When should we raise the stakes?

The Kelly Criterion

A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

TAKING ADVANTAGE OF THE ODDS

One of the most critical aspects of any betting strategy is determining the size of the bet when we believe the odds are in our favour. The answer to this pressing question was formalised by John L. Kelly, Jr., an engineer at Bell Labs by profession, as well as a hobbyist daredevil pilot and recreational gunslinger. His methodology, known as the Kelly Criterion, is a mathematical formula designed to establish the optimal bet size when we have an advantage, that is, when the odds favour us.

However, having the advantage doesn’t guarantee a successful outcome. Irrespective of our edge, excessive betting can lead to large losses and, in worst-case scenarios, to bankruptcy. This is where the Kelly Criterion comes into play. It takes account of both the size of our edge and the potentially damaging impact of volatility.

UNDERSTANDING THE KELLY CRITERION

The Kelly Criterion is essentially a mathematical formula that calculates the optimal amount to bet or invest when the odds are in our favour. The fundamental principle underpinning the Kelly Criterion is that the amount of capital wagered should be related to our advantage at the available odds. It emphasises the relationship between the size of our bet and our perceived edge.

Consider this simple illustration: Suppose you have a coin where the probability of getting a head (winning) is expected to be equal to the probability of a tail (losing). Now, suppose you have secret information that the next coin toss will certainly be heads. In this case, you have a 100% edge. According to the strict Kelly Criterion, you should bet your entire capital because you’re guaranteed to win. In real-life applications, even with very high confidence, betting one’s entire capital is risky, however, due to the possibility of unforeseen factors. This is more a thought experiment than a practical recommendation.

In any case, in most scenarios the outcomes are not binary, and the probability of winning is rarely 100%, even in theory.

Let’s consider a different situation: You’re still betting on the coin toss, but this time your secret information gives you a 60% chance of landing heads and a 40% chance of tails. Your edge is now 20%, and a very basic Kelly strategy is to stake 20% of your capital.

This example reflects the core concept of the Kelly Criterion. It’s not only about gauging when you have the advantage—it’s also important to understand precisely how much to stake when you do. In theory, this sounds simple, but in practice, accurately identifying that advantage can be complex.

On a broader scale, the Kelly Criterion can be employed in various fields beyond betting, such as investing and trading, to determine the optimal size of a series of bets or investments. Its aim is to maximise the exponential growth of the bettor’s or investor’s wealth over the long term.

A strength of the Kelly Criterion is its flexibility. It allows you to adjust the proportion of the capital that you bet based on how strong your edge is.

It’s important to note, however, that the Kelly Criterion assumes that the bettor can reinvest their winnings. This is crucial for the ‘compounding’ aspect of the strategy, which is what allows the wealth to grow faster than it would with other systems. This compound growth strategy is what differentiates Kelly betting from more static strategies, but also introduces higher volatility in the short term.

Ultimately, the Kelly Criterion offers a robust methodology for managing risk and maximising returns when the odds are in our favour. However, as with any strategy, understanding the core principles is just the beginning—it’s the accurate identification of the edge and the consistent application of the strategy that’s critical for long-term success. Misestimations can lead to over-betting and significant losses.

APPLYING THE KELLY CRITERION

The application of the Kelly Criterion can have profound implications for various fields beyond gambling, such as investing and trading. The crucial component is to understand that the Kelly Criterion isn’t just about betting when we have an edge; it’s about calculating the precise amount to bet to maximise compounded return over time.

This is where the Kelly formula can come into play:

F = Pw − (Pl/W)

where

F is the Kelly criterion fraction of capital to bet,

W is the amount won per amount wagered (i.e. win size, net of the stake, divided by loss size),

Pw is the probability of winning, and

Pl is the probability of losing.

When we apply this formula, we calculate the optimal fraction of our capital to bet, given our probability of winning (Pw), our probability of losing (Pl), and our potential return (W).

Consider a simple example: Suppose we have an even-money bet, i.e. the amount you stand to win, net of the stake, is the same as the amount you risk. In this scenario, the value of W is 1. If our chance of winning is 60% and our chance of losing is 40%, substituting these values into the simplified Kelly formula (F = Pw − Pl) gives us F = 0.60 − 0.40 = 0.20 or 20%. This means that in order to maximise our long-term return, we should bet 20% of our capital.

Let’s consider a slightly more complicated scenario: Suppose we have a bet where we stand to win double the amount we risk, i.e. W = 2, and the probability of winning and losing is both 50%. Substituting these values into the original Kelly formula gives us F = 0.50 − (0.50/2) = 0.50 − 0.25 = 0.25 or 25%. This means we should bet 25% of our working capital to maximise our long-term return.

The Kelly Criterion is designed to ensure that you never go bankrupt because the recommended bet size decreases as your capital decreases. However, this doesn’t mean you can’t lose money. The Kelly Criterion maximises long-term growth rather than short-term returns. This means that there will be times when you lose money, but over the long run, you should come out ahead.

It’s crucial to remember that the Kelly Criterion assumes you know the true probabilities of the outcomes, which is often not the case. In practice, we’re often working with estimated probabilities, which means there’s a risk that we could, for example, overestimate our edge and bet too much. Therefore, many investors and bettors use a fraction of the Kelly Criterion (betting a fixed fraction of the amount recommended by Kelly) to reduce their risk.

Lastly, while the Kelly Criterion offers a mathematical approach to betting and investing, it doesn’t account for the emotional aspect of risking money. Remember that the goal is not just to maximise returns, but also to sleep well at night.

POTENTIAL RISKS AND LIMITATIONS

While the Kelly Criterion can be an effective strategy for maximising the growth of capital in the long run, it is not without its potential risks and limitations. These should be understood and considered before applying the formula.

Estimation Errors

The effectiveness of the Kelly Criterion hinges on the accuracy of the probabilities used in the calculation. An overestimation of the probability of winning (Pw) can lead to excessive bet sizes and the risks associated with over-betting.

Minimum Bet Size

The Kelly Criterion presupposes that there is no minimum bet size, which is rarely the case in real-world scenarios, especially in investing and trading. In situations where a minimum bet size exists, the possibility of losing all of the capital becomes a reality if the amount falls below this threshold.

Risk Tolerance

The Kelly Criterion determines bet sizes purely based on mathematical calculations to maximise long-term growth. It does not take into account the individual bettor’s or investor’s risk tolerance. An aggressive bet size recommended by the Kelly Criterion may not be psychologically comfortable for some, causing stress and potentially leading to sub-optimal decisions.

Given these potential risks and limitations, it is common for many investors and bettors to use a fractional Kelly strategy, betting a fraction (like half or a third) of the amount recommended by the Kelly formula. This approach can help mitigate the risks associated with over-betting and inaccuracies in probability estimation while still providing the benefits of proportional betting and capital growth. However, even a fractional Kelly strategy should be tailored to individual circumstances, including risk tolerance and the ability to withstand potential losses.

CONCLUSION: TAKING ADVANTAGE OF OUR EDGE

The Kelly Criterion, devised by John L. Kelly, Jr., is a unique betting strategy that uses probability and potential payout to determine the optimal bet size when the odds are in our favour. The mathematical formula suggests betting a fraction of capital equivalent to the size of one’s advantage. However, it’s crucial to account for potential errors and uncertainties that can affect the real-world implementation of this strategy.

Uncertainty in the size of any actual edge at the odds and the potential for a bumpy ride due to volatility mean that we should always exercise caution. As a result, unless we’re prepared for potential high volatility and have unwavering confidence in our judgment, adopting a fractional Kelly strategy might be the most prudent approach. This strategy allows us to stake a defined fraction of the recommended Kelly amount, reducing risk while still taking advantage of our edge.