The stake that maximises long-run growth when you have an edge — and zero when you do not, which is every bet on a casino table.
The Kelly criterion gives the fraction of a bankroll to stake so that the bankroll grows fastest over many repetitions of the same bet. The formula is f = (bp − q) / b, where b is the payout to one, p is the chance of winning and q is the chance of losing. It is not a system for winning — it is a sizing rule that presupposes you are already winning.
That presupposition is the whole point on a casino site. Every bet on a table has a negative expectation, so bp is smaller than q, f comes out negative, and the correct stake is nothing. The calculator returns zero rather than a small positive number, because there is no stake size that makes a losing bet profitable.
Full-pay Deuces Wild returns slightly over 100% with flawless play. A bonus whose expected value after the wagering requirement exceeds its cost has a positive edge for as long as it lasts. Advantage play in sports betting or a mispriced market has one by definition. In those cases Kelly is the right tool, and the input that matters is an honest estimate of the edge.
Card counting is the classic textbook case and largely a historical one online, where continuous shuffling removes the dependency the count relies on.
Kelly is optimal only if your probability estimate is exactly right. Overstate your edge and full Kelly overbets sharply — staking at twice the correct fraction eliminates the growth advantage entirely, and beyond that the bankroll trends to zero even with a genuine edge.
Half Kelly gives up about a quarter of the growth rate and roughly halves the volatility, which is why practitioners who actually have an edge use a fraction rather than the full figure. Quarter Kelly is common where the edge estimate is soft.
Kelly maximises long-run growth given a known edge. The words doing the work are "known" — the formula is exquisitely sensitive to your probability estimate, and overestimating an edge by a little produces a stake that is too large by a lot.
Half or quarter Kelly gives up a modest amount of theoretical growth for a large reduction in drawdown, which is why practitioners with real money use fractions. Full Kelly also produces drawdowns most people cannot sit through, and a strategy you abandon mid-drawdown is not the strategy you modelled.
A formula for the fraction of a bankroll to stake, f = (bp − q) / b, that maximises the long-run growth rate of that bankroll when the bet has a positive expectation.
Only where a genuine edge exists — a positive-return paytable or a bonus worth more than it costs. On ordinary casino bets the formula returns zero, and that is the correct answer.
Staking half the Kelly fraction. It sacrifices about a quarter of the growth rate for roughly half the volatility, and protects against overestimating your edge.
Growth falls. At twice the Kelly fraction the advantage disappears entirely, and above that a bankroll trends toward zero even with a real edge.
No. Kelly sizes a bet that already has positive expectation. No staking pattern creates one, so on a negative-expectation bet Kelly and Martingale disagree about everything.