The Kelly Criterion

You have found a genuine edge. A coin that lands heads 60% of the time and pays even money; a blackjack table you can count; a mispriced bet where the true odds are in your favour. The hard question is not whether to bet — of course you bet, the edge is real — but how much. Stake too little and a lifetime of correct calls barely moves your wealth. Stake too much and you can win far more often than you lose and still go broke, wiped out by a run of bad luck you were always going to hit eventually. Somewhere between timid and reckless there is a single bet size that grows your money faster than any other. Finding it is the whole game, and the answer is more surprising — and more humbling — than most gamblers ever guess.

The trap that ruins people is a subtle one, and it hides inside the phrase “expected value”. Maximise the average outcome of a favourable bet and the maths screams bet everything, every time — because on any single round, more stake means more expected profit. But you do not live on the average of a thousand parallel universes; you live on one sequence of bets, multiplied together. Wealth compounds multiplicatively, and on a multiplicative process the average across all possible futures (the ensemble average) is a fantasy dragged upward by a handful of astronomically lucky paths, while the future you actually experience (the time average) quietly decays toward zero. This is the ergodicity gap, and it is why “maximise expected wealth” is exactly the wrong objective. The right one is to maximise the growth rate of a single, typical trajectory — which turns out to be maximising the expected logarithm of wealth.

Do that maths and out drops a clean, memorable rule. Call your win probability $p$, your loss probability $q = 1 - p$, and your net payoff odds $b$ (a winning stake returns $b$ times itself). The growth-maximising fraction of your bankroll to stake is $f^* = \frac{pb - q}{b} = p - \frac{q}{b}.$ In plain words: bet your edge divided by the odds, where your edge is $pb - q$. For an even-money bet ($b = 1$) it collapses to $f^* = 2p - 1 = p - q$ — literally bet your edge. The 60/40 coin? $f^* = 0.6 - 0.4 = 0.20$: stake exactly 20% of your bankroll each round, no more, no less. A bet paying 2-to-1 that you win half the time? $f^* = 0.5 - 0.5/2 = 0.25$, or 25%. Not a hunch, not a nerve level — a number.

Why not bet more than Kelly, since the edge is real? Because long-run growth as a function of bet size is a hump, not a ramp. Plot the growth rate against the fraction staked and it climbs from zero, peaks at exactly $f^*$, and then — this is the part that kills people — falls back to zero at around $2f^*$ and goes negative beyond. On the 60/40 coin, betting the Kelly 20% grows your money about +2.0% per round; doubling to 40% drags growth all the way back to zero; betting it all guarantees eventual ruin with probability one. Over-betting does not buy you extra growth for extra risk — past the peak it buys you less growth and more risk, and past $2f^*$ it turns your real, positive edge into a slow-motion bankruptcy. A winning player can bet themselves broke. That is the criterion’s darkest and most important lesson.

The hump is also why the honest practitioner bets less than Kelly, not more. The growth curve is gently rounded near its peak but plunges steeply past it, so erring low is cheap and erring high is catastrophic — the consequences are wildly asymmetric. And you never truly know your $p$ and $b$; everybody overestimates their edge, and an overestimated edge pushes full Kelly onto the ruinous side of the hump. The standard fix is fractional Kelly: bet half the Kelly fraction (or less). Half-Kelly keeps roughly 75% of the growth for about half the volatility — and full Kelly’s drawdowns are brutal enough (a coin-flip chance of your bankroll ever halving) that almost no human or institution can stomach the undiluted dose. Fractional Kelly is a margin of safety bolted onto a formula that assumes a certainty you will never have.

The idea traces to John Kelly Jr., a physicist at Bell Labs who in 1956 — inspired by his colleague Claude Shannon’s information theory — published “A New Interpretation of Information Rate” and, almost as a footnote, the growth-optimal betting fraction. It sat mostly ignored until Ed Thorp carried it from the blackjack tables of Las Vegas to Wall Street, running the hedge fund Princeton-Newport on Kelly-style position sizing. Handle the model with the humility it demands, though: it assumes you know the true odds (you don’t), it is indifferent to the short-run volatility and finite horizons that real people very much care about, it needs repeated, independent, divisible bets to work at all, and — read this twice — it is not a licence to bet big. Its deepest teaching is how easily a player with a real edge still goes broke. Survival first; growth second. Bet your edge, and respect the variance.