This book is about gambling systems with a particular emphasis on the Kelly system. A gambling system is a method for choosing bet sizes in order to maximize winnings and minimize the potential for loss. A good gambling system is a systematic method for managing money and risk.

Now you may ask if such a thing is even possible. Are there really money management methods that will increase your winnings? The answer for most gambling situations is no. The reason is that most of the gambling games that you will encounter in a casino for example, have what is called negative expectation. This means that in the long run you are almost always guaranteed to lose money playing these games. There is no gambling system that can make a negative expectation, or for that matter a zero expectation, game profitable.

A gambling system is only useful when confronted with a positive expectation game and such games are rare. The simplest example of a positive expectation gambling game is the simple coin toss bet with a biased coin. This is a coin where one of the sides, heads say, comes up more often than the other. If heads has a probability greater than 0.5 of coming up and you are able to bet on it repeatedly, then you will make money in the long run. In this case a gambling system can help you increase your winnings.

Probably the place where you are most likely to find a positive expectation gamble is in the financial markets. People do not usually associate investing with gambling but there really is no fundamental difference. Every investment is a gamble with the potential for losing or gaining money. If you had to draw a distinction between a gamble and an investment it would be that an investment usually has a small probability of total loss. This is one of the reasons why there are so many more positive expectation investments than gambling games.

Most of the gambling systems developed by gamblers base bet sizes on the size of previous bets and whether those bets were won or lost. These systems tend to produce large fluctuations in bankroll, both up and down. Since these systems do not take bankroll size into account when selecting a bet size, the chance of going bankrupt with such a system is high. The most common example of such a system is the Martingale system which is analyzed in chapter 2.

The Kelly system takes bankroll size into account by always betting a fixed fraction of the size of the bankroll. The fraction is selected so that the bankroll grows exponentially at the fastest possible rate in the long run. It is important to note that exponential growth only happens in the long run with the Kelly system. Considerable short term fluctuations are still possible with the Kelly system but since the amount wagered is based on the size of the bankroll, the probability of going bust is small. The Kelly system is discussed in detail in chapters 3 and 4.

In order to analyze gambling systems in detail, a significant amount of mathematics must be used. The treatment of gambling systems in this book is therefore highly mathematical. To get the most out of this book you should have a good familiarity with mathematical notation and have taken at least one calculus course. It would also be very helpful to have had some exposure to probability theory. The essential concepts from probability theory are reviewed in chapter 1 but to really understand the presentation it will help to have had a previous course in the subject.

Following the probability theory review in chapter 1, chapter 2 looks at some of the more commonly used gambling systems. Multiple bet state systems are analyzed as well as the Martingale and a simple cancellation system. For all these systems, the expectation and variance of the bankroll in a simple coin toss gambling game is calculated. For some systems it is not possible, or it is too complicated, to express the expectation and variance in a closed mathematical form. In this case Python code for performing the calculation is given.

Chapter 3 begins the analysis of the Kelly system. It starts with an analysis of fixed fraction betting in general. The question of how one chooses the fraction is considered next. Two approaches to answering this question are examined. One approach is to use the concept of a utility function which was first proposed by Daniel Bernoulli in 1738. The second approach is the one used by John Kelly. Kelly’s approach is to choose the fraction so as to maximize the long term exponential growth rate of the bankroll. A discussion of Kelly’s analogy between this growth rate and the rate of information transmission through a communications channel is included. Both the Kelly and Bernoulli approach lead to the conclusion that the optimal betting fraction is the one that maximizes the expectation of the logarithm of the bankroll. The rest of the chapter examines the calculation of the betting fraction, the expectation and variance of the bankroll, and the bankroll probabilities for single and multiple games. The chapter ends with a somewhat unusual application of the Kelly system to playing the Powerball Lottery.

The final chapter looks at the use of the Kelly system in investing. It starts out by looking at a single stock investment. This is more complicated than most gambling games since a stock investment can have many possible returns. Solving for the Kelly fraction in this case usually involves using numerical techniques. This is discussed in the text, and Python code for calculating the fraction is included. The calculation of the Kelly fraction when there is the possibility of including a risk free bond with the stock is taken up next. An interesting application of this is calculating the default probabilities of risky bonds based simply on their market interest rates. The chapter concludes by looking at the calculation of the Kelly fraction for investing in two stocks simultaneously. This must also be done numerically and Python code for the calculation is included.

This book is by no means an exhaustive treatment of the Kelly system but it should serve as a good introduction to the subject and a starting point for further investigation. Those wishing to delve further into the subject can consult the references at the end of the book.

This book has a website, where software and related information can be found:

http://quantwolf.com/betsmart.html

We can be reached by email at:

stefan[AT]exstrom DOT com

richard[AT]exstrom DOT com

Stefan Hollos and Richard Hollos

QuantWolf.com

Exstrom Laboratories LLC

Longmont, Colorado

November 2008