The Kelly Criterion: Maximizing Growth with Optimal Bet Sizing

The Kelly criterion is one of the most influential strategies in betting and investing for determining how much of your bankroll to wager when you believe you have an edge. In this post, we'll cover the traditional formula, derive it step by step, and introduce an alternative formulation that reframes the problem in a more intuitive way.

1. The Standard Kelly Criterion

The classical Kelly formula for a binary bet is expressed as:

$$f^* = \frac{p \cdot b - (1-p)}{b}$$

where:

• \(p\) is your probability of winning.
• \(1-p\) is your probability of losing.
• \(b\) represents the net profit per unit wagered (i.e., if you bet $1 and win, you gain \(b\) in profit).

Understanding the Terms

• Expected Net Winnings:
  When you wager $1, your expected net profit is:
  $$p \cdot b - (1-p)$$
  This figure represents your "edge"—how much advantage you have on each dollar bet.
• Optimal Fraction:
  Dividing the edge by \(b\) (the net gain per dollar bet) gives you the optimal fraction of your bankroll to bet, ensuring you maximize the logarithmic growth of your wealth over the long run.

2. Derivation of the Kelly Criterion

Understanding where the Kelly formula comes from can help clarify its assumptions and limitations. Here's how we derive it from first principles.

Setting Up the Scenario

Assume you have a bankroll \(W\) and you bet a fraction \(f\) of it. There are two outcomes:

• Win:
  Your wealth increases by the net profit on the wager. Your new wealth becomes:
  $$W_{\text{win}} = W \times (1 + f b)$$
• Loss:
  You lose the fraction of your bankroll you bet, so your wealth decreases to:
  $$W_{\text{lose}} = W \times (1 - f)$$

Maximizing Logarithmic Growth

The Kelly criterion aims to maximize the expected logarithmic growth of your wealth, which is more appropriate over many repeated bets. The expected log growth \(g(f)\) is:

$$g(f) = p \ln(1+fb) + (1-p) \ln(1-f)$$

When you reinvest your gains, your wealth grows multiplicatively rather than additively. The logarithm turns multiplication into addition, making it much easier to calculate long-term growth. Maximizing the expected log wealth is equivalent to maximizing the geometric mean return.

Finding the Optimal Fraction \(f^*\)

To find the optimal \(f^*\), we differentiate \(g(f)\) with respect to \(f\) and set the derivative equal to zero.

Step 1. Differentiate \(g(f)\):

$$g'(f) = \frac{d}{df}\left[ p \ln(1+fb) + (1-p) \ln(1-f) \right]$$

Using the chain rule:

$$g'(f) = \frac{pb}{1+fb} - \frac{1-p}{1-f}$$

Step 2. Set the Derivative to Zero:

For maximum growth, set \(g'(f) = 0\):

$$\frac{pb}{1+fb} = \frac{1-p}{1-f}$$

Step 3. Solve for \(f\):

1. Cross-multiply to eliminate fractions:
   $$pb(1-f) = (1-p)(1+fb)$$
2. Expand both sides:
   $$pb - pbf = 1-p + (1-p)fb$$
3. Collect like terms (group the terms with \(f\)):
   $$pb - (1-p) = fb(p + (1-p))$$
4. Simplify since \(p + (1-p) = 1\):
   $$pb - (1-p) = fb$$
5. Solve for \(f\):
   $$f = \frac{pb - (1-p)}{b}$$

This is the standard Kelly formula.

3. An Alternative Formulation

While the traditional form is mathematically robust, it can sometimes be mentally cumbersome. An alternative way to express the Kelly criterion is by using gross winnings instead of net winnings.

Switching to Gross Winnings

Define:

$$r = b + 1$$

Here, \(r\) represents the total payout multiplier on your wager. For example, if you win a double-or-nothing bet, then \(r = 2\).

The Break-even Point

In terms of \(r\), the break-even win probability is:

$$p = \frac{1}{r}$$

For instance, in a double-or-nothing scenario (\(r = 2\)), you must win at least 50% of the time to break even.

Reformulating the Kelly Fraction

Rewriting the standard formula using \(r\) (by substituting \(b = r-1\)) gives:

$$f^* = \frac{rp - 1}{r-1}$$

This can also be rearranged to:

$$f^* = \frac{p - \frac{1}{r}}{1 - \frac{1}{r}}$$

Intuitive Interpretation

Imagine a probability scale from 0 to 1:

• The break-even point is at \(1/r\).
• If your estimated probability \(p\) is just above \(1/r\), you have a small edge and should bet a small fraction of your bankroll.
• As \(p\) approaches 1 (a sure win), the fraction you bet increases proportionally.

For example, consider a bet where:

• \(r = 2\) (double-or-nothing, so the break-even probability is 0.50),
• And your estimated \(p = 0.80\).

Then:

$$f^* = \frac{0.80 - 0.50}{1 - 0.50} = \frac{0.30}{0.50} = 0.60$$

meaning you should bet 60% of your bankroll.

Alternatives

Conclusion

The Kelly criterion is a powerful framework for determining bet size, maximizing long-term growth while protecting against ruin. We started with the standard form:

$$f^* = \frac{p \cdot b - (1-p)}{b}$$

and then derived it step by step by maximizing the expected logarithm of your wealth. Finally, we introduced an alternative formulation using gross winnings:

$$f^* = \frac{p - \frac{1}{r}}{1 - \frac{1}{r}}$$

which provides a more intuitive perspective by relating your estimated win probability \(p\) to the break-even point \(1/r\).

Whether you prefer the traditional form or the alternative approach, understanding the derivation and intuition behind the Kelly criterion can empower you to make smarter decisions in situations of uncertainty. Feel free to share your thoughts or ask questions—exploring different perspectives on risk and reward only makes us better decision-makers in the long run!