Calculating the Kelly Criterion for More Than 2 Outcomes

The Kelly Criterion is a robust method developed for maximizing the growth of a bankroll over time, especially in scenarios involving multiple outcomes. This article delves into how to apply the Kelly Criterion when dealing with more than two possible outcomes. Understanding this concept is crucial for making informed betting decisions or investment strategies in fields like finance, sports betting, and lotteries.

Understanding the Kelly Criterion

The Kelly Criterion is a formula used to determine the fraction of a bankroll to bet in situations with known outcomes and probabilities. It is based on the idea that the best bet for a replayable game is the one that maximizes the expected growth of the bankroll over all possible sets of future outcomes.

Application to More Than Two Outcomes

In scenarios where there are more than two possible outcomes, the application of the Kelly Criterion becomes more complex but remains fundamentally the same. Let's consider a situation with three outcomes, given the following probabilities and payoffs:

Result 1: 1/3 chance of winning 2X

Result 2: 1/3 chance of losing X

Result 3: 1/3 chance of losing X/2

We need to calculate the expected growth to find the optimal bet size (X) that maximizes the growth of our bankroll. The expected growth is calculated as the product of the factors that each outcome contributes to the bankroll:

Step-by-Step Calculation

The first step is to determine how each outcome affects the bankroll:

1. Win 2X: Our bankroll changes by a factor of (frac{2X B}{B})

2. Loss X: Our bankroll changes by a factor of (frac{B - X}{B})

3. Loss X/2: Our bankroll changes by a factor of (frac{B - X/2}{B})

Our goal is to find the value of X that maximizes the product of these growth factors:

(frac{2X B}{B} times frac{B - X}{B} times frac{B - X/2}{B} text{Expected Growth})

The next step involves taking the natural logarithm of the expected growth to simplify the problem, as the logarithm of a product is the sum of logarithms:

(log left(frac{2X B}{B}right) log left(frac{B - X}{B}right) log left(frac{B - X/2}{B}right) G)

Maximizing the Expected Growth

To find the value of X that maximizes the expected growth, we take the derivative of G with respect to X and set it to zero:

[frac{dG}{dX} frac{2}{2X B} - frac{1}{B - X} - frac{1}{2B - X/2} 0]

Solving this equation for X involves a bit of algebraic manipulation. While solving analytically can be complex, numerical methods or software can provide accurate solutions.

Example Calculation

For a specific example, let's assume a bankroll (B) of $1000 and a potential bet (X) to be a significant fraction of the bankroll:

Firstly, let's rewrite the equation with the actual values:

[frac{2}{2X 1000} - frac{1}{1000 - X} - frac{1}{2000 - X/2} 0]

Using a numerical solver, we find that the optimal X is approximately $106 (10.6% of the bankroll).

Conclusion

The Kelly Criterion, when applied to more than two outcomes, requires more complex calculations but provides a clear framework for maximizing the expected growth of a bankroll. By carefully analyzing each outcome and using the properties of logarithms, one can find the optimal betting strategy to ensure long-term growth.

Keywords

Kelly Criterion, Expected Growth, Optimal Betting Strategy