The Kelly Criterion Applied to Sports Betting

How much should you bet when a betting opportunity appears particularly attractive?

Always 1% of your bankroll?

2%?

More if your Edge is significant?

This question lies at the heart of the Kelly Criterion, one of the best-known mathematical methods for determining optimal Stake Size.

Unlike Flat Betting, where each bet generally receives the same unit, the Kelly method adjusts the Stake according to two essential factors:

the available odds

and:

the bettor's estimated Edge.

In theory, the greater your advantage, the larger the fraction of your bankroll you can allocate to the bet.

But this method comes with one major difficulty.

To use Kelly correctly, you need to estimate with sufficient accuracy:

the true probability of an event occurring.

In sports betting, however, this probability is never known with certainty.

This is precisely what makes the Kelly Criterion both extremely interesting and potentially dangerous when used incorrectly.

As in the previous articles in this series, we will use the following reference:

Initial Bankroll: €5,000

1U = €50

or:

1% of the initial bankroll

What Is the Kelly Criterion?

The Kelly Criterion is a mathematical method used to determine what proportion of your capital should be wagered when you believe you have a statistical advantage.

Its objective is not simply to maximize the expected profit from your next bet.

Its objective is to:

maximize the long-term logarithmic growth of your capital.

In other words, Kelly seeks a balance between:

exploiting your Edge sufficiently

and:

avoiding risking an excessive proportion of your capital.

If the estimated Edge is small, Kelly recommends a small Stake.

If the estimated Edge is significant, Kelly may recommend a larger Stake.

And if no positive Edge is identified:

Kelly recommends not betting at all.

Where Does the Kelly Criterion Come From?

The Kelly Criterion is named after John L. Kelly Jr., a researcher at Bell Labs.

In 1956, he published a paper entitled A New Interpretation of Information Rate.

His work was not originally about sports betting.

The principle concerned the optimal growth of capital in repeated situations involving uncertainty.

The concept was later applied to several areas, including:

• gambling;
• betting;
• investing;
• portfolio management.

Today, the Kelly Criterion is particularly well known in quantitative circles and among bettors who want to mathematically connect their Stake to their Edge.

What Is the Kelly Criterion Formula?

In its classic form for a simple bet:

f = (bp - q) / b*

where:

f* = fraction of the bankroll to bet

b = net profit represented by decimal odds minus 1

p = estimated probability of winning

q = probability of losing, or 1 - p

Let's take an example.

You estimate that a team has:

a 55% probability of winning

The available odds are:

2.00

Therefore:

b = 2.00 - 1 = 1

p = 0.55

q = 0.45

The formula becomes:

f = (1 × 0.55 - 0.45) / 1*

Therefore:

f = 0.10*

Kelly theoretically recommends betting:

10% of the bankroll.

With a €5,000 bankroll:

Kelly Stake = €500

This is considerably more than our usual reference of:

1U = €50

Why Does Kelly Recommend 10% in This Example?

Because if your 55% estimate is perfectly accurate, odds of 2.00 represent a significant advantage.

The raw implied probability of odds of 2.00 is:

50%

Your estimate is:

55%

You therefore believe you have an advantage of 5 percentage points.

The Expected Value of the bet is:

(0.55 × 2.00) - 1 = +0.10

or:

+10% EV

If this estimate is genuinely accurate and repeatable, Kelly considers that this advantage justifies relatively significant exposure.

But the entire problem lies in this sentence:

if this estimate is genuinely accurate.

The Kelly Criterion Depends Entirely on Your Probability Estimate

Kelly does not know the true probability of winning.

It uses the probability you provide.

This is probably the most important point to understand.

Let's continue using odds of:

2.00

If you estimate the probability at 55%

Full Kelly:

10%

If the true probability is only 53%

Kelly:

6%

If it is 51%

Kelly:

2%

If it is 50%

Kelly:

0%

If it is 49%

Kelly becomes negative.

That means:

no bet.

A difference of only a few percentage points in your estimate can therefore completely change the recommended Stake.

Estimation Error Is the Main Danger of Kelly

Imagine that your model estimates:

55%

and you apply Full Kelly.

You therefore bet:

10% of your bankroll.

But suppose your model slightly overestimates the quality of the bet and the true probability is:

51%

The theoretically appropriate Kelly Stake for that probability would be only:

2%

You have therefore wagered five times more than Kelly would have recommended using the true probability.

And if the real probability were:

49%

you did not even have a positive Edge.

You have then committed 10% of your bankroll to a bet that actually had negative Expected Value.

This is why Kelly should never be viewed as a magic formula.

The quality of its output depends directly on the quality of its input.

Kelly Does Not Create an Edge

This distinction is fundamental.

The Kelly Criterion does not identify Value Bets.

It does not determine the true probability of an event.

It only answers the question:

"If my probability estimate is correct, what fraction of my bankroll should I theoretically bet?"

The Edge must therefore exist before Kelly is used.

A poor strategy combined with an excellent Stake Sizing system remains a poor strategy.

Kelly can optimize how an advantage is exploited.

It cannot create that advantage.

Example With Our €5,000 Bankroll

Let's consider several bets at odds of:

2.00

with different estimated probabilities.

This table immediately highlights an important characteristic of Kelly:

the Stake increases rapidly as the estimated Edge increases.

But it also means that any overestimation of your Edge can lead to an excessively large Stake.

Kelly vs Flat Betting: Two Different Philosophies

Flat Betting generally consists of using the same unit for every selection.

For example:

1U = €50

Whether your estimated Edge is small or large, your Stake remains the same.

Kelly works differently.

It attempts to adjust the size of each bet according to the estimated advantage.

Flat Betting

Advantages:

• simple;
• transparent;
• easy to follow;
• limits Stake Sizing errors;
• makes performance easier to compare;
• reduces the impact of errors in Edge estimation.

Disadvantage:

• does not differentiate opportunities according to their estimated Edge.

Kelly Criterion

Advantages:

• adjusts Stake according to Edge;
• provides a mathematical framework for capital growth;
• reduces the Stake when the advantage is small;
• recommends no bet when the Edge is zero or negative.

Main disadvantage:

• highly dependent on the accuracy of estimated probabilities.

The two approaches therefore serve different purposes.

Full Kelly: Why Can It Be So Aggressive?

Full Kelly means directly applying the fraction recommended by the formula.

If Kelly indicates:

8%

you bet:

8% of the bankroll.

If Kelly indicates:

12%

you bet:

12%.

Mathematically, this approach can make sense when the probabilities used are perfectly known.

But in sports betting, they almost never are.

A Stake of:

10%

means that only a few poor results can cause major fluctuations in your bankroll.

Even with a genuine positive Edge, the volatility can become psychologically and financially difficult to tolerate.

What Is Half Kelly?

One solution is to use only a fraction of the Stake calculated by Kelly.

Half Kelly means betting:

50% of the Full Kelly amount.

If Full Kelly recommends:

10%

Half Kelly recommends:

5%

With a €5,000 bankroll:

Full Kelly:

€500

Half Kelly:

€250

The objective is to preserve some of Kelly's logic while reducing:

• volatility;
• Drawdown;
• the impact of estimation errors;
• Risk of Ruin.

What Is Quarter Kelly?

Quarter Kelly uses:

25% of the Full Kelly Stake.

If Full Kelly indicates:

10%

Quarter Kelly becomes:

2.5%

With €5,000:

€125

We can therefore compare:

Fractional Kelly therefore makes it possible to reduce exposure considerably.

Why Use Fractional Kelly?

The main reason is uncertainty.

In a theoretical problem where the true probability is perfectly known, Full Kelly has particularly interesting mathematical properties.

But in sports betting:

p is estimated.

And that estimate can be wrong.

Using a fraction of Kelly therefore introduces a form of safety margin.

The greater the uncertainty surrounding your Edge, the more rational it may be to use a conservative Kelly fraction.

Kelly Is Extremely Sensitive to Overconfidence

Overconfidence is a major risk.

Suppose your model regularly assigns probabilities such as:

57%

61%

64%

But if your model is poorly calibrated, these probabilities may be too high.

An event predicted at 60% might actually occur only 54% of the time.

The difference may appear relatively small.

For Kelly, it can be substantial.

The question is therefore not only:

"Does my model identify winners?"

You also need to ask:

"Are the probabilities produced by my model properly calibrated?"

Probability Calibration Is Essential

A well-calibrated model means, in simplified terms, that:

among events to which it assigns approximately:

60% probability

around:

60% should actually occur

over a sufficiently large sample.

The same logic applies to:

55%

70%

80%

and so on.

To use Kelly seriously, it is therefore not enough to produce a ranking or a signal.

Ideally, you need:

sufficiently reliable and calibrated probabilities.

That is a much higher requirement.

Can You Use the Bookmaker's Implied Probabilities With Kelly?

Not directly to create an Edge.

Odds represent an implied market probability, but they generally include a margin.

For example, odds of:

2.00

mathematically correspond to:

50%

But if you simply use that same probability in Kelly:

p = 50%

with odds of:

2.00

Kelly recommends:

0%

Which makes sense.

You have identified no advantage.

To use Kelly, you therefore need a probability estimate that differs from, and is sufficiently higher than, the probability required to justify the available odds.

The Bookmaker Margin Must Also Be Considered

In a market with several outcomes, adding together the raw implied probabilities from all available odds generally produces a total above 100%.

The difference represents the Overround, or theoretical bookmaker margin.

Before comparing your model with the market, it can therefore be useful to calculate market probabilities adjusted for the margin.

This provides a better representation of the probability implied by the market.

But even after this adjustment:

the market probability is not automatically the true probability.

It remains a benchmark.

What Role Can Pinnacle Play in a Kelly Approach?

In a quantitative approach, odds from a low-margin bookmaker with significant market exposure can provide a particularly useful benchmark.

Opening Odds can help show the initial state of the market.

Closing Odds reflect the market after a much larger amount of information and betting activity has been incorporated.

This does not mean that the Closing Odd represents a perfect probability.

But it can provide a particularly useful reference for analyzing:

• your entry price;
• your CLV;
• the quality of your estimate;
• the ability of a strategy to identify Value.

Can CLV and Kelly Be Complementary?

Yes, conceptually.

Kelly uses your estimated Edge to determine the Stake.

CLV can then provide additional information about the quality of the price you obtained.

Suppose your model regularly recommends large Stakes because it detects substantial Edge.

But those selections consistently generate negative CLV.

That could be an important signal.

Your model may be estimating its Edge too aggressively.

Conversely, if selections consistently generate positive CLV, that may strengthen the case for analyzing the quality of the process more deeply.

CLV alone does not prove that your exact probability estimate is correct.

But it can provide a complementary indicator.

Kelly and Sample Size

A strategy shows:

+15% ROI after 60 bets.

Can we conclude that it has a sufficiently large Edge to justify aggressive Kelly staking?

No.

The Sample Size is too small to draw a strong conclusion about the exact size of the Edge.

This issue is particularly important with Kelly.

Why?

Because the Stake depends directly on the estimated Edge.

If you overestimate the Edge based on a small sample, you may also overestimate the appropriate Stake.

Statistical uncertainty therefore needs to be part of the analysis.

Kelly and Variance

Kelly obviously does not eliminate Variance.

Even with a perfect calculation, a sequence of losses remains possible.

Suppose your model correctly identifies a series of Value Bets.

That does not mean the next selections will win.

Kelly seeks to optimize long-term growth.

It does not attempt to make short-term results smooth or consistent.

This distinction is essential.

Kelly and Drawdown

The larger the Stakes, the larger the fluctuations in bankroll can become.

Full Kelly can therefore produce Drawdowns that are difficult to tolerate.

Even if the strategy remains theoretically valid, a bettor may abandon it after a:

-20%

-30%

or:

-40%

Drawdown.

There is therefore a difference between:

mathematical optimality

and:

actual risk tolerance.

A strategy that is psychologically impossible to follow is not necessarily appropriate for the user.

Kelly and Risk of Ruin

The Kelly Criterion is directly connected to Risk of Ruin management.

With perfectly known probabilities and under the appropriate theoretical assumptions, Kelly seeks to avoid overbetting while maximizing long-term logarithmic growth.

But in the real world, the central problem remains:

estimation error.

If you believe you have a 10% Edge when you actually have only 2%, your Stake can be far too large.

And if your Edge is actually negative, Kelly calculated from an incorrect estimate can accelerate losses rather than protect your bankroll.

The Danger of Overbetting

Overbetting means wagering more than your advantage justifies.

It is one of the major risks in bankroll management.

Suppose the correct theoretical Kelly is:

2%

but you bet:

8%

You are committing four times more capital than your true Edge would justify.

Even with a winning strategy, this can significantly worsen your growth profile and increase volatility.

Having an Edge does not justify any Stake size.

Underbetting: Can You Bet Too Little?

Yes, from the perspective of theoretically maximizing growth.

If the true optimal Kelly is:

5%

and you bet:

0.5%

you are exploiting only a small part of your advantage.

Your Risk of Ruin and volatility will be lower, but your potential growth will also be slower.

This is the fundamental trade-off:

growth versus risk.

In practice, however, underbetting may be intentional when greater importance is placed on stability and capital preservation.

Why Betting Above Kelly Is Particularly Dangerous

There is an important difference between betting less than Kelly and betting more than Kelly.

Betting less generally reduces theoretical growth, but it also reduces volatility.

Betting more increases risk.

As you move significantly above the optimal Kelly fraction, the risk-return profile deteriorates.

The Kelly Criterion can therefore also be used as a framework for understanding:

when a Stake becomes too aggressive.

Should Kelly Be Recalculated After Every Bet?

In its dynamic form, Kelly uses the current bankroll.

Suppose:

Initial Bankroll: €5,000

Kelly recommends:

2%

Stake:

€100

If the bankroll subsequently increases to:

€5,500

2% becomes:

€110

If it falls to:

€4,000

2% becomes:

€80

The Stake therefore changes with the capital.

This makes it possible to:

gradually increase Stakes as the bankroll grows

and:

reduce them when it declines.

Kelly Is Therefore a Proportional Staking Method

Unlike strict Flat Betting with a fixed monetary amount, Kelly naturally operates as a proportional staking method.

The Stake changes according to:

the current bankroll

and:

the estimated Edge.

Two variables can therefore change the amount wagered.

This flexibility is a strength.

But it also makes the method more complex.

What Happens When Several Bets Are Available at the Same Time?

Simple Kelly is particularly easy to understand when considering one isolated bet.

The situation becomes more complex when several bets are open simultaneously.

Suppose your model identifies five opportunities and recommends:

4%

3%

5%

2%

4%

If each Stake is applied independently, your total exposure can reach:

18% of the bankroll.

And if the bets are correlated, the real risk can be even greater.

Portfolio-level risk management then becomes necessary.

Correlation Is an Important Issue With Kelly

Two bets are not necessarily independent.

Imagine several selections linked to:

• the same match;
• the same team;
• the same league;
• the same market conditions;
• the same statistical model.

If those bets are strongly correlated, treating them as completely independent opportunities can create excessive exposure.

Kelly applied to an entire portfolio is mathematically more complex than simple Kelly applied bet by bet.

Should You Set a Maximum Stake?

In practical use, setting a cap can be relevant.

For example:

Maximum Stake = 2%

or:

Maximum Stake = 3%

even when the Kelly calculation recommends more.

Why?

Because this limit can protect against:

• model errors;
• poorly calibrated probabilities;
• incorrect data;
• exceptional events;
• undetected correlations;
• Overconfidence.

A Stake Cap can therefore act as an additional layer of risk management.

Example of Kelly With a Stake Cap

Suppose your bankroll is:

€5,000

and your maximum Stake is:

2%

or:

€100

Your model produces the following recommendations:

You preserve Kelly's logic for smaller Edges while preventing the most aggressive estimates from generating very large Stakes.

Can Kelly Be Combined With Units?

Yes.

The two concepts are compatible.

If:

1U = 1% of the bankroll

then a Kelly recommendation of:

0.5% = 0.5U

1% = 1U

1.5% = 1.5U

2% = 2U

and so on.

This makes it possible to maintain performance reporting in Units while adapting the Stake to the Edge.

But you must then accept that results are influenced by both:

the quality of the selections

and:

the quality of the Stake Sizing.

Why Does the Flat Stakes Summary (1U) Remain Essential?

Imagine a strategy using Kelly.

Its actual results are excellent.

But what caused that performance?

Were the selections genuinely strong?

Or did the Stake Sizing system simply allocate larger Stakes to the winning bets in the observed sample?

The Flat Stakes Summary (1U) recalculates performance assuming:

1U on every selection.

This helps isolate the quality of the selections themselves.

Comparing:

actual results using Kelly

with:

Flat Stakes Summary (1U)

can therefore provide particularly useful information.

Kelly Should Not Hide a Poor Strategy

Suppose:

Actual performance with variable Stakes: +25U

but:

Flat Stakes Summary (1U): -4U

This situation deserves careful analysis.

The positive result may be heavily dependent on Stake Sizing.

Conversely:

Actual performance: +30U

and:

Flat Stakes Summary (1U): +24U

suggest that the quality of the selections is also making a significant contribution.

It is therefore important to distinguish between:

Selection Edge

and:

Sizing Edge.

Can You Apply Kelly Without a Probability Model?

It is difficult to do so rigorously.

To calculate Kelly, you need an estimate of:

p

meaning the estimated true probability of the event.

Simply saying:

"I really like this bet"

or:

"This is my best bet of the day"

does not provide a probability that can be used mathematically.

A serious Kelly approach therefore ideally requires:

• a probability model;
• historical data;
• a reproducible methodology;
• probability calibration;
• validation on a sufficiently large sample.

Kelly Is Particularly Suited to Quantitative Approaches

The Kelly Criterion makes the most sense when a strategy can produce a probability estimate for each opportunity.

For example:

Model Probability: 54%

Adjusted Market Implied Probability: 50%

Estimated Edge: positive

Calculated Kelly: X%

The process then becomes systematic.

It is much harder to mathematically justify Kelly when probabilities are mainly based on subjective judgment.

The Strategy Builder and the Logic of Edge

In a data-driven approach, the objective is to identify historical situations with potentially repeatable characteristics.

A strategy can notably be analyzed through:

• ROI;
• Sample Size;
• Drawdown;
• Odds;
• CLV;
• consistency;
• behavior across different periods.

These elements help evaluate the robustness of a potential Edge.

But even a historically successful strategy can never provide certainty about the exact probability of the next bet.

This is why Stake Sizing must always account for uncertainty.

Kelly and Historical Data: Beware of Overfitting

A model can perform extremely well on historical data while failing on new data.

This is the problem of Overfitting.

If a system is too heavily optimized on past results, it may produce an exaggerated estimate of its Edge.

Applying aggressive Kelly staking can then amplify the problem.

You combine:

an overestimated Edge

with:

an excessive Stake.

The result can be particularly dangerous.

Backtesting and Kelly: A Combination to Use Carefully

An excellent Backtest does not guarantee identical future performance.

You should notably examine:

• Sample Size;
• stability of results;
• different time periods;
• different markets;
• data quality;
• Overfitting risk;
• Maximum Drawdown;
• CLV when available.

Kelly calculated from an overly optimistic Backtest can lead to significant Overbetting.

A Conservative Approach to Kelly

A cautious application can involve several layers of protection.

For example:

1. Estimate the Edge.
2. Verify the quality and calibration of the model.
3. Calculate Full Kelly.
4. Use only a fraction of Kelly.
5. Apply a Stake Cap if necessary.
6. Control total exposure.
7. Account for correlations.
8. Regularly reassess probability estimates.
9. Compare results with the Flat Stakes Summary (1U).
10. Monitor CLV and Drawdown.

The objective is to prevent a single overly optimistic estimate from putting a significant part of the bankroll at risk.

Example of a Conservative Kelly Approach

Suppose:

Bankroll: €5,000

Your model estimates:

Probability: 55%

Odds:

2.00

Full Kelly:

10%

or:

€500

But you decide to use:

Quarter Kelly

The Stake becomes:

2.5%

or:

€125

You also apply a Stake Cap of:

2%

The final Stake therefore becomes:

€100

You have transformed a theoretical recommendation of:

€500

into actual exposure of:

€100

You are still exploiting your estimated Edge, but with a much larger safety margin.

The Optimal Kelly Is Not Necessarily the Right Kelly for Every Bettor

Mathematical optimality is only one dimension of the problem.

A bettor must also consider:

• tolerance for Drawdown;
• time horizon;
• model reliability;
• confidence in the estimated Edge;
• strategy Variance;
• number of simultaneous bets;
• correlation;
• Liquidity constraints.

The theoretically optimal formula is therefore not necessarily the most appropriate practical configuration for every user.

Which Indicators Should You Analyze Before Using Kelly?

Before seriously applying the Kelly Criterion, it is useful to analyze:

Sample Size

ROI

CLV

Maximum Drawdown

Variance

Average Odds

Hit Rate

Liquidity

Flat Stakes Summary (1U)

Probability Calibration

Correlation

Simultaneous Exposure

Model Stability Over Time

The greater the uncertainty surrounding these factors, the stronger the case for a conservative approach.

Mistakes to Avoid With the Kelly Criterion

Several mistakes can make Kelly dangerous:

1. Overestimating your Edge.
2. Using subjective probabilities without calibration.
3. Applying Full Kelly without accounting for uncertainty.
4. Ignoring Variance.
5. Ignoring Maximum Drawdown.
6. Neglecting correlations between bets.
7. Accumulating several Kelly Stakes simultaneously.
8. Using an overfitted Backtest.
9. Changing probabilities emotionally.
10. Confusing past performance with future probability.
11. Failing to set exposure limits.
12. Believing Kelly can turn a strategy with no Edge into a winning strategy.

Flat Betting or Kelly: Which One Should You Choose?

There is no universal answer.

Flat Betting prioritizes:

simplicity

transparency

comparability

and:

control over the risk created by estimation errors.

Kelly prioritizes:

adapting the Stake to the estimated Edge

and:

the theoretical optimization of capital growth.

But it requires much greater precision.

For a bettor who cannot estimate probabilities reliably:

Flat Betting may be much more robust.

For a quantitative approach with sufficiently reliable and calibrated probability estimates:

Kelly can become a particularly useful Stake Sizing tool.

Conclusion

The Kelly Criterion is one of the most elegant methods for mathematically connecting:

Edge

Odds

Bankroll

and:

Stake.

Its principle is powerful.

The greater your estimated advantage, the larger the proportion of capital you can theoretically allocate.

When the Edge disappears:

the Stake should disappear as well.

But Kelly's strength is also its main weakness.

The formula depends directly on:

the quality of your probability estimate.

If your probability estimate is accurate, Kelly can provide a particularly interesting framework for optimizing capital growth.

If your Edge is overestimated, the Stake can become excessive.

This is why, in real-world sports betting, approaches such as:

Half Kelly

Quarter Kelly

or:

Kelly combined with a Stake Cap

can be used to introduce an additional margin of safety.

With our reference bankroll of:

€5,000

and:

1U = €50 = 1%

Kelly should not be viewed simply as a method for betting more when you feel confident.

It is a mathematical Stake Sizing system that requires rigorous probability estimates.

The fundamental question is therefore not:

"How confident am I in this bet?"

but:

"What is my best estimate of the true probability, how uncertain is that estimate, and what fraction of my bankroll is rationally justified by that Edge?"

That distinction separates a quantitative application of Kelly from simply varying Stakes subjectively.

FAQ: Kelly Criterion and Sports Betting

What Is the Kelly Criterion?

The Kelly Criterion is a formula used to determine the theoretical fraction of a bankroll to wager based on the available odds and estimated Edge.

What Is the Kelly Criterion Formula?

In its classic form:

f = (bp - q) / b*

where b represents the net profit associated with the odds, p is the estimated probability of winning and q = 1 - p.

What Does Full Kelly Mean?

Full Kelly means betting exactly the fraction of the bankroll recommended by the Kelly formula.

What Is Half Kelly?

Half Kelly means betting 50% of the Stake recommended by Full Kelly.

What Is Quarter Kelly?

Quarter Kelly means betting 25% of the Stake recommended by Full Kelly.

Why Can Kelly Be Dangerous?

Because the formula depends directly on your probability estimate. If the Edge is overestimated, the recommended Stake can become far too large.

Can Kelly Create an Edge?

No. Kelly sizes a bet based on an estimated Edge. It does not create that Edge.

Can You Use Kelly With a 100U Bankroll?

Yes. If 1U represents 1% of the bankroll, a 2% Kelly recommendation corresponds to 2U, for example.

Does Kelly Eliminate Risk of Ruin?

No. In practice, probabilities and Edge are estimated with uncertainty. An inaccurate estimate can lead to excessive Stakes and increased risk.

Flat Betting or Kelly?

Flat Betting is simpler and less dependent on precise probability estimates. Kelly adjusts Stake according to estimated Edge but requires much more robust probability estimates.