Staking
Flat vs Kelly: which staking plan copes better with error
Flat staking is the same amount on every bet, while Kelly is a percentage of the bankroll worked out from the estimate p and the odds. When the estimate is right, Kelly grows faster. When p is overestimated, the Kelly stake is too big: in the example at odds 2.10, when the true probability is 48% instead of 50%, full Kelly halves the bankroll in a typical run of 1,000 bets, while flat staking stays in profit.
Contents
Key terms
- Flat staking (level stakes)
- The same amount on every bet, for example 200, whether the bankroll is going up or down.
- Percentage of bankroll
- The stake is a fixed percentage of the current bankroll, for example 2%. After a loss the stake shrinks by itself.
- Kelly stake
- A percentage of the bankroll from the formula f = (p × odds − 1) / (odds − 1) (see Kelly criterion). It depends on the estimate p.
- Drawdown
- How far the bankroll has fallen from its highest point, for example to half.
- Median
- The typical result: half of all runs end above it and half below.
Three staking plans
- Flat: 200 per bet, which is 2% of a starting bankroll of 10,000.
- Percentage: 2% of the current bankroll.
- Kelly: full Kelly and half Kelly, worked out from the estimate p = 0.50 and odds of 2.10. Full Kelly is 4.55% and half Kelly 2.27%.
All three plans place the same bet at odds of 2.10. Only the true probability, which the bettor cannot see, is different.
What if the estimate is right, and what if it is not?
1,000 bets at odds 2.10, starting bankroll 10,000. The bettor estimates p = 0.50, and the true probability is 0.50, 0.48 or 0.46.
| Staking plan | True p = 0.50 | True p = 0.48 | True p = 0.46 |
|---|---|---|---|
| Flat 200 | 20,000 | 11,600 | 3,200 |
| 2% of current bankroll | 21,801 | 9,419 | 4,069 |
| Half Kelly (2.27%) | 23,433 | 9,029 | 3,479 |
| Full Kelly (4.55%) | 31,114 | 4,625 | 687 |
The medians are exact, with no simulation: at p = 0.50, 0.48 and 0.46 the median number of wins is exactly 500, 480 and 460.
When the estimate is right (p = 0.50), full Kelly ends highest. At p = 0.48 the bet still has a positive EV of +0.8%, but the true Kelly stake is only 0.73%. A stake of 4.55% is 6.25 times too big, so full Kelly halves the bankroll in a typical run. Flat staking still ends in profit, at 11,600.
The same goes for 2% of the current bankroll: at p = 0.48 that is 2.75 times the true Kelly stake, and anything above roughly double Kelly has negative growth. At p = 0.46 the EV is −3.4% and every plan loses. No staking plan turns a negative EV into a positive one.
How deep are the drawdowns?
Even with the right estimate, Kelly goes through deep drawdowns. An approximate formula for a continuous model (Thorp): the probability that the bankroll at some point falls to a fraction x of the start is x^(2/c − 1), where c is the fraction of the Kelly stake being bet.
| Plan | Chance the bankroll ever falls to half |
|---|---|
| Full Kelly (c = 1) | 50.0% |
| Half Kelly (c = 0.5) | 12.5% |
Half Kelly keeps 75% of the full Kelly growth rate and cuts the chance of halving the bankroll from 50% to 12.5%.
Pros and cons of each plan
- Flat staking: simple and the least sensitive to an error in p, because the stake size does not depend on the estimate. It does not follow the bankroll, so a long losing run can use it all up.
- Percentage of bankroll: after losses the stake shrinks by itself, so the bankroll rarely runs down to zero. When the percentage is more than double the true Kelly stake, the typical result goes down.
- Full Kelly: the fastest long-run growth, but only with the right p. It swings the most and loses the most when p is overestimated.
- Half Kelly: slower growth, much smaller drawdowns and more room for error in the estimate.
The same holds for every plan: the result depends on how accurate the estimate p is. The accuracy of the model's estimates is measured on the AI model performance page.
Limitations
- The example uses made-up probabilities and one set of odds for every bet. In reality odds and estimates change from bet to bet.
- A median does not show the spread: flat staking at p = 0.46 ends at 3,200 in a typical run, but many runs lose the whole bankroll before the end.
- The chances of falling to half are approximate (continuous model) and hold only when p is right.
- No staking plan changes the expected value of a bet. It only decides how much the bankroll swings along the way.
Related guides
Sources
- Thorp, E. O. (2006). The Kelly criterion in blackjack, sports betting and the stock market. Handbook of Asset and Liability Management, Vol. 1, 385–428.