was based on the knowledge of the fact that some cards would already not appear in the game!

But what does it mean to have “thoroughly shuffled” cards? Usually in gambling houses the process of “thoroughly shuffling” presupposes the process when a croupier, one of the gamblers or, that is still oftener seen of late, a special automatic device makes a certain number of more or less monotonous movements with a pack (the number of which varies from 10 to 20-25, as a rule). Each of these movements changes the arrangement of cards in a pack. As mathematicians say, as a result of each movement with cards a kind of “substitution” is made. But is it really so that as a result of such 10-25 movements a pack is thoroughly shuffled, and in particular, if there are 52 cards in a pack then a probability of the fact that, for instance, an upper card will appear to be a queen will be equal to 1/13? In other words, if we will, thus, for example, shuffle cards 130 times, then the quality of our shuffling will turn out to be more “thorough” if the number of times of the queen’s appearance on top out of these 130 times will be closer to 10.

Strictly mathematically it is possible to prove that in case our movements appear to be exactly similar (monotonous) then such a method of shuffling cards is not satisfactory. At this it is still worse if the so called “order of substitution” is less, i.e. less is the number of these movements (substitutions) after which the cards are located in the same order they were from the start of a pack shuffling. In fact, if this number equals to t, then repeating exactly similar movements any number of times we, for all our wish, can not get more different positioning of cards in a pack, or, using mathematical terms, not more t different combinations of cards.

Certainly, in reality, shuffling of cards does not come down to recurrence of the same movements. But even if we assume that a shuffling person (or an automatic device) makes casual movements at which there can appear with a certain probability all possible arrangements of cards in a pack at each single movement, the question of “quality” of such mixing turns out to be far from simple. This question is especially interesting from the practical point of view that the majority of notorious crooked gamblers achieve phenomenal success using the circumstance, that seemingly “careful shuffling” of cards actually is not such!

Mathematics helps to clear a situation with regard to this issue as well. In the work “Gambling and Probability Theory” A.Reni presents mathematical calculations allowing him to draw the following practical conclusion: “If all movements of a shuffling person are casual, so, basically, while shuffling a pack there can be any substitution of cards, and if the number of such movements is large enough, reasonably it is possible to consider a pack “carefully reshuffled”. Analyzing these words, it is possible to notice, that, firstly, the conclusion about “quality” of shuffling has an essentially likelihood character (“reasonably”), and, secondly, that the number of movements should be rather large (A.Reni prefers not to consider a question of what is understood as “rather a large number”). It is clear, however, that the necessary number at least a sequence higher than those 10-25 movements usually applied in a real game situation. Besides, it is not that simple “to test” movements of a shuffling person (let alone the automatic device) for “accidence”!

Summing it all up, let’s come back to a question which has been the headline of the article. Certainly, it would be reckless to think that knowledge of maths can help a gambler work out a winning strategy even in such an easy game like twenty-one. Thorp succeeded in doing it only by using imperfection (temporary!) of the then used rules. We can also point out that one shouldn’t expect that maths will be able to provide a gambler at least with a nonlosing strategy. But on the other hand, understanding of mathematical aspects connected with gambling games will undoubtedly help a gambler to avoid the most unprofitable situations, in particular, not to become a victim of fraud as it takes place with the problem of “cards shuffling”, for example. Apart from that, an impossibility of creation of a winning strategy for all “cases” not in the least prevents “a mathematically advanced” gambler to choose whenever possible “the best” decision in each particular game situation and within the bounds allowed by “Dame Fortune” not only to enjoy the very process of the Game, as well as its result.