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Gambling Theory and Other Topics

by Mason Malmuth

Non-Self-Weighting Strategies

A self-weighting gambling experience is when your expectation and variance are the same from play to play. But what does that mean, and why is it so important? To answer these questions, let's start with a simple example.

Suppose you make a trip to Las Vegas, walk up to a craps table, and start betting $1 every time you roll the dice. You do this for 10,000 rolls, and then suddenly on the next roll you pull $1 million out of your pocket and put it all into action. Now how many times have you rolled the dice? Well, there are two answers, both correct. The mathematicians will say 10,001 times, but the statisticians will say one time simply because your results are all clustered around that one big bet. Notice, that during the first 10,000 rolls of the dice, we were looking at a self-weighting gambling experience. But after that $1 million bet came along, we had a very non-self-weighting gambling experience. Incidentally, and this is important, all successful gamblers are statisticians, not mathematicians.

Why are all successful gamblers statisticians? To answer this, think about all the games that can be beat. This includes poker, blackjack, sports betting, and progressive slot machines, just to name a few. It does not include roulette, craps, baccarat, super pan nine, and non-progressive slot machines. The reason the games in the first group are beatable is that successful non-self-weighting strategies are available. This is not true of the second group. Their self-weighting characteristics mean that the player will never have the best of of it. To understand this, let's take a closer look at poker.

Suppose you are a Southern California poker player. Your game is high draw, jacks or better to open, played with a joker that counts either as an ace or can be used to complete straights and flushes. You are dealt a pair of aces. How should they be played?

The typical player plays this hand in a very straightforward manner. If no one has yet opened, he will open; If someone already has opened, he will call. Notice that this is a self-weighting strategy. In other words, there is very little difference in how this hand is played. Also notice that self-weighting players are very predictable opponents, and very predictable opponents are the easiest to beat. This is true in all forms of poker.

Now what about the expert high draw player? How does he play a pair of aces? Well most of the time he will open when he is in an early position, but not all the time, especially if his other cards are small. (The small cards maximize the chances that someone else will be able to open the pot.) If someone else already has opened, the expert will call most of the time, but not all the time. If his opponent has opened very late, where most players are likely to play any minimum opener, such as a pair of jacks, he will raise. If a tight player opens early, the expert knows that he is (1) up against a quality hand and (2) less likely to be against a pair of aces, since the two aces in his hand reduce the chances that his opponent is also holding a pair of aces by 70 percent. (Not 40 percent as many people think. This is because there are 10 ways you can pick two aces out of five, while there are only three ways you can pick two aces out of three.) Consequently, there is a good chance the opener has a pair of aces badly beat. The expert player would throw his hand away in this situation, especially if several players still remain to act behind him.

It should be obvious from this example that the expert player is following a non-self-weighting strategy. Even though he does not play as many hands as his self-weighting counterpart, the expert still puts about the same amount of money in the pot. That is because he gets full value for some of the hands he plays, while his typical self-weighting opponent does not.

By the way, it doesn't matter what kind of poker game it is. The expert players, who consistently take home the money, all follow non-self-weighting strategies.

One claim I've made is that the low-limit jackpot games in California can easily be beaten. Most people believe just the opposite, simply because the house takes a lot of money out of the pot to pay not only for the game, but for the jackpot as well. (For those readers unfamiliar with jackpot poker, a player wins the jackpot when he has a powerful hand beaten such as aces full or better in high draw or a six-four lowball.) These jackpots become quite large, somtimes as high as $40,000 in certain games.

The jackpots have a unique effect on almost all players, turning them into self-weighting opponents. What happens is that typical players now have an incentive to play many more hands. Also, they don't want to make marginal (value) bets and raises -- where the real money is won in limit games -- simply because they are more interested in survival and shooting for the jackpot.

Remember, the more self-weighting your opponent is, the poorer he plays. An extension of this is that the more self-weighting opponents you have, the easier a game is to beat. As we have seen, in the typical jackpot game, almost everyone follows self-weighting strategies. This is why I agure that an expert player, who follows non-self-weighting strategies, can easily beat these games. The drop and the rake should be a small price to pay for the privilege of going home a winner. Summarizing, we see that in games with a jackpot, there are non-self-weighting strategies available that allow the expert player to compete with a positive expectation. The non-self-weighting player waits until he has the best of it and then takes maximum advantage of the situation. That is, he will make as big a bet as possible and take advantage of a very small edge if that is all the edge he has.

However, even though you have the best of it, this doesn't mean that you are going to win. In addition, following a non-self-weighting strategy has the statistical effect of decreasing the overall sample size. This leads to two related topics that will be covered in Part Two of this book: fluctuations, which every skilled gambler suffers from, and the "extremely silly subject of money management."