The Kelly Criterion: How Much to Stake (and Why Fractional)

You found an edge. Now the hard part: how much do you bet? Bet too little and you barely grow. Bet too much and one cold streak ends you. The Kelly criterion answers the question exactly — and then quietly hands you a loaded gun.

The formula

Kelly picks the one stake that grows your bankroll fastest over the long run. For a bet that wins with probability and pays decimal profit per dollar risked:

That fraction of your bankroll is the stake. Bet more and growth stalls while the swings turn violent. Bet less and you leave money on the table. It’s the rare formula that’s both provably optimal and genuinely useful.

Why full Kelly will wreck you

Here’s the catch, and it’s a big one. Kelly is optimal only if is exactly right — and yours never is. Every edge you estimate is a guess with error bars.

And Kelly doesn’t punish symmetrically. Overbet by some amount and the damage far outweighs the reward of underbetting by that same amount. Feed the formula an edge you’ve overestimated and full Kelly marches you straight into the drawdown that ends most bankrolls. The math is right; it just assumes a certainty you don’t have.

Fractional Kelly: the fix

The fix is boring, and it works: bet a fixed fraction of the Kelly stake — a half (the default here), or a quarter if your edge is especially uncertain. -Kelly keeps most of the long-run growth while slashing volatility and drawdowns, and it buys a margin of safety against your own bad estimates. When the edge is a guess — and it always is — size down.

Betting several games at once? Naive per-bet Kelly overbets, badly, especially when outcomes are correlated (think two bets in the same game). Size them together, not one at a time. More on that below.

Where the formula comes from

No rulebook, no magic — just one page of calculus anyone can follow. Here’s the whole argument, the way it would go up on a board.

The shape of the answer

Growth as a function of stake size is a hill with a single peak, and that peak is . Walk left of it and you give up a little growth. Walk right and growth drops fast — bet double Kelly and it hits zero. Past that, a bet with a genuine edge loses money over time from oversizing alone. That lopsidedness is the whole case for betting half: your estimate is fuzzy, and the right side of this hill is a cliff.

Watch it work: one season, three bettors

The proof is asymptotic. The intuition is visceral. Below, three bettors face the exact same season — same wins, same losses, a 55% edge at even money — and differ only in how they size each bet. One bets Kelly (10% of the current bankroll). One bets a flat $200 every time, the way most people do. One runs a Martingale, doubling after every loss to chase it back. Same luck, three rules. Watch.

Advanced: the growth curve, and betting several games at once

Everything above falls out of one function. Stake the same fraction of your bankroll on every bet with win probability and net odds , and your expected log-growth per bet is:

A win multiplies your bankroll by ; a loss by . Average the logs, weighted by probability, and you have the compound growth rate. Kelly’s is simply where this curve peaks.

It’s a lopsided hill. Left of the peak, growth fades gently. Right of it, the term falls off a cliff — big stakes get punished without limit.

Set the derivative to zero and the peak lands on the formula from the top, written in its edge-over-odds form:

The numerator is the bet’s expected profit per dollar — its EV. Kelly is that edge divided by the odds: the same edge at longer odds means a smaller stake, because the losses land more often.

Now the harder case: several bets running at once. The single-bet formula assumes each bet is the only claim on your bankroll. Stack up per-bet Kelly stakes and every one sizes itself as if the others don’t exist — too much total exposure, and shared losing outcomes get double-counted. The fix is to optimize all the stakes jointly, over every way the slate can land:

where runs over all win/loss combinations of the bets. It’s exactly what the app’s joint sizer computes.

Every formula here lives on the Formula Sheet for quick reference.