Kelly criterion betting sizes each bet as a share of your bankroll with one formula: stake = edge / (decimal odds - 1), where edge = your chance of winning x the odds - 1. An 8% edge at $2.16 suggests 0.08 / 1.16 = 6.9% of your bank.
Staked that way bet after bet, a bank grows as fast as it can, but only if the edge you feed the formula is real. Your edge is an estimate, and full Kelly punishes estimates that run high, which is why staking a half or a quarter of it is common.
The Kelly formula for betting
Three lines, in decimal odds:
- edge = your chance x decimal odds - 1, which is the same as offered price / your fair price - 1
- Kelly share = edge / (decimal odds - 1)
- Kelly stake = current bankroll x Kelly share
The edge is what $1 staked earns on average, and the odds minus 1 is what it wins when it wins. Dividing one by the other scales the stake to the price. The same edge earns a bigger share at a short price and a smaller one at a long price, where losing runs are longer.
In textbook notation it reads f = (b x p - q) / b, with b = odds - 1, p = your chance and q = 1 - p, which works out to the same sum.
John L. Kelly Jr. published the rule in 1956, in a Bell System Technical Journal paper called A New Interpretation of Information Rate. If your chance is right, it picks the stake that makes a bank grow fastest over many bets, which is also the stake that makes the typical, or median, bank largest. It does not make the average bank largest, and that difference is the whole case against staking more.
Risk: Betting involves risk. Fastest growth is a long-run result, and even with a correct chance a bank staked at full Kelly can fall a long way before it grows. See responsible gambling for limits and support.
Worked example: an 8% edge at $2.16
Example: Illustrative figures, no real match: you give a team a 50% chance, a bookmaker offers $2.16, and your bankroll is $2,400.
- edge = 0.50 x 2.16 - 1 = 0.08, or 8%
- Kelly share = 0.08 / (2.16 - 1) = 0.068966, or about 6.9% of the bank
- Kelly stake = $2,400 x 0.068966 = $165.52
Kelly re-sizes from the bank after every result:
| Result | Change | Bank after | Next Kelly stake |
|---|---|---|---|
| Win | +$165.52 x 1.16 = +$192.00 | $2,592.00 | $178.76 |
| Loss | -$165.52 | $2,234.48 | $154.10 |
A win multiplies the bank by 1 + 0.068966 x 1.16 = 1.08 and a loss by 1 - 0.068966 = 0.931034. One of each, in either order, gives 1.08 x 0.931034 = 1.0055, so the bank ends at $2,413.24.
The expected profit is $165.52 x 0.08 = $13.24 a bet. The typical bank grows by about half that, because wins and losses multiply rather than add: the square root of 1.0055172 is 1.0027548, a gain of 0.28% or $6.61 a bet. The Kelly criterion calculator runs these steps for any bank, price and chance.
Risk: Betting involves risk. At a true 50% chance this bet still loses half the time, and every loss cuts the Kelly stakes that follow. See responsible gambling for limits and support.
Why staking more than Kelly gives growth back
Take the same bet, a 50% chance at $2.16, made 1,000 times from a $2,400 bank at different shares (illustrative):
| Stake per bet | Times the Kelly share | Expected profit per bet, share of the bank | Typical (median) bank after 1,000 bets | Chance of finishing below $2,400 |
|---|---|---|---|---|
| 1.72% (quarter Kelly) | 0.25 | 0.14% | $8,012 | 2.0% |
| 3.45% (half Kelly) | 0.5 | 0.28% | $18,919 | 4.1% |
| 6.90% (Kelly) | 1 | 0.55% | $37,581 | 12.1% |
| 13.79% | 2 | 1.10% | $2,400 | 48.7% |
| 20.69% | 3 | 1.66% | $0.57 | 87.9% |
The median run has 500 winners and 500 losers, so the typical bank = $2,400 x ((1 + 1.16 x share) x (1 - share)) to the power 500. At the Kelly share that is $2,400 x 1.0055172 to the power 500 = $37,581. At twice the Kelly share it is (1 + 0.16) x (1 - 0.137931) = 1.0000, so the typical bank goes nowhere. Between once and twice the Kelly share the typical bank still grows, only more slowly; past twice it shrinks.
The last column counts the runs with too few winners, from the binomial distribution; at twice Kelly a further 2.5% finish exactly level.
Expected profit keeps rising with the stake, and so does the average bank across all possible runs. That average is $2,400 x (1 + 0.08 x share) to the power 1,000, about $32 billion at three times Kelly. It is carried by a handful of extreme winning streaks, while the typical run at that stake ends at 57 cents. Kelly is the stake that makes the typical result as large as it can be, and every stake above it gives growth back while adding risk.
Risk: Betting involves risk. These figures assume the 50% chance is exactly right and every bet is accepted at $2.16. With a wrong estimate a Kelly stake can shrink a bank, and there is no guarantee of profit. See responsible gambling for limits and support.
Why estimated edges make full Kelly too aggressive
Kelly treats your chance as exact, but yours is an estimate. The bets you actually place are not a fair sample of your estimates either: you bet only where an estimate shows an edge, so those bets lean towards estimates that came out too high.
Example: Illustrative assumptions. You rate runners that a bookmaker offers at $5.00, and each of your ratings is either 2 points too high or 2 points too low, equally often, so your ratings are right on average. There are equal numbers of two kinds of runner: those with a true 20% chance (no edge at $5.00) and those with a true 22% chance (a 10% edge).
| Runners at $5.00 | True chance | Your rating | Edge you see: rating x 5.00 - 1 | Kelly share you stake | True Kelly share |
|---|---|---|---|---|---|
| 20% runners, rated high | 20% | 22% | +10% | 2.5% | 0% |
| 20% runners, rated low | 20% | 18% | -10% | No bet | No bet |
| 22% runners, rated high | 22% | 24% | +20% | 5.0% | 2.5% |
| 22% runners, rated low | 22% | 20% | 0% | No bet | No bet |
Every bet you place comes from a rating 2 points too high. On those bets you see an average edge of 15% and stake an average of (2.5% + 5.0%) / 2 = 3.75% of the bank. The true edge averages 5%, and the true Kelly share (0% + 2.5%) / 2 = 1.25%. You stake three times the true Kelly share, the multiple that shrank the typical bank in the table above, without a single biased rating.
The same pull comes from a fair price built with some of the margin left in, a price that moves before your bet is accepted, and news you have not seen. Longer prices make it worse, because the error in the edge is the error in your chance multiplied by the price: a 2-point rating error at $5.00 is a 10-point error in the edge.
Staking less than the full Kelly share
Staking a fixed fraction of the Kelly share, usually a half or a quarter, is the common answer to estimates that run high. In the example above, half Kelly would stake 1.875% on average, still 1.5 times the true Kelly share, and quarter Kelly 0.94%, three quarters of it.
Below the true share you give up some growth; above it you give up growth and add risk, so erring low costs less. Fractional Kelly works out how much growth each fraction keeps and the swings that come with it.
How to use the Kelly criterion, step by step
With illustrative numbers: a $1,800 bank, and a team you rate at 35% that a bookmaker prices at $3.10.
- Set the bank: money you can afford to lose, kept apart from everything else, as bankroll management explains.
- Write down your chance before you look at the prices: 35%, a fair price of 1 / 0.35 = $2.86.
- Work out the edge: 0.35 x 3.10 - 1 = 0.085, or 8.5%. If the edge is zero or negative, stop here: there is no bet.
- Work out the Kelly share: 0.085 / (3.10 - 1) = 0.0405, or 4.05%.
- Apply your fraction: a quarter of 4.05% is 1.01%.
- Size the stake from the current bank: $1,800 x 0.010119 = $18.21, then round down to $18 and cap it at your own maximum and at whatever the bookmaker accepts.
- Record the price, the stake, your fair price and the closing price, and re-size from the new bank once the bet settles.
The closing price is the check on step 2. If your prices keep beating it, your estimates are probably sound; if they do not, the edge you are sizing may not be there, and closing line value explains the test.
Note: Kelly is maths, not a product feature. B337 does not size bets with Kelly: a Kelly stake is a sum you work out yourself, by hand or with the calculator.
When the Kelly formula does not fit
Kelly needs a chance of your own, which comes from the price work at the heart of a betting strategy. Some bets do not suit it at all:
| Situation | The problem | What to do instead |
|---|---|---|
| You have no chance of your own, only the price | The price's own chance gives an edge of zero, so Kelly stakes nothing | Small level stakes, until you have estimates you can test |
| The bookmaker accepts less than the Kelly stake | The bet goes on smaller than the formula says | Bet what is accepted; never use another person's account or open a new one to place the rest |
| Several bets are open at once | Each single share assumes nothing else is riding | Size them together, as fractional Kelly shows |
| A bonus bet | The stake is not your money, so a loss costs your bank nothing | Value it by what it converts to, with the bonus bet converter |
| A tote bet | The dividend is not known until after the result, and your own money lowers it | Treat any approximate as rough, and stake a small fraction |
Kelly errors that change the stake
| Mistake | What it does to the stake |
|---|---|
| Sizing from the starting bank | After the $2,400 bank falls to $1,800, Kelly on the $2.16 bet is $124.14, not $165.52: a third too much |
| Topping the bank back up to keep stakes up | It undoes the one thing Kelly does after a loss, which is cut the stake, and it is chasing losses |
| Nudging the price's own chance up by feel | At $4.50 the price implies 22.2%; calling it 25% creates a 12.5% edge and a 3.57% Kelly share from nothing measured |
| Adding single Kelly shares for runners in the same race | The bets are linked, since a win for one is a loss for the other, so their stakes have to be worked out together |
| Rounding up | The $18.21 quarter-Kelly stake rounded up to $25 is 1.39% of the bank, about 1.4 times the quarter-Kelly share |