Log Utility and the Kelly Criterion Connection

Maximizing Growth: An Introduction to the Kelly Criterion

In the world of betting and investing, one of the most important decisions you can make is how much of your capital to risk on a particular opportunity. Bet too little, and you miss out on potential gains. Bet too much, and you risk ruin. The Kelly Criterion is a mathematical formula that provides a solution to this problem, offering a systematic way to determine the optimal size of your bets to maximize the long-term growth of your capital.

Developed by John L. Kelly, Jr., a scientist at Bell Labs, the Kelly Criterion was originally used to analyze noise in long-distance telephone signals. However, its applications extend far beyond telecommunications, and it has become a popular tool for gamblers and investors alike.

The Formula for Optimal Bet Sizing

The Kelly Criterion formula is as follows:

Kelly % = (BP - Q) / B

Where:

• Kelly % is the percentage of your capital you should bet.
• B is the decimal odds of the bet minus 1.
• P is the probability of winning.
• Q is the probability of losing (which is 1 - P).

Let's consider an example. Suppose you have a biased coin that has a 60% chance of landing on heads and a 40% chance of landing on tails. You are offered even money (2.0 in decimal odds) on a bet that the coin will land on heads.

In this case:

• B = 2.0 - 1 = 1
• P = 0.60
• Q = 0.40

Plugging these values into the Kelly Criterion formula, we get:

Kelly % = (1 * 0.60 - 0.40) / 1 = 0.20*

This means you should bet 20% of your capital on this opportunity. If you have a bankroll of $100, you should bet $20.

The Connection to Logarithmic Utility

The Kelly Criterion is closely related to the concept of logarithmic utility. In economics, utility is a measure of the satisfaction or value that a person gets from a good or service. A utility function is a mathematical representation of this relationship.

A logarithmic utility function assumes that the marginal utility of wealth decreases as wealth increases. In other words, the more money you have, the less satisfaction you get from each additional dollar. This is a realistic assumption for most people, as the difference between having $1 million and $2 million is much more significant than the difference between having $100 million and $101 million.

The Kelly Criterion can be derived by maximizing the expected value of the logarithm of wealth. This is why it is often said that the Kelly Criterion is optimal for investors with a logarithmic utility function. By using the Kelly Criterion, you are essentially making bets that maximize the long-term growth rate of your capital, which is consistent with the goal of a logarithmic utility investor.

Practical Considerations and Limitations

While the Kelly Criterion is a powerful tool, it's important to be aware of its limitations. The formula assumes that you know the exact probability of winning and the exact odds of the bet, which is rarely the case in the real world. If you overestimate your edge, you can end up betting too much and increasing your risk of ruin.

For this reason, many investors use a fractional Kelly strategy, where they bet a fraction of the amount recommended by the Kelly Criterion. This provides a margin of safety and reduces the volatility of your returns.

Conclusion: A Disciplined Approach to Bankroll Management

The Kelly Criterion provides a disciplined and mathematical approach to bankroll management. By using the Kelly Criterion, you can avoid the emotional pitfalls of betting and make decisions that are based on a sound mathematical framework. While it's not a magic formula that will guarantee you win every bet, it can help you maximize the long-term growth of your capital and increase your chances of success in the world of betting and investing.