Kelly Criterion Betting: How Much to Stake

A bettor with a real edge and a bad stake size still loses money. Kelly criterion betting is the standard answer to that problem. The formula takes the price you are getting and your own estimate of the chance, and returns the fraction of your bankroll to put behind the bet. The arithmetic is easy. The difficulty sits entirely in the number you feed it.

The formula, and the one input that matters

Kelly is written f = (bp - q) / b. Here b is the decimal odds minus one, so the profit per unit staked. p is your estimated chance of winning, and q is 1 minus p. The output f is the share of your bankroll to stake. Edward Thorp, who carried the idea out of Bell Labs into blackjack and then into markets, set out the case in The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market: over a long enough run, staking this way maximises the growth rate of the bankroll.

Notice that p is yours. The sportsbook has its own number built into the price, and if you feed the formula the book's implied probability it returns zero every time, because the vig pushes the implied chances across a market above 100% in total. Kelly only tells you to bet when your read differs from the market's, in your favour.

A worked example at +150

Say you have a $2,000 bankroll and a price of +150, which is 2.50 in decimal, so b = 1.5. You put the true chance at 45%.

• Break-even chance at 2.50 is 1 / 2.50 = 40%.
• Your edge is 5 percentage points.
• f = (1.5 x 0.45 - 0.55) / 1.5 = 0.125 / 1.5 = 0.083

Full Kelly says stake 8.3% of the bankroll, or $167, off a 5 point edge. That is a big bet by most standards. The formula is describing what the maximum growth rate costs you in volatility, and most people decline to pay it. Our Kelly calculator runs this and the fractional versions if you want to put your own numbers through it.

A three point error in your read is a 2.5x error in your stake

Change nothing but p. Suppose the true chance was 42% rather than 45%. You were three points optimistic, which is a small miss by the standards of hand-built probability estimates.

f = (1.5 x 0.42 - 0.58) / 1.5 = 0.05 / 1.5 = 0.033

The correct stake was 3.3% of bankroll, or $67. You bet $167. You overbet by a factor of two and a half, and you did it while feeling precise.

Push the error out to five points and it gets worse. At a true 40% you sit exactly at break-even, f = 0, and the right stake is nothing at all. The bet you sized at $167 had no edge in it.

That sensitivity is the real argument against full Kelly. It also explains why experienced bettors spend far more time on their probability estimates than on their staking plan. If you are unsure whether your reads actually beat the market, closing line value is the cheapest way to find out.

What half Kelly buys you

Staking a fraction of the Kelly amount is normal practice. Half Kelly on the example above is $83, and quarter Kelly is $42.

The trade is a good one. Under the standard continuous approximation, staking a fraction c of the Kelly amount gives you c(2 - c) of the optimal growth rate:

• Half Kelly: 0.5 x 1.5 = 0.75, so roughly three quarters of the growth for half the bet size.
• Quarter Kelly: 0.25 x 1.75 = 0.44, so roughly 44% of the growth for a quarter of the bet size.

Half Kelly surrenders about a quarter of the theoretical growth rate and halves every stake. Since your p is an estimate rather than a fact, the smaller number is frequently closer to correct anyway, as the 42% case showed. The Kelly criterion summary puts the practitioner's reasoning in the same terms: fractional Kelly reduces the chance of ruin, reduces volatility, and leaves room for model error.

Where Kelly stops being the right tool

Arbitrage changes the question. When you back both sides of a market at prices that already fix the return, the profit comes from the prices rather than from your estimate of anything. There is no p to be wrong about. Sizing is capped instead by what the books will accept and by how much you are willing to have tied up at once. The arbitrage calculator handles that case, and arbitrage betting 101 covers the mechanics.

The middle ground is where care is needed. If you are taking positive expected value bets off a model you have tested, Kelly is the right frame for sizing them. If you are taking them off a feeling, the formula will faithfully convert your overconfidence into an oversized bet.

The constraint the formula ignores

Kelly assumes you can stake whatever it tells you to. Sportsbooks disagree. Put 8% of a serious bankroll into the same soft market week after week and you will get limited, at which point the growth rate question turns academic. Most workable staking plans are Kelly capped by two other numbers: the book's maximum, and the largest bet you can place without attracting a trader's attention.

Bankroll management covers the rest of that frame, including how much of a bankroll you need before any of this is worth the effort.

Common questions

What Kelly fraction should I use?

Half Kelly is the common starting point, and quarter Kelly is normal for anyone whose probability estimates are new or untested. The less confidence you have in your p, the smaller the fraction should be. Full Kelly is only correct if your estimate is exactly right, which it almost never is.

Can I use Kelly without knowing my true win probability?

Not usefully. Kelly converts an edge into a stake, so with no reliable edge estimate it simply amplifies whatever error is in your read. Build the estimate first, check it against closing lines for a few hundred bets, then let the formula size things.

Does Kelly apply to arbitrage?

No. An arbitrage return is determined by the two prices you take, not by a probability you estimated, so there is no edge for the formula to work from. Stake sizing there is a question of book limits and how much capital you want in play at once.

Disclaimer: BetSuite is a data tool, not a sportsbook. All betting involves risk, and legality varies by state and province. Nothing here is a promise of profit. Please bet responsibly.