## Introduction to the Kelly System

The Kelly system is a betting system. For any gambling game with two or more random outcomes, it will tell you how much of your money to bet on each play of the game. The goal is to maximize your winnings as you play the game repeatedly. As with any gambling system, it will only work for what are called positive expectation games.

A positive expectation game is one where you win more than you lose, on average, as you play the game for a long time. To calculate the expectation, multiply the amounts that you can win (positive) or lose (negative) by their corresponding probabilities and add up the products. If the sum is a positive number, then you have a positive expectation game.

The simplest example is a coin toss bet where you can
either win or lose a dollar. If **p** is the probability of
winning, and 1-**p** is the probability of losing, then the
expectation is equal to 2**p**-1. In order for this to be a
positive expectation game, **p** must be greater than 1/2.

In a slightly more general case, suppose you can win **W**
dollars with probability **p**, and lose **L**
dollars with probability 1-**p**. The expectation in this case is
(**W**+**L**)**p**-**L**
which is positive when **p** is greater than
**L**/(**W**+**L**).

Now how do you apply the Kelly system to a game like this? In the Kelly
system you always bet a fixed fraction of your total bankroll on each
game. The fraction is fixed as long as the payoffs and probabilities do not
change. For the game described above, the fraction you should bet is

**f** =
(**p**(**W**+**L**)-**L**)/(**LW**).

If you bet this fraction on each game then your bankroll should grow exponentially, on average, if you play the game for a very long time. It is important to keep in mind that this is long term average behavior, and that in the short term there can be very large fluctuations in the size of your bankroll both up and down.

You can also calculate Kelly fractions for games with more than two outcomes. In this case, it is usually not possible to come up with a simple formula for the fraction and numerical techniques have to be used.

If you are interested in an in depth introduction to the Kelly system, see the book:
**Bet Smart: The Kelly System for
Gambling and Investing**.