Kelly criterion
How much of the bankroll to put behind an edge, what that costs you in growth if you size it differently, and how deep a hole it can dig on the way.
Above the 50.0% these odds imply, so there is an edge.
A win returns your stake plus 100 per 100 staked.
Share of the stake you give up. 100% is an ordinary bet.
Only used to turn the fraction into an amount.
Most people who use this bet a fraction of it.
A 5.00% stake grows the bankroll +0.38% per bet on average, which is 75% of the fastest rate available. Staking more than 19.9% turns that negative: every bet is still worth taking on expected value, but the bankroll shrinks anyway, because a loss takes away a larger share than the same-sized win puts back.
| Fraction of Kelly | Stake | Growth per bet | Of the best rate | Ever down 50% |
|---|---|---|---|---|
| Full | 10.00% · 1,000 | +0.50% | 100% | 50.0% |
| Three quarter | 7.50% · 750 | +0.47% | 94% | 31.5% |
| Half | 5.00% · 500 | +0.38% | 75% | 12.5% |
| Quarter | 2.50% · 250 | +0.22% | 44% | 0.8% |
| Tenth | 1.00% · 100 | +0.10% | 19% | 0.0% |
What a bad run looks like
Betting 50% of Kelly, there is a 12.5% chance the bankroll is down 50% at some point along the way, even while it grows in the long run. Full Kelly has a coin flip's chance of halving at some point, which is why almost nobody bets it straight.
Kelly maximises the expected log of the bankroll rather than the bankroll itself: g(f) = p·ln(1 + f·b) + q·ln(1 − f·a), which peaks at f* = p/a − q/b. With a full loss on the downside that is the familiar (p·b − q)/b. Maximising expected money instead would tell you to stake everything on any favourable bet, which goes to zero almost surely.
The drawdown numbers come from the diffusion approximation, where the chance of ever falling to a level α while betting a multiple k of Kelly is α^((2−k)/k). It holds when each bet is small relative to the bankroll, and gets optimistic when it is not.
The real weak point is not the maths, it is the first input. Kelly assumes you know the win probability, and a confident estimate that is a few points too high will happily hand you a stake well past the ruin line. Betting a fraction of Kelly is mostly insurance against your own estimate.
Trying to size the confidence in a number rather than a bet? The significance tool gives you the posterior on a conversion rate.