In the fifties and sixties, Las Vegas was a very different place from what it is now. In the fifties it still carried a good amount of wild west flavor. No carpets, no dress code at all, cowboy boots and hats worn. In the sixties, casinos moved towards the "strip" and became more elegant, when the mafia took over many of the casinos. And of course, alcohol and other illegal drugs, as well as prostitution was never far away. As was cheating and violence.
A world very far away from the academical world. Except for the academical interest of some mathematicians in gambling and parlor games. As discussed in the France 1654 chapter, Probability Theory started from questions over games of luck, and games of some skill were discussed first in Zermelo's paper on chess [Z 1910], and later in papers on simplified version of poker by Borel, von Neumann, and Kuhn [VN 1928], [K1950]. From then on, parlor games, in most cases simplified, were taken as examples of simple games. Mathematicians, in particular the young Princeton graph theorists, also played these games extensively, together with other parlor games they invented. Although such games are usually too complex for a complete analysis, the mathematical treatment is accurate, since there is no modeling involved. The rules are firm, and the payoffs are given. Except for one problems with the payoffs as we will see below.
Most professional gamblers of that time were very far away from every academic background, and they obviously were not interested in what insight academia had to offer. Except for the very simple comparison of odds and payoff ratio, which was standard and distinguished professional gamblers from the so-called "suckers". Many gamblers played both in casinos as well as gambled on sports events, like horse racing, and their expertise was not in mathematics but rather in the sports involved---the key was to estimate the odds correctly.Note that casino games are by definition games of luck. You don't need a license to set up a chess tournament with a high prize for the winner, but you need a license for roulette, slot machines, black jack, or poker. Although, in reality the amount of luck involved differs. There is no skill in roulette, it doesn't matter much what you do, but in poker, on the long run, skilled players will always win against beginners. This is where the skill of game theory may be helpful.
During the fifties and sixties, mathematicians and mathematics very slowly started to influence casino play and gambling. Starting point was Black Jack and the insight that both your odds as well as your best strategy depends on what's left in the deck of cards. "Card counting" means that players try to estimate what's left by keeping track, more or less, on the cards drawn so far. Jess Marcum, who was employed at RAND (see this chapter) before becoming a professional gambler was one of the first applying this technique. There is an early paper on card counting and the odds [BCMM 1957]. There is also an approach on the optimal betting size by J.L. Kelly Jr. [K 1956]. These ideas were further developed and popularized by mathematics professor Edward O. Thorp.
Thorp wrote a paper where he did improve the results of [BCMM 1957], using a computer for the tedious calculations. After presenting its contents in a talk with the promising title "Fortune's formula: A winning strategy for blackjack", a reporter got interested and interviewed Thorp. The article appeared in local newspapers and as a result, Thorp got thousands of letters asking for the system, or offering to invest money. He accepted the offer from Emmanuel Kimmel, a bookmaker and gambler with mob connections. Together they went to Reno and made thousands of dollars, also using Kelly's criterion for the betting size [P2005]. This weekend winning spree in Reno got them a lot of attention from casino owners in Reno, but only in 1962, when Thorp published his bestseller book "Beat the dealer" [T 1962] selling 700,000 times, both gamblers and casinos really got aware that sophisticated mathematics can be really useful for them.
Card counting is explained in the chapter on Mini Blackjack. Let us rather explain the Kelly criterion:
The game is obviously favorable to you. If you bet an amount of a in the first round, with 10% probability you have 1000 + 100a after the first round, and with 90% probability you have an amount of 1000 - a. Therefore the expected value after one round equals 0.1 · (1000 + 100a) + 0.9 · (1000 - a) = 1000 + 9.1a. The higher the bet, the higher the expected value after each round. So you would always bet as much as you have, right?
Or rather not? When applying the risky strategy of always betting everything, when you bet the whole $1000 in the first round, with 90% probability all is gone after the first round and you can go home. Even if you win in the first round and again bet all your money, the whole $100000, in the second round, it is very likely that you lose then. And so on. Playing this risky strategy, with probability (0.1)5 = 0.00001, you win in all five rounds and take $10,000,000,000,000 home, to become the richest man in the world, about 200 times as rich as Bill Gates. With the remaining probability 0.99999 you lose all your money at some point and go home without any money left. Still, the expected value for the game when playing this risky strategy equals 0.00001 · 10,000,000,000,000 + 0.99999 · 0 = 100,000,000. Not bad at all, but maybe you would prefer a somewhat higher probability for a smaller amount of money. Wouldn't it be nicer to have $1,000,000 after the 5 rounds, with probability 90%? But of course, the expected value of this option would only be $900,000, much less than in the risky strategy.
In fact the expected value approach gets problematic when high numbers are involved. I bet that most of us would prefer $1,000,000 on the hand over a 50-50 chance of winning $3,000,000 or nothing. Still the second option has an expected value of $1,500,000. Actually not the expected value calculation is flawed, but most people don't consider $3,000,000 to be three times as valuable as $1,000,000. Usually an additional $1000 is less valuable for a rich person than for a poor person. This problem can be overcome by introducing a so-called utility function. Not the money itself is the payoff, but the satisfaction u(x) derived from possessing an amount of x dollars. Possible utility functions are the square root function u(x) = sqrt(x), where four times the money is only producing twice the happiness, or u(x) = log(x+1), where the logarithm is (roughly) counting the number of digits of a number. The reason why we use log(x+1) instead of the simpler log(x) is that log(x) is not defined for x=0 and for negative numbers. Using log(x+1) attaches a utility of log(1)=0 to the value x=0, and then slowly increasing utilities to positive x. For instance, u(9)=1, u(99)=2, u(999)=3, and so on with this utility function.
What's the reason for choosing the logarithm, or maybe the square root as the utility function? Well, both have certain properties that utility is supposed to have: They are increasing---more money means more utility--- but they increase less and less, which matches the observation that an increase of $1 means less to somebody having already a lot than to somebody having little. The reason why the logarithm is often proposed as utility function goes back to Daniel Bernoulli, who claimed that the increase of utility of a dollar would be inversely proportional to the amount of money already owned. An additional dollar is twice as valuable to somebody having $1000 than to somebody having $2000. This property is fulfilled only by the logarithm. But keep in mind that this claim is simple, and therefore attractive to mathematicians, but does not necessarily reflect reality in full.
The Kelly criterion is that betting choice that produces the highest expected utility for the logarithm as utility function in one round. Remember that in our example, if we bet an amount of a in the first round, with 10% probability you have the utility log(1000 + 100a +1) after the first round, and with 90% probability you have the utility of log(1000 - a +1). So the expected utility is 0.1 · log(1001 + 100a) + 0.9 · log(1001 - a), which is, according to some laws of the logarithmic function, equal to log((1001 + 100a)0.1· (1001 - a)0.9). This value is highest if (1001 + 100a)0.1· (1001 - a)0.9 is highest. Experiment a little with your calculator or with Excel to verify that a=91 gives you the highest value of about 1158 for this expression, and the logarithm of this value is about 3.06357. So according to Kelly you would bet $91 in the first round. But in each subsequent round, the reasoning is similar, and you would, with these parameters, always bet about 9.1% of your available money. Note that Kelly used the function log(x) instead of log(x+1), which yields a slightly different recommendation, still rounding to 9.1% for our parameters.
The game can be simulated in the "one choice" sheet of the Excel sheet Kelly.xls. In cell A36 you tell how much of your wealth you risk in each round, and see the probabilities for the different outcomes to the right. If you, for example, risk 50% of your money in each round, you end, after 5 rounds, with
But you should keep in mind that the Kelly criterion depends on the chosen utility function. If we select another utility function, as the square root function, we get a different conclusion---with our parameters to bet about 58% of your money in each round. As long as you don't have specified your utility function, how much you value huge amounts of money, the outcome of the calculation is not very precise.
The Kelly criterion can be used for all games where the player has an advantage. Games where there is a disadvantage, like roulette or the slot machines, a rational player would better not even play at all. But of course, the player's disadvantage is small enough. And poker has a special role, since one just needs to be a better player than the others to have an advantage, and most people think that they are above average.
Card counting in Black Jack became more and more an issue in the casinos. Eventually casinos took countermeasures by changing the rules: More decks of cards were used in the shoe, which was also shuffled more frequently, limits on the maximum bet (or better, the ratio of maximum and minimum bet). Although recognizing what goes on and using your brain cannot reasonable be forbidden, card counters were still banned from the casinos, and lists of card counters were created and sold. Card counters had to reveal what they were doing, and since the most telling sign for card counters is the sudden change between very low and very high bets, they started to work in groups, with someone betting always high, but sitting down at a desk when signaled by a low bet player that the odds are favorable. Ken Uston was very successful with this method, and a famous MIT student group worked in this way during the 80s and 90s. The movie "21" is based on their story.
A lot of things have changed since the 60s. Las Vegas is now family-friendly and non-smoking. There may still be drugs or organized crime, but they are not visible on the surface. And through the internet, online poker and other online games have become more popular than ever. The casino's and the professional gambler's opinion towards mathematics has also changed. Nowadays, gamblers know and appreciate what game theory can offer not only to Black Jack, but also to poker and other games, Even though poker has not been mathematically "solved" yet, some mathematicians create competitive poker programs based not on poker cracks wisdom but on Game Theory [http://www.blackjackhero.com/blackjack/strategy/card-counting/history/]. One of the best players, Chris ("Jesus") Ferguson, has even a Ph.D. degree in Computer Science and publishes papers on game theory.
Whether card counter do still have an advantage depends on these parameters, which differ from casino to casino. Some authors claim that still optimal playing card counters have a slight advantage. And why not---it would certainly be rational for the casinos to keep at least the rumor that there is something to win alive. This way more people would come and play, and of course, most of them would not be able to play optimal anyway.
http://www.blackjackhero.com/blackjack/strategy/card-counting/history/ .
Roger Baldwin, Wilbert Cantey, Herbert Maisel, and James McDermott, Playing Blackjack to Win (1957): Hints on how to adjust plazing strategy, based on the distribution in the stack. Didn't consider changes in bets based on expectations. .
Example: Jess Marcum. born 1919. electical engineering BS degree. worked for RAND from 1947- Las Vegas, early 50s. At that time, blackjack was played with one deck of cards, dealing to last card. Didn't publish anything to keep his advantage. Was banned from all Las Vegas casinos. Then Reno, was banned there soon too. Then different other locations. Compare thew rather subjective story here: http://www.blackjackforumonline.com/content/JessMarcumEarlyDaysofCardCounting.HTML .
1971, Keith Taft and son Marty Taft build a portable computer, operated by toes. 1977 Marty was caught. later law changed, banned all computing devices. http://www.bbc.co.uk/sn/tvradio/programmes/horizon/million_prog_summary.shtml http://www.bbc.co.uk/sn/tvradio/programmes/horizon/million_trans.shtml
Ken Uston, 1935-1987, math genius, entered Yale with 16, Vice President of the Pacific Stock Exchange, faszinated by Thorp's book. early 70s team card counting, play caller counts, permanently bets low. big player enters if count gets very favorable avoids bet spread. Al Francesco's t
eam. banned from most casinos, used disguise. his book revealed team play techniques. lawsuit ---Atlantic City cannot bar card counters.