Other Betting Systems

Betting systems have existed for as long as gambling has. A betting system is either bogus or clever depending upon whether it is based on a sufficiently deep understanding of the given game so that there is some method to the madness.

Gambling systems, even bogus ones, are always interesting to hear about, because they say something about how people perceive (or misperceive) probability. My favorite bogus systems include:

* Doubling up in Casino Gambling - Consider the following strategy for gambling in roulette. Walk into the casino and bet a dollar on black. If it wins, boldly pocket your earnings. If not, bet $1 again on black. If it wins, your are back to where you started. If it loses, bet $2 on black to recoup your losses. After each loss, keep doubling up. Inevitably, you are going to win sometime, and at that point you are all caught up. Now you can start again from the beginning. You can't ever lose money with this scheme, can you?

What's the problem? Nothing really, so long as you have an infinitely deep pocket and are playing on a table without a betting limit. If your table has a betting limit or you are not able to print money, you will eventually reach a point where the house will not let you bet as much as you need in order to play by this system. At this point you have been completely wiped out.

This doubling or Martingale system offers you a high probability of small returns in exchange for a small possibility of becoming homeless. Casinos are more than happy to let you take this chance. After all, Donald Trump has a much deeper pocket than either you or I have.

* The O'Hare Straddle - An alternate doubling scheme is as follows. Borrow a large amount of cash on a short term basis. Set aside enough money for a ticket on the next plane to South America. Bet the rest on one spin of the roulette wheel at even money. If you win, return the principal and retire on the rest. Otherwise, use the plane ticket.

Mathematically, the key to making this work is being bold enough to wager all the money on a single bet, rather than making multiple smaller bets. The casino extracts a tax on each `even-money' wager via the 0 and 00 slots on the wheel. You pay more tax each time you re-bet the winnings, thus lowering your chances of a big killing. However, the most likely result of playing the O'Hare straddle will be a sudden need to increase your fluency in Spanish.

* Collecting Statistics on Lottery Numbers - Some people carefully chart the frequency with which lottery numbers have come in recently, and then play the numbers that are either ``due'' or ``hot''. Unfortunately, the notion of a number ``being due'' or ``being hot'' violates all laws of probability, technically the assumption that the numbers arise from independent Bernoulli trials. Lottery numbers are selected by drawing numbered balls from a jar, or some equivalent method. Provided that the balls have been thoroughly mixed up, there is no way a ball can know that it has not been selected for a while, and hence is due. Similarly, the notion of a number ``being hot'' makes sense only when the numbers have been drawn according to a non-uniform random number generator.

As we will see, poor random number generators certainly exist; I will talk more about this in Chapter 4. There is also historical precedent for poorly mixed-up balls. During the Vietnam War, the United States military draft selected soldiers by lottery according to birthday. Each of the 365 birthdays for the year were stamped on a ball and tossed into a jar, and unlucky 19 year olds mustered into the army if their birth date was selected. In 1970, several newspapers observed that December's children had a startlingly high chance of being drafted, and indeed, the lottery selection procedure turned out to be flawed. It was fixed for the next year, presumably small consolation to those left marching in the rice paddies.

Although each lottery combination is just as likely to come in as any other, there is one formally justifiable criterion you can use in picking lottery numbers. It makes a great deal of sense to try to pick a set of numbers which nobody else has selected, since if your ticket is a winner you won't have to share the prize with anybody else who is a winner. For this reason, playing any ticket with a simple pattern of numbers is likely a mistake, since someone else might stumble across the same simple pattern. I would avoid such patterns as 2-4-6-8-10-12, and even such numerical sequences as the primes 2-3-5-7-11-13 and the Fibonacci numbers 1-2-3-5-8-13, because there are just too many mathematicians out there for you to keep the prize to yourself.

There are probably too many of whatever-you-are-interested-in as well, so stick to truly random sequences of numbers unless you like to share. Indeed, my favorite idea for a movie would be to have one of the very simple and popular patterns of lottery numbers come up a winner; say, the numbers resulting from filling in the entire top row on the ticket form. As a result, several hundred people will honestly think they won the big prize, only later to learn it is not really so big (say only $5,000 or so). This will not be enough to get members of the star-studded, ensemble cast out of the trouble they got into the instant they thought they became millionaires.

On the other hand, there are well-founded betting systems available for certain games, if you know what your are doing:

* Card counting in Blackjack - Blackjack is unique among casino games in that a sufficiently clever player can indeed have an advantage over the house. In blackjack, each player starts with two cards, and the goal is to collect a set of cards whose total points are as close to 21 as possible without going over. The key decision for any player is whether to accept an additional, unknown card from the house. This card will increase your point total, which is good, unless it takes you over 21, which is bad. You win money if your total is closer to 21 than the dealer's, who must play according to a well-defined strategy.

If you know nothing about the cards which you are to be dealt, then the dealer's strategy is sufficient to guarantee the house a nice advantage. However, a sufficiently clever player does know something about the hand which she will be dealt. Why? Suppose in the previous hand she saw that all four aces had been dealt out. If the cards have not been reshuffled, all of those aces are now sitting in the discard pile. Assuming that only one deck of cards is being dealt from, there is no possibility of seeing an ace in the next hand, and a clever player can bet accordingly. By keeping track of what she has seen (card counting) and properly interpreting the results, she knows the true odds of each possible card showing up and thus adjusts her strategy accordingly. Card counters theoretically have a inherent advantage of up to 1.5% against the casino, depending upon which system they use.

Edward Thorp's book Beat the Dealer started the card-counting craze in 1962. Equipped with computer-generated counting charts and a fair amount of chutzpah, Thorp took on the casinos. Once it became clear (1) that he was winning, and (2) it wasn't just luck, the casinos became quite unfriendly. Most states permit casinos to expel any player they want, and it is usually fairly easy for a casino to detect and expel a successful card counter. Even without expulsion, casinos have made things more difficult for card counters by increasing the number of decks in play at one time. If there are ten decks in play, seeing four aces means that there are still 36 aces to go, greatly decreasing the potential advantage of counting.

For these reasons, the most successful card counters are the ones who write books which less successful players buy. Thorp himself was driven out of casino gambling in Wall Street, where he was reduced to running a hedge fund worth hundreds of millions of dollars. Still almost every mathematically-oriented gambler has been intrigued by card counting at one point or another. Gene Stark, a colleague of mine who you'll read more about later, devised his own card counting system, and used it successfully a few times in Atlantic City. However, he discovered that making significant money off a 1.5% advantage over the house requires a large investment of either time or money. It isn't any more fun making $5.50 an hour counting cards than it is tending a cash register.

* The Eudaemonic Pie - Physicists tend to be good at mathematics. A few years ago, the American Physical Society had its annual convention in Las Vegas, during which the conference hotel/casino took a serious financial hit. The hotel rented out rooms to the conference at below cost, planning to make the difference back and more from the gambling losses of conference goers. However, the physicists just would not gamble. They knew that the only way to win was not to play the game.

But another group of physicists did once develop a sound way to beat the game of roulette. A roulette wheel consists of two parts, a moving inner-wheel and a stationary outer-wheel. To determine the next ``random'' number, the inner wheel is set spinning, and then the ball sent rolling along the rim of the outer wheel. Things rattle around for several seconds before the ball drops down into its slot, and people are allowed to bet over this interval. However, in theory, the winning number is preordained from the speed of the ball, the speed of the wheel, and the starting position of each. All you have to do is measure these quantities to sufficient accuracy and work through the physics.

As reported in Thomas Bass's entertaining book The Eudaemonic Pie, this team built a computer small enough to fit in the heel of a shoe, and programmed in the necessary equations. Finger or toe presses at reference points on the wheel were used to enter the observed speed of the ball. It was necessary to carefully conceal this computer because otherwise casinos would be certain to ban the players to moment they started winning.

Did it work? Yes, although they never quite made the big score in roulette. Like Thorp, the principals behind this scheme were eventually driven to Wall Street, building systems to bet on stocks and commodities instead of following the bouncing ball. Their latter adventures are reported in the sequel, The Predictors.

* Flooding Large Lottery Pools - Lotteries in the United States keep getting bigger. The bigger a jackpot, the more people that want to play. Many states have switched to systems of accumulated pools, where if no grand prize winner emerges in a given week, the money rolls over to supplement next week's prize. The pool grows very large whenever a few weeks goes by without a winner. Whenever the pool gets large enough (say $100 million), it starts a betting frenzy which draws national attention.

The interesting aspect of large pools is that any wager, no matter how small the probability of success, can yield positive expected returns given a sufficiently high payoff. Most state lotteries are obligated to pay some fraction (say 50%) of all betting receipts back to the bettors. If nobody guesses right for a sufficiently long time, the potential payoff for a winning ticket can overcome the vanishingly small odds of winning. For any lottery, there exists a pool size sufficient to ensure a positive expected return assuming a given number of tickets sold.

But once it pays to buy one lottery ticket, then it pays to buy all of them. This has not escaped the attention of large syndicates which place bets totaling millions of dollars on all possible combinations, thus ensuring themselves of a winning ticket.

State lottery agents frown on such betting syndicates, not because they lose money (the cost of the large pool has been paid by the lack of winners over the previous few week) but because printing millions of tickets ties up agents throughout the state and discourages the rest of the betting public. Still, these syndicates like a discouraged public. The only danger they face are other bettors who also pick the winning numbers, since the pool must be shared with these other parties. Given an estimate of how many tickets will be bought by the public, this risk can be accurately measured by the syndicate to determine whether to go for it.

Syndicate betting has also occurred in jai-alai, in a big way. Palm Beach Jai-Alai ran an accumulated Pick-6 pool, which paid off only if a bettor correctly picked the winners of six designated matches. This was quite a challenge, since each two dollar bet was a 86 = 262,144-to-one shot for the jackpot.

On March 1, 1983, the pool stood at $551,332, after accumulating over 147 nights. This amount was more than it would cost to buy one of every possible ticket. That day an anonymous syndicate invested an additional $524,288 to guarantee itself a large profit, but only if it didn't have to share. Only $21,956 was wagered on Pick 6 that night by other bettors, giving the syndicate an almost 96% chance of keeping the entire pot to itself, terrific odds in its favor. Indeed, only the syndicate held the winning combo of 4-7-7-6-2-1, a ticket worth $790,662.20.