In 1956 John L. Kelly Jr., working at Bell Labs on noise in telephone lines, noticed that the same mathematics that bounds a noisy channel also bounds a gambler’s fortune. The result — a staking rule that maximizes the expected logarithm of wealth — remains the only betting system with a clean conscience: it cannot be beaten in the long run by any other fixed fraction, and it cannot be budged from the outside.

The rule

Stake a fraction f of your bankroll on a bet that wins b per unit staked with probability p, and loses your stake with probability q = 1 − p. Your expected log-growth per bet is

G(f) = p·ln(1 + b·f) + q·ln(1 − c·f)

and the fraction that maximizes it — the Kelly stake — is

f* = (p·b − q·c) ⁄ (b·c)

with c the loss per unit staked (usually 1; less if you cut losses). If the numerator is negative there is no edge, and the correct stake is nothing at all.

Why the logarithm

Wealth compounds multiplicatively, so after n bets your fortune is a product of outcomes. Products are sums in log-space — and by the law of large numbers the sum is dragged, inevitably, toward its expectation n·G(f). Maximizing G is therefore not one preference among many: in the long run, almost every path generated by the G-maximizing fraction ends above almost every path generated by any other fraction. The logarithm is the arithmetic of compounding.

The cliff

Growth rises to its peak at f* and falls to zero well before 2f* — then turns negative. Betting twice the Kelly stake does not halve your growth; it reverses it. A strategy with a genuine edge, staked too aggressively, loses money with each repetition. The app’s growth-curve markers make the cliff visible: terracotta is the curve, and the quiet dashed line to its right is where winning starts to bankrupt you.

Why fractions of Kelly

Full Kelly is the fastest median path and a violent ride: half-Kelly surrenders only a quarter of the growth rate while removing roughly half the variance. Practitioners also distrust their own inputs. If your true win rate is a little below the one you typed, you have silently moved rightward past f* — and the penalty for overbetting is asymmetric and steep, while the penalty for underbetting is gentle. Estimation error, therefore, argues for fractions: bet half-Kelly and an error in p mostly just makes you richer more slowly.

Signals, confidence, calibration

The Signal Market tab sizes every trade from a stated confidence: p̃ = 50% + (confidence − 50%) × calibration. The calibration dial is the entire moral of the story. A signal engine that says 70% and delivers 55% is not slightly wrong — through Kelly’s arithmetic it is aggressively overbet, and the ledger shows the bleed. Watch the two “realized” vitals: hit rate and payoff, measured from the tape, against the numbers you claimed.

The duel

The Duel tab hands you the dice and the true odds, and asks only one question, over and over: how much? Kelly plays beside you, staking f* without emotion. Most players discover their inner discipline lasts about four wins in a row — then the stake creeps up, and the cliff finds them. The reckoning tiles keep the score in growth rates, which is the only currency that compounds.

What the lab computes exactly

Nothing in the Growth Lab is simulated. After n bets with k wins, wealth is exactly V₀(1+bf)^k(1−cf)^(n−k), and k follows a binomial law — so the percentile fans, the probability of finishing down, and the probability of doubling are computed in closed form from the binomial distribution. Only the Signal Market rolls dice, because there the market itself is the die.

Caveats, honestly stated

• Kelly assumes you know p and b. You do not. Treat f* as a ceiling, not a target.
• The drawdown probability in the vitals uses a Brownian (diffusion) approximation — marked “≈” in spirit; the fans are exact.
• Simultaneous, correlated bets require the matrix form of Kelly — a different beast entirely.
• Fees, slippage and finite lifetimes all push the honest stake downward.
• Growth-optimal is not utility-optimal for every soul. Kelly maximizes the median; it does not maximize your sleep.

Reading

J.L. Kelly, A New Interpretation of Information Rate (1956) · E.O. Thorp, The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market · MacLean, Thorp & Ziemba, The Kelly Capital Growth Investment Criterion.